Scattering in Quantum Dots via Noncommutative Rational Functions

Abstract

In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via $N\ll M$ channels, the density $\rho$ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio $\phi := N/M\le 1$; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $\phi \rightarrow 0$, we recover the formula for the density $\rho$ that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any $\phi <1$ but in the borderline case $\phi =1$ an anomalous $\lambda ^{-2/3}$ singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.

1 Introduction and Results

Since the pioneering discovery of E. Wigner on the universality of eigenvalue statistics of large random matrices [38], random matrix theory has become one of the most successful phenomenological theories to study disordered quantum systems, see [2] for a broad overview. Among many other applications, it has been used for open quantum systems and quantum transport, in particular to predict the distribution of transmission eigenvalues of scattering in quantum dots and wires. The theory has been developed over many excellent works starting with the ground-breaking papers by Mello et al. [30] and by Verbaarschot et al. [37]; for a complete overview with extensive references, see reviews by Beenakker [7, 8], Fyodorov and Savin [21] and Schomerus [35].

We will focus on quantum dots, i.e., systems without internal structure, coupled to an environment (electron reservoir) via scattering channels. Quantum wires, with a typically quasi one-dimensional internal structure, will be left for further works. In the simplest setup the quantum dot is described by a self-adjoint Hamiltonian (complex Hermitian matrix¹) $H\in {\mathbb {C}}^{M\times M}$ acting on an M-dimensional state space ${\mathbb {C}}^M$. It is coupled to an environment with $N_0$ effective degrees of freedom via an $M\times N_0$ complex coupling matrix W. Following Wigner’s paradigm, both the Hamiltonian H and the coupling matrix W are drawn from random matrix ensembles respecting the basic symmetries of the model. Typically, the entries of W are independent, identically distributed (i.i.d.), while H is a Wigner matrix, i.e., it has i.i.d. entries on and above the diagonal. We allow for general distributions in contrast to most existing works in the literature that assume H has Gaussian or Lorentzian distribution.

The Hamiltonian of the total system at Fermi energy $E\in {\mathbb {R}}$ is given by (see [7, Eq. (80)])

$$ {{\mathcal {H}}} = \sum _{a=1}^{N_0} | a \rangle E \langle a | + \sum _{\mu ,\nu =1}^M |\mu \rangle H_{\mu \nu } \langle \nu | + \sum _{\mu =1}^M\sum _{a=1}^{N_0} \Big [ |\mu \rangle W_{\mu a} \langle a| + |a\rangle W_{\mu a}^* \langle \mu | \Big ]. $$

One common assumption is that the interaction W is independent of the Fermi energy E.

At any fixed energy $E\in {\mathbb {R}}$, we define the scattering matrix (see [7, Eq. (81)])

$$\begin{aligned} S(E) := I - 2\pi \mathrm {i}\,W^* (E\cdot I - H + \mathrm {i}\,\pi W W^*)^{-1} W \in {\mathbb {C}}^{N_0\times N_0}. \end{aligned}$$

(1.1)

This is the finite-dimensional analogue of the Mahaux–Weidenmüller formula in nuclear physics [28] that can be derived from ${{\mathcal {H}}}$ in the $N_0\rightarrow \infty$ limit. The definition (1.1) will be the starting point of our mathematical analysis. Since $H=H^*$, one can easily check that S(E) is unitary.

To distinguish between transmission and reflection, we assume that the scattering channels are split into two groups, left and right channels, with dimensions $N_1+N_2=N_0$ and the interaction Hamiltonian is also split accordingly; $W = (W_1 , W_2) \in {\mathbb {C}}^{M\times (N_1+N_2)}$. Therefore, S(E) has a natural $2\times 2$ block structure and we can write it as (see [7, Eq. (23)])

$$\begin{aligned} S(E) = \left( \begin{array}{cc} R &{}\quad T' \\ T &{}\quad R' \end{array} \right) , \end{aligned}$$

(1.2)

where $R\in {\mathbb {C}}^{N_1 \times N_1}$, $R'\in {\mathbb {C}}^{N_2 \times N_2}$ are the reflection matrices and $T\in {\mathbb {C}}^{N_2 \times N_1}$, $T' \in {\mathbb {C}}^{N_1 \times N_2}$ are the transmission matrices.

As a consequence of unitarity, one finds that $TT^*$, $T'(T')^*$, $I-RR^*$ and $I -R' (R')^*$ have the same set of nonzero eigenvalues. For simplicity, we assume that $N_1=N_2=N$, i.e., generically these four matrices have no zero eigenvalues, the general $N_1 \ne N_2$ case has been considered in [11]. We denote these transmission eigenvalues by $\lambda _1, \lambda _2, \ldots , \lambda _N$. They express the rate of the transmission through each channel. By unitarity of S, clearly $\lambda _i\in [0,1]$ for all i; $\lambda _i=0$ means the channel is closed, while $\lambda _i=1$ corresponds to a fully open channel. The transmission eigenvalues carry important physical properties of the system. For example, ${\mathrm {Tr}}TT^* =\sum _i \lambda _i$ gives the zero temperature conductance (Landauer formula [7, Eq. (33)]), while

$$\begin{aligned} \sum _i \lambda _i (1-\lambda _i) = {\mathrm {Tr}}TT^* - {\mathrm {Tr}}(TT^*)^2 \end{aligned}$$

(1.3)

is the shot noise power, giving the zero temperature fluctuation of the current (Büttiker’s formula, [12, 7, Eq. (35)]). The dimensionless ratio of the shot noise power and the conductance is called the Fano factor (see [8, Eq. (2.15)])

$$\begin{aligned} F: =\frac{ \sum _i \lambda _i (1-\lambda _i)}{\sum _i \lambda _i}. \end{aligned}$$

(1.4)

The current fluctuation is therefore given by a certain linear statistics of the transmission eigenvalues, and thus, it can be computed from the density $\rho$ of these eigenvalues. Therefore, determining $\rho$ is a main task in the theory of quantum dots.

In many physical situations, it is found that $\rho$ has a bimodal structure with a peak at zero and a peak at unit transmission rates. Furthermore, $\rho$ exhibits a power law singularity at the edges of its support [0, 1]. One main result of the theory in [7] is that in the $M\gg N\gg 1$ regime at energy $E=0$ the density of transmission eigenvalues for a quantum dot is given by

$$\begin{aligned} \rho _{ \text{ Bee }}(\lambda ) = \frac{1}{\pi \sqrt{\lambda (1-\lambda )}}, \end{aligned}$$

(1.5)

(the answer is different for quantum wires), i.e., it has an inverse square root singularity at both edges, see [8, Eq. (3.12)]. In this case, the Fano factor is $F=1/4$ which fits well the experimental data.

The goal of this paper is to revisit and substantially generalize the problem of transmission eigenvalues with very different methods than Beenakker and collaborators used. While those works used invariant matrix ensembles for H and relied on explicit computations for the circular ensemble, we consider very general distribution for the matrix elements of H. In particular, we show that in the regime $\phi := N/M \in {(}0,1{]}$, $M\rightarrow \infty$, the empirical density of transmission eigenvalues has a deterministic limit $\rho =\rho _{\phi }$ and we give a simple algebraic equation to compute it. The solution can be continuously extended as $\phi \rightarrow 0$, the equation simplifies for $\phi =0$, the density becomes explicitly computable and it coincides with (1.5); $\lim _{\phi \rightarrow 0} \rho _{\phi } =\rho _{\phi =0}= \rho _{ \text{ Bee }}$ for $E=0$ and our formula holds for any E in the bulk spectrum of H. While no short explicit formula is available for the general case $\phi \in (0, 1]$, we can analyze the singularities of $\rho _{\phi }$ for any fixed $\phi \in (0, 1]$ in detail.

More precisely, we rigorously prove that for any fixed $\phi \in (0,1)$ the density $\rho _{\phi }$ has an inverse square root singularity both at 0 and at 1,

$$\begin{aligned} \begin{aligned}&\rho _{\phi } (\lambda ) \sim \lambda ^{-1/2},&\;\quad \text { for } 0<\lambda \ll 1, \;\;\quad&\qquad \text { and } \\&\rho _{\phi } (\lambda ) \sim (1-\lambda )^{-1/2},&\text { for } 0<1-\lambda \ll 1,&\quad \end{aligned} \end{aligned}$$

(1.6)

qualitatively in line with $\rho _{ \text{ Bee }}$ from (1.5). However, $\rho _{\phi }$ is not symmetric around $\lambda =\frac{1}{2}$ and the Fano factor at $E=0$ slightly differs from $\frac{1}{4}$ when $\phi \ne 0$. Figure 1 shows the Fano factor for different values of $\phi$ numerically computed from our theory. We mention that the deviation from 1/4 is within $2\%$ for the entire range of $\phi \in [0,1)$ which is well below the error bar of the experimental results presented in Fig. 6 of [7] and adapted from [31].

Fig. 1

Fano factor for $E=0$ and $\phi \in (0,1)$

The value $\phi =1$ is special, since the singularity of $\rho$ at $\lambda \approx 0$ changes to

$$\begin{aligned} \rho _{\phi =1}(\lambda ) \sim \lambda ^{-2/3}, \;\quad \text{ for } 0<\lambda \ll 1 \end{aligned}$$

(1.7)

from $\lambda ^{-1/2}$ in (1.6), while the inverse square root singularity at 1 persists. This enhancement of singularity signals the emergence of a $\delta$-function component at 0 in $\rho _{\phi }$ as $\phi$ becomes larger than 1, which is a direct consequence of $TT^*$ not having full rank when $N>M$. We remark that this regime is quite unphysical since it corresponds to more scattering channels than the total number of internal states of the quantum dot. In realistic quantum dots, N is smaller than M since not every mode of the dot may participate in scattering. We therefore do not pursue the detailed analysis of $\rho _{\phi }$ for $\phi >1$, although our method can easily be extended to these $\phi$’s as well.

There are two main differences between our model and that of Beenakker et al. First, the distribution imposed on the random matrix H is different and we consider random W. Second, our current method works in the entire regime $\phi = N/M\in {(}0,1{]}$, while Beenakker assumes $M\gg N$, i.e., $\phi \ll 1$. We now explain both differences.

Following Wigner’s original vision, any relevant distribution must respect the basic symmetry of the model; in our case, this demands that H be complex Hermitian, while no symmetry constraint is imposed on W. Respecting basic symmetries, one may define ensembles essentially in two ways. Invariant ensembles are defined by imposing that the entire distribution be invariant; it is typically achieved via a global Gibbs factor times the natural flat measure on the space of matrices satisfying the basic symmetry. Wigner ensembles and their generalizations impose distributions directly on the matrix elements and often demand independence (up to the basic symmetry constraint). These two procedures typically yield different ensembles.

While in the simplest case of random Hermitian matrices both types of ensembles have been actively investigated, for ensembles with more complicated structure, like our S that is a rational function of the basic ingredients H and W (1.1), up to recently only the invariant approach was available. Sophisticated explicit formulas have been developed to find the joint distribution of eigenvalues for more and more complicated structured ensembles (see [18]), which could then be combined with orthogonal polynomial methods to obtain local correlation functions. The heavy reliance on explicit formulas, however, imposes a serious limitation on how complicated functions of random matrices, as well as how general distributions on these matrices can be considered. For example, the Gibbs factor is often restricted to Gaussian or closely related ensembles to remain in the realm of explicitly computable orthogonal polynomials.

There have been considerable developments in the other type of ensembles in the recent years. Departing from the invariant world, about 10 years ago the Wigner–Dyson–Mehta universality of local eigenvalue statistics has been proven for Wigner matrices with i.i.d. entries, see [16] and references therein. Later, the i.i.d. condition was relaxed and even matrices with quite general correlation structures among their entries can be handled [14]. One of the key ingredients was to better understand the Matrix Dyson equation (MDE), the basic equation governing the density of states [1]. Together with the linearization trick, this allows us to handle arbitrary polynomials in i.i.d. random matrices [13] and in the current work we extend our method to a large class of rational functions. Note that even if the building block matrices have independent entries, the linearization of their rational expressions will have dependence, but the general MDE can handle it [see (2.13)]. In our work, we deal only with bounded rational functions and relatively straightforward regularizations of unbounded rational functions, the general theory of unbounded rational functions is still in development, see [29] and references therein. We stress that the distribution of the matrix elements of H and W can be practically arbitrary. In particular, our result is not restricted to the Gaussian world.

In comparison, the result of Beenakker et al. [8, Sect. III.B], see also [10, Sect. IV], postulates that H is GUE, while W may even be deterministic and only its singular values are relevant. For example, if all nonzero singular values are equal one and $N_0=2N\le M$, i.e., $\phi \le \frac{1}{2}$, then S can be written as

$$ S = \frac{I + i\pi {{\widetilde{H}}}}{I- i\pi {{\widetilde{H}}}}, \qquad {{\widetilde{H}}}: = Q^T(H-E \cdot I )^{-1} Q, $$

where $Q\in {\mathbb {C}}^{M\times 2N}$ with $Q_{ij}=\delta _{ij}$. In fact, for the sake of explicit calculations, it is necessary to replace the GUE by a Lorentzian distribution (irrelevant constants ignored),

$$\begin{aligned} P(H) \sim \text{ det }[ I + H^2]^{-M}, \end{aligned}$$

(1.8)

since in this way ${{\widetilde{H}}}$ is also Lorentzian, and argue separately that in the sense of correlation functions (1.8) is close to a GUE when M is very large [10, Section III]. Under these conditions S becomes Haar distributed on U(2N) as $M\rightarrow \infty$ and N is fixed; this is the step where $M\gg N$ is necessary. Furthermore, one can verify [7, Section II.A.1] that at energy $E=0$ the transmission eigenvalues of a Haar distributed scattering matrix follow the circular ensemble on [0, 1]. Therefore, $\lambda _i$’s have a well-known joint distribution

$$ P\big (\{\lambda _i\}\big )\sim \prod _{i<j} (\lambda _j-\lambda _i)^2 \quad \text{ on } \;\;\; [0,1]^N, $$

and their density can be easily computed, yielding (1.5) in the $N\rightarrow \infty$ limit.

While Beenakker’s result relies on an impressive identity, it allows little flexibility in the inputs: H needs to be Lorentzian with very large dimension; moreover, $M\gg N$ and $E=0$ are required. In contrast, our setup allows for a large freedom in choosing the distribution of H, it covers the entire range $M\ge N$ and also applies to any E in the bulk; however, for computational simplicity we model the contacts W also by a random matrix. While the most relevant regime for scattering on quantum dots is still $M\gg N$, as scattering involves surface states only, a very recent work [20] introduces absorbing channels well inside the quantum dot that leads to physical models with $M\sim N$.

The flexibility in our result stems from the fact that our method directly aims at the density of states via an extension of the MDE theory to linearizations of rational functions of random matrices. It seems unnecessarily ambitious, hence requiring too restrictive conditions, to attempt to find the joint distribution of all eigenvalues. Even for Wigner matrices, this is a hopeless task beyond the Gaussian regime. We remark that the present analysis of the density of transmission eigenvalues for the quantum dot is only one convenient application of our approach. This method is powerful enough to answer many related questions concerning the density of states such as the analysis of the scattering poles [19] as well as extensions from quantum dots to quantum wires that we will address in future work.

1.1 Model and Main Theorem

In order to accommodate all parameters that will appear in our analysis, we introduce a generalized version of the scattering matrix (1.1)

$$\begin{aligned} S(w):= I - 2 \,\mathrm {i}\,\gamma \, W^*(w\cdot I -H + \mathrm {i}\,\gamma \, WW^*)^{-1} W \, \end{aligned}$$

(1.9)

with $\gamma >0$ and $w\in {\mathbb {C}}_{+}$ for $\phi \in {(}0,1/2{]}$ and $w\in {\mathbb {C}}_+ \cup {\mathbb {R}}$ for $\phi \in {(}1/2,1{]}$, where ${\mathbb {C}}_+ : =\{z \in {\mathbb {C}}: \mathrm {Im}z >0\}$ denotes the complex upper half plane. The constant $\gamma >0$ quantifies the coupling between the internal quantum dot Hamiltonian H and the external leads $W_1,W_2$, i.e., the effective Hamiltonian of the open quantum systems becomes $H-\mathrm {i}\,\gamma W W^{*}$. The spectral parameter $w$ is used to regularize the potentially unstable inverse in (1.9). Note that for $\phi \in (0,1/2)$ the matrix $WW^*$ has a nontrivial kernel (and even for $\phi =1/2$ it has a very small eigenvalue), hence initially a regularization with a small positive imaginary part is necessary that will later be carefully removed. For the technically easier $\phi \in {(}1/2,1{]}$ regime, we may directly choose $w$ to be real since the eigenvalues of $WW^*$ are bounded away from zero with very high probability.

The general formula (1.1) recovers the scattering matrix (1.1) by setting $\gamma = \pi$, $\mathrm {Re}w= E$ and $\mathrm {Im}w= 0$. The regularized scattering matrix still has the $2\times 2$-block structure from (1.2) with $T \in {\mathbb {C}}^{N \times N}$. We consider the following random matrix model [see (1.1) and (1.2)]

$$\begin{aligned} \varvec{T}_{w,\phi , \gamma } := T T^* = 4 \gamma ^2 W_{2}^*\frac{1}{w-H + \mathrm {i}\,\gamma W W^{*}} W_{1} W_{1}^* \frac{1}{{\overline{w}}-H - \mathrm {i}\,\gamma W W^{*}} W_{2} , \end{aligned}$$

(1.10)

where the triple of parameters $(w, \phi , \gamma )$ belongs to the same set as for (1.9). Furthermore, for $M,N\in {\mathbb {N}}$, $\phi := N/M$ the matrices $H\in {\mathbb {C}}^{M \times M}$ and $W_{1},W_{2} \in {\mathbb {C}}^{M\times N}$ are three independent random matrix ensembles satisfying the following assumptions

• (H1qd) H is a Hermitian random matrix having independent (up to symmetry constraints) centered entries of variance 1/M;

• (H2qd) $W_{1}$ and $W_{2}$ are (non-Hermitian) random matrices having independent centered entries of variance 1/M;

• (H3qd) entries of H, $W_{1}$ and $W_{2}$, denoted by H(i, j), $W_{1}(i,j)$ and $W_{2}(i,j)$ correspondingly, have finite moments of all orders, i.e., there exist $\varphi _{n} >0$, $n\in {\mathbb {N}}$, such that

  $$\begin{aligned} \begin{aligned}&\max _{\begin{array}{c} 1\le i,j \le M \end{array}} {\mathbb {E}}\big [\,|\sqrt{M}H(i,j)|^{n}\big ] \\&\quad + \max _{\begin{array}{c} 1\le i \le M \\ 1\le j \le N \end{array}}\Big ({\mathbb {E}}\big [\,|\sqrt{M}W_{1}(i,j)|^{n} \big ]+{\mathbb {E}}\big [\,|\sqrt{M}W_{2}(i,j)|^{n}\big ]\Big ) \le \varphi _{n} . \end{aligned} \end{aligned}$$

  (1.11)

Remark 1.1

(Constant matrices) In (1.10) and later in the paper, for $B \in {\mathbb {C}}$, $n\in {\mathbb {N}}$ and $I_{n}\in {\mathbb {C}}^{n\times n}$ the identity matrix of size n, we use the shorthand notation $B \cdot I_{n} = B$. This notation is used only when the dimension of $I_{n}$ can be unambiguously determined from the context.

Remark 1.2

In the sequel, we consider the coupling constant $\gamma$ to be a fixed positive number; therefore, we will omit the dependence on $\gamma$ in our notation.

Denote by $\mu _{\varvec{T}_{w,\phi }}(d\lambda ):=\frac{1}{N}\sum _{i=1}^{N} \delta _{\lambda _{i}}$ the empirical spectral measure of $\varvec{T}_{w,\phi }$, where $\lambda _{i}$ are the eigenvalues of the Hermitian matrix $\varvec{T}_{w,\phi }$. To simplify the presentation, we will assume in this paper that the dimensions of the matrices H, $W_1$ and $W_2$ grow over a subsequence $(N, M)= (kn, ln)$, $n\in {\mathbb {N}}$, i.e., $N/M=\phi$ is kept fixed. One could easily extend our argument to include the general situation when one considers two sequences $N=N_n$ and $M=M_n$ tending to infinity such that $\phi _n = N_n/M_n \rightarrow \phi$.

We now formulate our two main results. The first one, Theorem 1.3, is the global law for the empirical eigenvalue density; it shows that $\mu _{\varvec{T}_{w,\phi }}(d\lambda )$ has a deterministic limit denoted by $\rho _{w,\phi }(d\lambda )$. The second result, Theorem 1.4, contains key properties of $\rho _{w,\phi }(d\lambda )$. It shows that the regularization in the spectral parameter $w$ can be removed and that $\rho _{w,\phi }$ can be continuously extended to $\phi \rightarrow 0$; moreover, it explicitly identifies its singularities at zero and one.

Theorem 1.3

(Global law). Fix a positive real number $\gamma >0$ and a rational number $\phi \in {(}0,1{]}\cap {\mathbb {Q}}$. Let $w$ be a fixed spectral parameter: for $\phi \in {(}0,1/2{]}$ we assume that $w\in {\mathbb {C}}_+$, while for $\phi \in {(}1/2,1{]}$ we assume $w\in {\mathbb {C}}_+ \cup {\mathbb {R}}$. Then, there exists a deterministic probability measure $\rho _{w,\phi }(d\lambda )$ with $\mathrm {supp}\, \rho _{w,\phi } \subset [0,1]$ such that $\mu _{\varvec{T}_{w,\phi }}(d\lambda )$ converges weakly to $\rho _{w,\phi }(d\lambda )$ in probability (and almost surely) as $M,N \rightarrow \infty$ with $N/M = \phi$.

Theorem 1.4

(Properties of the transmission eigenvalue density). Let the numbers $\gamma , w,\phi$ and the measure $\rho _{w,\phi }(d\lambda )$ be as in Theorem 1.3.

• (i) The function

  $$\begin{aligned} \Big ( \big ({(}0,1/2{]}\cap {\mathbb {Q}}\big ) \times {\mathbb {C}}_+ \Big )\cup \Big ( \big ( {(}1/2,1{]}\cap {\mathbb {Q}}\big )\times \big ( {\mathbb {C}}_+ \cup {\mathbb {R}}\big )\Big ) \ni (\phi ,w) \mapsto \rho _{w,\phi }(d\lambda ) \end{aligned}$$

  from Theorem 1.3 with values in probability measures on [0, 1] can be extended to a function on ${(}0,1{]} \times ( {\mathbb {C}}_+ \cup {\mathbb {R}})$ that is continuous in the weak topology. In particular, the weak limit

  $$\begin{aligned} \rho _{E,\phi }(d\lambda )=\lim _{\, {\mathbb {C}}_+ \ni w\rightarrow E}\rho _{w,\phi }(d\lambda ) \end{aligned}$$

  (1.12)

  exists for all $E \in {\mathbb {R}}$. For every $E \in {\mathbb {R}}$ and $\phi \in {(}0,1{]}$ the measure $\rho _{E,\phi }(d\lambda )$ is absolutely continuous, i.e., $\rho _{E,\phi }(d\lambda )=\rho _{E,\phi }(\lambda )d \lambda$. The function $\rho _{E,\phi }(\lambda )$ is bounded when $\lambda$ is away from 0 and 1. Furthermore, the weak limit $\rho _{E,0}(d\lambda ):=\lim _{\phi \downarrow 0}\rho _{E,\phi }(d\lambda )$ exists for every $E \in {\mathbb {R}}$.

• (ii) If $\phi = 1$ and $\gamma >0$, then $\rho _{E,\phi }(\lambda )$ has the following asymptotics near 0 and 1:

  • (a) for $E\in {\mathbb {R}}$

    $$\begin{aligned} \rho _{E, 1}(\lambda ) = \frac{1}{\pi }\root 3 \of {\frac{1+E^2}{4 \gamma ^{2} }} \, \lambda ^{-2/3} + O\left( \lambda ^{-1/3}\right) \text{ as } \lambda \rightarrow 0_{+}, \end{aligned}$$

    (1.13)

  • (b) for $|E| < \frac{1}{ \gamma } (2 \sqrt{1 + 6 \gamma ^2 + \gamma ^4 }-2 \gamma ^2 - 2 )^{1/2}$

    $$\begin{aligned} \rho _{E, 1}(\lambda ) = \frac{1}{\pi } \, \frac{ -4 \xi _{0}}{\xi _{0}^{2}+ \gamma ^2 E^2 + 4 }\, (1-\lambda )^{-1/2} + O\left( 1\right) \text{ as } \lambda \rightarrow 1_{-} , \end{aligned}$$

    (1.14)

    where $\xi _{0} = -\sqrt{2 \sqrt{1 + 6 \gamma ^2 + \gamma ^4 }-2 \gamma ^2 -2- \gamma ^2 E^2}$.

• (iii) If $\phi \in (0,1)$ and $\gamma >0$, then

  • (a) for $E\in {\mathbb {R}}$

    $$\begin{aligned} \rho _{E,\phi }(\lambda ) = \frac{1}{\pi } \, \frac{ \gamma ^2 E^2 \xi _0^4+ \big (\xi _0^2+2\phi \big )^2\big (4+ \gamma ^2 \xi _0^2\big )}{4 \gamma \xi _0\big (\xi _0^2+2\phi \big )^2}\, \lambda ^{-1/2} + O\left( 1\right) \text{ as } \lambda \rightarrow 0_{+}, \end{aligned}$$

    (1.15)

    where $\xi _{0}>0$ is the unique positive solution of

    $$\begin{aligned} \xi _0^6 + \big (E^2+8\phi -4\big ) \xi _0^4+ 4 \phi \big (E^2+ 5\phi -4\big ) \xi _0^2+16(\phi -1)\phi ^2=0. \end{aligned}$$

    (1.16)

  • (b) for $|E| < E_*:=\frac{1}{ \gamma }\Big (2 \sqrt{ 1 + 2 (1+2\phi ) \gamma ^2 + (2 \phi -1)^2 \gamma ^4 } - 2 \gamma ^2 (2\phi -1) -2\Big )^{1/2}$ the density satisfies

    $$\begin{aligned} \rho _{E, \phi }(\lambda ) = \frac{1}{\pi } \, \frac{-4 \xi _1}{\xi _1^2 + \gamma ^2 E^2+4 }\,(1-\lambda )^{-1/2} + O\left( 1\right) \text{ as } \lambda \rightarrow 1_{-} , \end{aligned}$$

    (1.17)

    where

    $$\begin{aligned} \xi _1 = - \gamma \sqrt{E_*^2- E^2}. \end{aligned}$$

    (1.18)

• (iv) If $\phi =0$, $\gamma >0$ and $|E| < 2$ the density is given globally by the explicit formula

  $$\begin{aligned} \rho _{E, 0 }(\lambda ) = \frac{1}{ \pi }\frac{\gamma \big (1+ \gamma ^2 \big ) }{\big (1+ \gamma ^2 \big )^2\lambda + \gamma ^2 \big (4-E^2\big )(1-\lambda )} \,\sqrt{\frac{4-E^2}{\lambda (1-\lambda )}}, \qquad \lambda \in (0,1). \end{aligned}$$

  (1.19)

Proofs of Theorems 1.3 and 1.4

We now explain the structure of the paper that contains the proof of both theorems. The density function $\rho _{E, \phi }(\lambda )$ will be derived from a deterministic equation, the Dyson equation of the linearizaton of (1.10) [see Eqs. (2.13) and (2.27) for $\phi =1$, as well as (3.7) and (3.18) for $\phi \in (0,1)$ ]. This equation plays a central role in our analysis. Sections 2 and 3 contain the proofs of the model specific parts of Theorem 1.4 which use some key conclusions of the general theory on noncommutative rational functions developed in Appendix A. The $\phi =1$ case is treated in Sect. 2 with full details, while in Sect. 3 we explain the modifications for the general rational $\phi \in (0,1)$ case. Theorem 1.3 is proven in Lemma 2.5 for $\phi = 1$ and Lemma 3.4 for general rational $\phi \in (0,1)$ using the general global law from Appendix A. Part (ii) is proven in Sect. 2.5 after having established key properties of the solution for the Dyson equation. Section 3 follows the same structure as Sect. 2 and proves parts (i), (iii) and (iv) of Theorem 1.4 for the $\phi \in [0,1)$ case. $\square$

Remark 1.5

The restriction $|E|<E_*$ in Theorem 1.4 is used to identify the regime in which the density $\rho _{E,\phi }$ has two singularities. For $|E|>E_*$, the singularity at $\lambda = 1$ disappears and the support of the density becomes bounded away from $\lambda =1$. Physically, this indicates that there are no fully open channels in this regime. This effect is most severe in the case $\phi =0$, where the system becomes completely reflective as |E| increases above the threshold $E_*=2$. Our approach is also applicable for $|E|>E_*$, but for the simplicity of the presentation, we restrict our analysis to the physically more relevant situation when two singularities exist simultaneously.

We formulated Theorem 1.3 about the specific matrix (1.10). However, our method works for very general noncommutative (NC) rational expressions in large matrices with i.i.d. entries (with or without Hermitian symmetry) generalizing our previous work [13] on polynomials. For convenience of the readers interested only in the concrete scattering problem, the main part of our paper focuses on this model and we defer the general theory to Appendix A. The details of this appendix are not necessary in order to follow the main arguments in Sects. 2 and 3 provided some basic facts from the appendix are accepted. These facts will be re-stated for the specialization to our concrete operator $\varvec{T}_{ w,\phi }$ from (1.10). On the other hand, Appendix A is written in a self-contained form, so readers interested in the general theory can directly go to it, skipping the concrete application.

In Appendix A, we first give the precise definition of the set of admissible rational expressions that requires some technical preparation, see Sect. A.1 for details. Roughly speaking, we can consider any rational expression whose denominators are stably invertible with overwhelming probability. We remark that this property holds for (1.10) since the imaginary part of $w-H + \mathrm {i}\,\gamma W W^{*}$ has a positive lower bound as long as $\phi >1/2$ or $w\in {\mathbb {C}}_+$. With this definition at hand, we develop the theory of global and local laws as well as the identification of the pseudospectrum for such rational expressions in Sects. A.6 and A.5, respectively.

2 Proof of Theorem 1.4 for the Special Case $\phi = 1$

In this section, we study the model (1.10) for fixed $\gamma >0$, $\phi = 1$ and $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$. This special choice of parameter $\phi$ ensures that the linearization of $\varvec{T}_{ w,\phi }$ has a fairly simple structure, which makes the proof of Theorems 1.3 and 1.4 more transparent and streamlined. Generalization to $\phi \in (0,1)$ is postponed to Sect. 3. Since the parameter $\phi$ is fixed to be equal to 1, we will omit the dependence on $\phi$ in the notation throughout the current section. The information about linearizations of general rational functions is collected in the Appendix A.1. Here, we often refer to specialization of these results to $\varvec{T}_{ w}$.

2.1 Linearization Trick and the Dyson Equation for Linearization

We consider the random matrix model $\varvec{T}_{ w}$ defined in (1.10) as a self-adjoint rational function of random matrices H, $W_1$ and $W_2$. In order to study its eigenvalues, we introduce the self-adjoint linearization matrix $\varvec{H}_{ w}\in {\mathbb {C}}^{8N\times 8N}$

$$\begin{aligned} \varvec{H}_{ w} := \left( \begin{array}{ccc|ccc|cc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad W_{2}^* \\ 0 &{}\quad 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 &{}\quad 0 &{} \quad 0 &{} \quad 0 &{}\quad W_{2}^* \\ 0 &{}\quad -\frac{\mathrm {i}\,}{ \gamma } &{} \quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad W_{2}^* &{}\quad 0 \\ \hline 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 &{}\quad W_{1}^* \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{1}{4 \gamma ^2} &{}\quad 0 &{}\quad W_{1}^* &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{\mathrm {i}\,}{ \gamma }&{}\quad 0 &{}\quad 0 &{}\quad W_{1}^* &{}\quad 0 \\ \hline 0 &{}\quad 0 &{}\quad W_{2} &{}\quad 0 &{}\quad W_{1} &{}\quad W_{1} &{}\quad 0 &{} \quad w-H \\ W_{2} &{}\quad W_{2} &{}\quad 0 &{}\quad W_{1} &{}\quad 0 &{}\quad 0 &{}\quad {\overline{w}} -H &{}\quad 0 \\ \end{array} \right) \end{aligned}$$

(2.1)

with $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$. Denote by $J_{m} \in {\mathbb {C}}^{m\times m}$, $m\in {\mathbb {N}}$, a matrix whose (1, 1) entry is equal to 1 and all other entries are equal to 0. For any $n\in {\mathbb {N}}$ and $R\in {\mathbb {C}}^{n \times n}$, we define $\Vert R\Vert$ to be the operator norm of R. The following definition is a specialization of the notion of generalized resolvent from Appendix A.3.

Definition 2.1

(Generalized resolvent). We call the matrix-valued function $z \mapsto (\varvec{H}_{w} - z J_{8}\otimes I_N)^{-1}$ defined for $z\in {\mathbb {C}}_+$ the generalized resolvent of $\varvec{H}_{w}$.

The results below are formulated using the notion of asymptotically overwhelming probability.

Definition 2.2

We say that a sequence of events $\{\Omega _{N}\}_{N\in {\mathbb {N}}}$ holds asymptotically with overwhelming probability (a.w.o.p. for short) if for any $D>0$ there exists $C_{D}>0$ such that

$$\begin{aligned} {\mathbb {P}}[\,\Omega _{N}] \ge 1-\frac{C_{D}}{N^{D}} . \end{aligned}$$

(2.2)

Consider the set

$$\begin{aligned} \Theta _N := \Big \{\Vert H\Vert \le 3,\, \Vert W_{1}\Vert \le 3, \, \Vert W_2\Vert \le 3, \,\Vert (W W^*)^{-1}\Vert \le 12 \Big \}\; . \end{aligned}$$

(2.3)

Note that $W W^*$ is a sample covariance matrix with concentration ratio 1/2, hence its spectrum is asymptotically supported on an arbitrarily small neighborhood of $[(1-\frac{1}{\sqrt{2}})^{2}, (1+\frac{1}{\sqrt{2}})^{2}]$ with very high probability. The limiting support follows from the classical Bai–Yin theorem [6], the corresponding large deviation result under somewhat different conditions follows, e.g., from Corollary V.2.1 of [17]. In fact, the boundedness of $\Vert (WW^*)^{-1}\Vert$ and $\Vert W_i\Vert$ also follows from [33], at least for subgaussian distributions. Alternatively, under our conditions (1.11) this result also follows from Lemma B.1 in the Appendix by choosing $n=M$, $l=1$ and $m=2$. Similarly, from the properties of classical Wigner and iid ensembles (see, e.g., [5, Section 5]), we obtain that the event $\{\,\Vert H\Vert \le 3,\, \Vert W_{1}\Vert \le 3, \, \Vert W_2\Vert \le 3 \, \}$ also holds a.w.o.p., and we conclude that for any $D>0$ there exists $C_D>$ such that

$$\begin{aligned} {\mathbb {P}}[\Theta _{N}] \ge 1- \frac{C_{D}}{N^{D}} . \end{aligned}$$

(2.4)

In the rest of this section, we consider the random matrix models (1.9) and (1.10) restricted to the set $\Theta _N$. Since on this set the smallest eigenvalue of $WW^*$ cannot be smaller that 1/12, we have that $\Vert (w-H + \mathrm {i}\,\gamma W W^{*})^{-1} \Vert \le \frac{1}{12 \gamma }$ for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$, and thus, the model (1.10) is well defined on $\Theta _N$. In the next lemma, we establish an a priori bound for the generalized resolvent $(\varvec{H}_{w} - z J_{8}\otimes I_N)^{-1}$.

Lemma 2.3

(Basic properties of the generalized resolvent). 

• (i) For any $\gamma >0$ there exists $C_{\gamma }>0$ such that a.w.o.p.

  $$\begin{aligned} \big \Vert (\varvec{H}_{ w} - z J_{8} \otimes I_N)^{-1}\big \Vert \le C_{\gamma } \bigg (1 + \frac{1}{\mathrm {Im}z}\bigg ) \end{aligned}$$

  (2.5)

  uniformly for all $z\in {\mathbb {C}}_{+}$ and $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$.

• (ii) For all $w\in {\mathbb {C}}_+ \cup {\mathbb {R}}$, $z\in {\mathbb {C}}_{+}$ and $1\le i,j\le N$

  $$\begin{aligned} \Big [\big (\varvec{H}_{ w}-zJ_{8}\otimes I_N\big )^{-1}\Big ]_{ij} = \Big [\big (\varvec{T}_{ w}- zI_N\big )^{-1}\Big ]_{ij} ,\quad 1\le i,j \le N . \end{aligned}$$

  (2.6)

Proof

Let $\varvec{H}_{ w}^{\mathrm {(init)}}$ be the linearization matrix obtained via the linearization algorithm described in Appendix A.2

$$\begin{aligned} \varvec{H}_{ w}^{\mathrm {(init)}} = \left( \begin{array}{cccccccc} 0&{}0&{}0&{}0&{}0&{}2 \gamma W_{2}^{*}&{}0&{}0 \\ 0&{}0&{}0&{}0&{}W_{1}&{} -(w-H) &{} -\sqrt{ \gamma } W_{1} &{} -\sqrt{ \gamma } W_{2} \\ 0&{}0&{}0&{}0&{}0&{} -\sqrt{ \gamma } W_{1}^{*} &{} -\mathrm {i}\,&{} 0 \\ 0&{}0&{}0&{}0&{}0&{} -\sqrt{ \gamma } W_{2}^{*} &{} 0 &{} -\mathrm {i}\,\\ 0&{} W_{1}^{*} &{}0&{}0&{}-1&{}0&{}0&{}0 \\ 2 \gamma W_{2} &{} -( {\overline{w}} -H) &{} -\sqrt{ \gamma } W_{1} &{} - \sqrt{ \gamma } W_{2} &{}0&{}0&{}0&{}0 \\ 0&{}-\sqrt{ \gamma } W_{1}^{*} &{} \mathrm {i}\,&{}0&{}0&{}0&{}0&{}0 \\ 0&{}-\sqrt{ \gamma } W_{2}^{*} &{} 0&{}\mathrm {i}\,&{}0&{}0&{}0&{}0 \end{array} \right) . \end{aligned}$$

(2.7)

Denote by $\{E_{ij}, 1\le i,j \le 8 \,\}$ the standard basis of ${\mathbb {C}}^{\,8\times 8}$

$$\begin{aligned} E_{ij} = \big (\delta _{ki}\delta _{jl}\big )_{k,l=1}^{8} , \end{aligned}$$

where $\delta _{\alpha \beta }$ is the Kronecker delta. Then, one can easily check that $\varvec{H}_{ w}$ in (2.1) can be obtained from $\varvec{H}_{ w}^{\mathrm {(init)}}$ by applying the following transformation

$$\begin{aligned} \varvec{H}_{ w} = {\widetilde{T}}{\widetilde{P}}_{78}{\widetilde{P}}_{23}{\widetilde{P}}_{34}{\widetilde{P}}_{67}{\widetilde{P}}_{28} \varvec{H}_{ w}^{\mathrm {(init)}} {\widetilde{P}}_{28}{\widetilde{P}}_{67}{\widetilde{P}}_{34}{\widetilde{P}}_{23}{\widetilde{P}}_{78} {\widetilde{T}} \end{aligned}$$

(2.8)

with ${\widetilde{T}} = \mathrm {diag}\big (1,-2\sqrt{ \gamma },\frac{1}{2 \gamma ^{3/2}} , -2\sqrt{ \gamma },-\frac{1}{2 \gamma } , \frac{1}{2 \gamma ^{3/2}}, -2 \gamma , \frac{1}{2 \gamma }\big ) \otimes I_{N}$, ${\widetilde{P}}_{ij}= \big ( E_{ij} + E_{ji} + \sum _{l \notin \{i,j\}} E_{ll} \big ) \otimes I_N$. Note that all transposition matrices ${\widetilde{P}}_{ij}$ in (2.8) leave the upper-left $N\times N$ block intact. Thus, (2.6) follows from the Definition A.5 of linearization and the Schur complement formula [see, e.g., (A.20)] by taking ${\mathcal {A}}= {\mathbb {C}}^{N\times N}$.

In order to prove the bound (2.5), consider the set $\Theta _N$ defined in (2.3). One can see that $\varvec{H}_{ w}^{\mathrm {(init)}}$ satisfies the bound (2.5) by specializing Lemma A.10 for ${\mathcal {A}}= {\mathbb {C}}^{N\times N}$, $x_1 = H$, $y_{1} = W_{1}$, $y_{2} = W_{2}$, $C = 12/\gamma$ and the rational expression q being (1.10) on the set $\Theta _N$, as well as using the standard relation between the operator and max norms, similarly as in, e.g., (A.47). On the other hand, $\varvec{H}_{ w}^{\mathrm {(init)}}$ and $\varvec{H}_{ w}$ are related by (2.8). For any $R\in {\mathbb {C}}^{8\times 8}$, applying the transformation $R \mapsto {\widetilde{P}}_{ij} R {\widetilde{P}}_{ij}$ does not change the norm of R, while applying the map $R \mapsto {\widetilde{T}} R \, {\widetilde{T}}$ might change the norm by an irrelevant nonzero constant factor only. We thus conclude that $\varvec{H}_{ w}$ also satisfies the bound (2.5) with a constant $C_{\gamma }$ being possibly different than the one for $\varvec{H}_{ w}^{\mathrm {(init)}}$. Since the set $\Theta _N$ satisfies (2.4), the bound (2.5) holds a.w.o.p. $\square$

Define

$$\begin{aligned} \kappa _{1} = \left( \begin{array}{ccc} 0&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma } \\ 0 &{}\quad -\frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 \end{array} \right) ,\quad \kappa _{2} = \left( \begin{array}{ccc} 0&{}\quad 0&{}\quad \frac{\mathrm {i}\,}{ \gamma } \\ 0 &{}\quad -\frac{1}{4 \gamma ^2} &{}\quad 0 \\ -\frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 &{}\quad 0 \end{array} \right) \end{aligned}$$

(2.9)

and

$$\begin{aligned} \kappa _{3} = \left( \begin{array}{cc} 0&{}\quad w\\ {\overline{w}} &{}\quad 0 \end{array} \right) ,\quad \kappa _{4} = \left( \begin{array}{ccc} 0&{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0 \end{array} \right) ,\quad \kappa _{5} = \left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 0 \end{array} \right) . \end{aligned}$$

(2.10)

With this notation $\varvec{H}_{ w}$ can be rewritten as

$$\begin{aligned} \varvec{H}_{ w} = K_0( w)\otimes I_N + K_1 \otimes H + L_1\otimes W_1 + L_1^{*}\otimes W_1^* + L_2\otimes W_2 + L_2^{*}\otimes W_2^* , \end{aligned}$$

(2.11)

where $K_0=K_0( w),K_1,L_1,L_2 \in {\mathbb {C}}^{8\times 8}$ are given by their block structures as

$$\begin{aligned}&K_0 = \left( \begin{array}{ccc} \kappa _{1}&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \kappa _{2} &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad \kappa _{3} \end{array} \right) ,\quad K_1 = \left( \begin{array}{ccc} 0&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad - \sigma _{1} \end{array} \right) ,\quad L_1 = \left( \begin{array}{ccc} 0&{}\quad 0&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{} \kappa _4 &{} 0 \end{array} \right) ,\nonumber \\&L_2 = \left( \begin{array}{ccc} 0&{}\quad 0&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ \kappa _5 &{}\quad 0 &{}\quad 0 \end{array} \right) , \end{aligned}$$

(2.12)

and $\sigma _{1} = \left( \begin{array}{cc} 0 &{} \quad 1 \\ 1 &{} \quad 0 \end{array} \right)$ is the usual Pauli matrix. Consider the Dyson equation for linearization (DEL)

$$\begin{aligned} -\frac{1}{M} = z J_{8} - K_0 (w) + \Gamma \big [ M\big ] \end{aligned}$$

(2.13)

with a linear map $\Gamma : {\mathbb {C}}^{\,8\times 8} \rightarrow {\mathbb {C}}^{\,8\times 8}$ given by

$$\begin{aligned} \Gamma \big [R\big ] := K_{1} R K_{1} + L_1 R L_1^* + L_1^* R L_1+ L_2 R L_2^* + L_2^* R L_2 ,\qquad R\in {\mathbb {C}}^{\,8\times 8}. \end{aligned}$$

(2.14)

Lemma 2.4

(Existence and basic properties of the solution to the DEL (2.13)) For any $\gamma >0$, $w\in {\mathbb {C}}_{+} \cup {\mathbb {R}}$ and $z \in {\mathbb {C}}_{+}$ define $M_{z, w} \in {\mathbb {C}}^{\,8\times 8}$ as

$$\begin{aligned} M_{z, w}:= & {} ({\mathrm {id}}_8 \otimes \, \tau _{{\mathcal {S}}}) \Big ( \big (K_0( w) - z J_{8}\big )\otimes \mathbb {1}_{{\mathcal {S}}}+ K_{1}\otimes s\nonumber \\&+ L_1\otimes c_1 +L_1^{*}\otimes c_1^* + L_2 \otimes c_2 + L_2^{*} \otimes c_2^*\Big )^{-1} , \end{aligned}$$

(2.15)

where $s,c_1,c_2$ are freely independent semicircular and circular elements and $\tau _{{\mathcal {S}}}$ is a tracial state on a $C^*$-probability space $({\mathcal {S}}, \tau _{{\mathcal {S}}})$ with the unit element $\mathbb {1}_{{\mathcal {S}}}$. Then,

• (i) For any $\gamma >0$ there exists $C_{\gamma }>0$ such that $M_{z, w}$ satisfies the a priori bound

  $$\begin{aligned} \Vert M_{z, w}\Vert \le C_{\gamma } \Big (1 + \frac{1}{\mathrm {Im}z}\Big ) \end{aligned}$$

  (2.16)

  uniformly for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$.

• (ii) For any $\gamma >0$, $w\in {\mathbb {C}}_{+} \cup {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$, matrix $M_{z, w}$ satisfies the DEL (2.13) and has positive semidefinite imaginary part, $\mathrm {Im}M_{z, w} \ge 0$. Moreover, for all $\gamma >0$ and $w\in {\mathbb {C}}_{+} \cup {\mathbb {R}}$, the matrix-valued function $z\mapsto M_{z, w}$ is analytic on ${\mathbb {C}}_{+}$.

• (iii) For any $\gamma >0$, $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$ function $z\mapsto M_{z, w}$ admits the representation

  $$\begin{aligned} M_{z, w} = M^{\infty }_{ w} + \int _{{\mathbb {R}}} \frac{V_{ w}(d\lambda )}{\lambda - z} , \end{aligned}$$

  (2.17)

  where $M_{w}^{\infty } \in {\mathbb {C}}^{\,8\times 8}$, and $V_{ w}(d\lambda )$ is a positive semidefinite matrix-valued measure on ${\mathbb {R}}$ with compact support. In particular, $\lim _{z\rightarrow \infty } M_{z, w} = M^{\infty }_{ w}$.

Proof

Fix $\gamma >0$ and denote the noncommutative rational (in fact, polynomial) expression

$$\begin{aligned} q_{1,w}(x,y_1,y_2,y_1^*,y_2^*) := w-x + \mathrm {i}\,\gamma (y_1y_1^* + y_2 y_2^*) . \end{aligned}$$

(2.18)

Using the fact that $c_1 c_1^* + c_2 c_2^*$ has free Poisson distribution of rate 2, which in particular implies that $\Vert (c_1 c_1^* + c_2 c_2^*)^{-1}\Vert \le (1-\frac{1}{\sqrt{2}})^{-1}$, we conclude that the triple $(s,c_1,c_2)$ belongs to the effective domain ${\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ with $C = \gamma ^{-1} (1-\frac{1}{\sqrt{2}})^{-1}$ for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ (see Sect. A.1 for the corresponding definitions).

Following the proofs of Lemmas A.7 and A.10 and specializing them to our concrete case, we obtain that for any fixed $\gamma >0$ there exists $C_{\gamma }>0$ such that

$$\begin{aligned}&\Big \Vert \big ( (K_0(w) {-} z J_{8})\otimes \mathbb {1}_{{\mathcal {S}}}{+} K_{1}\otimes s{+} L_1\otimes c_1 {+}L_1^{*}\otimes c_1^* \nonumber \\&\quad + L_2 \otimes c_2 {+} L_2^{*} \otimes c_2^*\big )^{-1} \Big \Vert _{{\mathbb {C}}^{\,8\times 8} \,\otimes \,{\mathcal {S}}} \le C_{\gamma } \Big (1 + \frac{1}{\mathrm {Im}z}\Big ) \end{aligned}$$

(2.19)

uniformly for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$, which yields the a priori bound (2.16). Part (ii) of Lemma 2.4 now follows directly from parts (ii)–(iv) of Lemma A.12. Finally, (2.17) follows from the representation (A.27) in Lemma A.12. $\square$

With DEL (2.13), we associate the corresponding stability operator ${\mathscr {L}}_{z, w}:{\mathbb {C}}^{\,8\times 8} \rightarrow {\mathbb {C}}^{\,8\times 8}$ given by

$$\begin{aligned} {\mathscr {L}}_{z, w}\big [R \,\big ] := R - M_{z, w} \, \Gamma \big [R\,\big ] \,M_{z, w} ,\quad R\in {\mathbb {C}}^{\,8\times 8} . \end{aligned}$$

(2.20)

The following lemma is directly obtained from Proposition A.19 and establishes Theorem 1.3 and the weak limits in the part (i) of Theorem 1.4 for $\phi =1$.

Lemma 2.5

(Global law for $\mu _{\varvec{T}_{ w}}$). For $w\in {\mathbb {C}}_{+} \cup {\mathbb {R}}$ the empirical spectral measure $\mu _{\varvec{T}_{ w}}(d\lambda )$ converges weakly in probability (and almost surely) to $\rho _{ w}(d\lambda )$, where $\rho _{ w}(d\lambda ):=\langle e_1, V_{ w}(d\lambda ) \, e_1 \rangle$ is the (1, 1) component of the matrix-valued measure $V_{ w}(d\lambda )$ from (2.17). Moreover, $\mathrm {supp} \, \rho _{ w}(d\lambda ) \subset [0,1]$ for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ and $\rho _{ w}(d\lambda )$ converges weakly to $\rho _{ E }(d\lambda )$ as $w\in {\mathbb {C}}_+$ tends to $E\in {\mathbb {R}}$.

Proof

We apply Proposition A.19 to the rational expression in random matrices (1.10). By (A.78), for any $w\in {\mathbb {C}}_{+}$ and fixed $\theta >0$ the generalized resolvent of the linearization $(\varvec{H}_{ w} - z J_{8} \otimes I_N)^{-1}$, corresponding to $\varvec{G}_{z}$ in (A.78), converges uniformly on $\{\,z : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$ to $M_{z,w} \otimes I_N$, corresponding to $M_{z}^{(\mathrm {sc})} \otimes I_N$ in (A.78). In particular, the identity (2.6), similarly to (A.79) (see also Definition A.17), implies the pointwise convergence of the Stieltjes transform of the empirical spectral measure of $\varvec{T}_{w}$: for any $\theta , \varepsilon , D > 0$ there exists a constant $C_{\theta , \varepsilon ,D}>0$ such that

$$\begin{aligned} {\mathbb {P}}\bigg [\,\Big |\frac{1}{N}{\mathrm {Tr}}\, \Big (\varvec{T}_{w} - z I_N \Big )^{-1} - M_{z, w}(1,1)\Big | \ge \frac{N^{\varepsilon }}{N} \bigg ] \le \frac{C_{\theta , \varepsilon ,D}}{N^{D}} \end{aligned}$$

(2.21)

uniformly on $\{\,z : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$ and $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$. The convergence in (2.21) together with (2.17) imply that the weak convergence

$$\begin{aligned} \lim _{N\rightarrow \infty } \mu _{\varvec{T}_{w}}(d\lambda ) = \rho _{ w}(d\lambda ) \end{aligned}$$

(2.22)

holds in probability (see, e.g., [4, Theorem 2.4.4]).

Now we prove the almost sure convergence. Take any $z_0\in {\mathbb {C}}_{+}$, a sequence $\{z_i\}\subset {\mathbb {C}}_{+}$ of different complex numbers with $z_{i} \rightarrow z_{0}$ such that $z_0$ is the accumulation point of $\{z_i\}$. Define the sequence of events

$$\begin{aligned} A_N:= \bigg \{\max _{1\le i \le N}\Big \{\Big |\frac{1}{N}{\mathrm {Tr}}\, \Big (\varvec{T}_{w} - z_i I_N \Big )^{-1} - M_{z_{i}, w}(1,1)\Big |\Big \} \ge \frac{1}{\sqrt{N}}\bigg \} . \end{aligned}$$

(2.23)

Then it follows from (2.21) and the Borel–Cantelli lemma applied to $\{A_N\}$ that with probability 1

$$\begin{aligned} \lim _{N\rightarrow \infty }\frac{1}{N}{\mathrm {Tr}}\, \Big (\varvec{T}_{w} - z_i I_N \Big )^{-1} = M_{z_i, w}(1,1) \end{aligned}$$

(2.24)

for all $i\in {\mathbb {N}}$. Finally, applying the Vitali–Porter theorem we conclude that the weak convergence (2.22) holds almost surely.

To prove the bound on the support of $\rho _{ w}(d\lambda )$ note that the scattering matrix $S( w)$, related to $\varvec{T}_{ w}= T T^{*}$ via (1.2), is unitary. This implies that all singular values of T are located in the interval [0, 1], thus $\mathrm {supp}\, \mu _{\varvec{T}_{ w}} \subset [0,1]$. But from (2.22), we know that the empirical spectral measure of $T T^{*}$ converges weakly to $\rho _{w}(d\lambda )$, which yields $\mathrm {supp} \rho _{w} \subset [0,1]$.

It follows from (2.19) and the definition of $M_{z,w}$ in (2.15) that for any $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$, $E\in {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$

$$\begin{aligned} \Vert M_{z,w} - M_{z,E}\Vert \le 2\, C_{\gamma }^{2} \,\big |w- E\big |\, \Big (1 + \frac{1}{\mathrm {Im}z}\Big )^2 . \end{aligned}$$

(2.25)

This implies the pointwise convergence

$$\begin{aligned} \lim _{w\rightarrow E} M_{z,w} = M_{z,E} \end{aligned}$$

(2.26)

for all $z\in {\mathbb {C}}_{+}$. By (2.17) and $\rho _{ w}(d\lambda )=\langle e_1, V_{ w}(d\lambda ) \, e_1 \rangle$, the (1, 1)-components $M_{z,w}(1,1)$ and $M_{z,E}(1,1)$ define the Stieltjes transforms of the measures $\rho _{w}(d\lambda )$ and $\rho _{E}(d\lambda )$ correspondingly, from which we conclude that (2.26) yields the weak convergence of $\rho _{w}(d\lambda )$ to $\rho _{E}(d\lambda )$ as $w\rightarrow E$. $\square$

Note that $M_{z, w}$ is a matrix-valued Herglotz function. Therefore, from the properties of the (matrix-valued) Herglotz functions (see, e.g., [22, Theorems 2.2 and 5.4]), the absolutely continuous part of $\rho _{ w}(d\lambda )$ is given by the inverse Stieltjes transform of $M_{z, w}(1,1)$ (see Lemma A.12)

$$\begin{aligned} \rho _{ w}(\lambda ) := \lim _{\eta \downarrow 0} \frac{1}{\pi } \mathrm {Im}M_{\lambda +\mathrm {i}\,\eta , w}(1,1) . \end{aligned}$$

(2.27)

We call the function $\rho _{ w}(\lambda )$, defined in (2.27), the self-consistent density of states of the solution to the DEL (2.13). It will be shown in Sect. 2.5 that $\rho _{ w}(d\lambda )$ is in fact purely absolutely continuous, i.e., $\rho _{ w}(d\lambda ) = \rho _{ w}(\lambda ) d\lambda$. The statements (a) and (b) of part (ii) of Theorem 1.4 will be derived from the study of $M_{z, w}$ for the spectral parameter z being close to the real line.

Note that our particular choice of linearization (2.1) allows rewriting the original DEL (2.13) in a slightly simpler form. More precisely, if

$$\begin{aligned} R = \left( \begin{array}{ccc} R_{11}&{}\quad R_{12}&{} \quad R_{13} \\ R_{21}&{}\quad R_{22}&{}\quad R_{23} \\ R_{31}&{}\quad R_{32}&{}\quad R_{33} \end{array} \right) \in {\mathbb {C}}^{8\times 8} \end{aligned}$$

(2.28)

with $R_{11}, R_{22} \in {\mathbb {C}}^{3\times 3}$ and $R_{33} \in {\mathbb {C}}^{2\times 2}$, then (2.12) yields

$$\begin{aligned} \Gamma [R] = \left( \begin{array}{ccc} \kappa _{5}^{t} R_{33} \kappa _{5}&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \kappa _{4}^{t} R_{33} \kappa _{4}&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \sigma _{1} R_{33} \sigma _{1} + \kappa _{4} R_{22} \kappa _{4}^{t} + \kappa _{5} R_{11} \kappa _{5}^{t} \end{array} \right) , \end{aligned}$$

(2.29)

so that the image $\Gamma [R]$ is a block-diagonal matrix. Together with the definition of $K_0$ in (2.12), this implies that the right-hand side in (2.13) is a block-diagonal matrix with blocks of sizes 3, 3, and 2 correspondingly. We conclude that any solution to the DEL (2.13) has a block-diagonal form, which, in particular, allows us to write

$$\begin{aligned} M_{z, w} = \left( \begin{array}{ccc} M_{1} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad M_{2} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad M_{3} \end{array} \right) \end{aligned}$$

(2.30)

with $M_{1},M_{2} \in {\mathbb {C}}^{3\times 3}$ and $M_{3} \in {\mathbb {C}}^{2\times 2}$, where we omit the dependence of the blocks on z and $w$. Now DEL (2.13) can be decomposed into a system of three matrix equations of smaller dimensions

$$\begin{aligned} -\frac{1}{M_{1}} = zJ_{3}-\kappa _{1}+ \kappa _{5}^t M_{3} \kappa _{5} ,\quad -\frac{1}{M_{2}} = -\kappa _{2}+ \kappa _{4}^t M_{3} \kappa _{4} \end{aligned}$$

(2.31)

and

$$\begin{aligned} -\frac{1}{M_{3}}= & {} - \kappa _{3} (w) - \frac{ 1 }{ -\frac{1}{2 \gamma ^2z}(I_2 + \sigma _3)-\frac{1}{ \gamma }\sigma _2 + M_{3}} \nonumber \\&- \frac{1}{-2(I_2 - \sigma _3)-\frac{1}{ \gamma }\sigma _2+ M_{3} } + \sigma _{1} M_{3} \sigma _{1} , \end{aligned}$$

(2.32)

where $\sigma _{1} = \left( \begin{array}{cc} 0 &{}\quad 1 \\ 1 &{}\quad 0 \end{array} \right)$, $\sigma _{2}= \left( \begin{array}{cc} 0 &{}\quad -\mathrm {i}\,\\ \mathrm {i}\,&{}\quad 0 \end{array} \right)$ and $\sigma _{3}= \left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad -1 \end{array} \right)$ are the standard Pauli matrices. The proof of Theorem 1.4 is based on the study of matrices $M_{1}$, $M_{2}$ and $M_{3}$.

2.2 Useful Identities

From now on and until the end of Sect. 2, we study the matrix-valued function $M_{z,w}$ with $w= E \in {\mathbb {R}}$. We start by collecting several important relations between the components of $M_{z,E}$.

Lemma 2.6

For all $E\in {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$

$$\begin{aligned} M_{z,-E} = ( Q^{-}\,M_{z,E} \, Q^{-})^{t} , \end{aligned}$$

(2.33)

where $Q^{-} = \mathrm {diag} (-1,-1,1,-1,1,1,-1,1) \in {\mathbb {C}}^{8\times 8}$. In particular, for all $E\in {\mathbb {R}}$, $z\in {\mathbb {C}}_{+}$ and $1\le k\le 8$

$$\begin{aligned} M_{z,E}(k,k) = M_{z,-E}(k,k) . \end{aligned}$$

(2.34)

Proof

Let $\varvec{H}_{E}^{(\mathrm {sc})} \in {\mathcal {S}}^{8\times 8}$ be given by

$$\begin{aligned} \varvec{H}_{E}^{(\mathrm {sc})} := \left( \begin{array}{ccc|ccc|cc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad c_{2}^* \\ 0 &{}\quad 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma } \mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad c_{2}^* \\ 0 &{} \quad -\frac{\mathrm {i}\,}{ \gamma } \mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad c_{2}^* &{}\quad 0 \\ \hline 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma }\mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad c_{1}^* \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{1}{4 \gamma ^2} \mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad c_{1}^* &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{} \quad -\frac{\mathrm {i}\,}{ \gamma } \mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad 0 &{}\quad c_{1}^* &{} \quad 0 \\ \hline 0 &{}\quad 0 &{} \quad c_{2} &{}\quad 0 &{}\quad c_{1} &{}\quad c_{1} &{} \quad 0 &{}\quad E \mathbb {1}_{{\mathcal {S}}}-s\\ c_{2} &{}\quad c_{2} &{} \quad 0 &{}\quad c_{1} &{}\quad 0 &{}\quad 0 &{}\quad E \mathbb {1}_{{\mathcal {S}}}-s&{}\quad 0 \\ \end{array} \right) , \end{aligned}$$

(2.35)

where $s$ is a semicircular element, $c_1,c_2$ are circular elements, all freely independent in a $C^*$-probability space $({\mathcal {S}},\tau _{{\mathcal {S}}})$, so that

$$\begin{aligned} M_{z,E} := ({\mathrm {id}}_{8}\otimes \,\tau _{{\mathcal {S}}}) \big (\varvec{H}_{E}^{(\mathrm {sc})} - zJ_{8}\otimes \mathbb {1}_{{\mathcal {S}}}\big )^{-1} . \end{aligned}$$

(2.36)

Using the fact that $-s$, $-c_{1}^*$ and $-c_{2}^*$ form again a freely independent family of one semicircular and two circular elements, we can easily check that (here $\times$ denotes multiplication in ${\mathcal {S}}^{8\times 8}$)

$$\begin{aligned} ({\mathrm {id}}_{8}\otimes \,\tau _{{\mathcal {S}}})\Big (\big ((Q^{-}\otimes \mathbb {1}_{{\mathcal {S}}}) \times \big (\varvec{H}_{-E}^{(\mathrm {sc})} - zJ_{8}\otimes \mathbb {1}_{{\mathcal {S}}}\big ) \times (Q^{-}\otimes \mathbb {1}_{{\mathcal {S}}})\big )^{t}\Big )^{-1} = M_{z,E} , \end{aligned}$$

(2.37)

from which (2.33) follows after factorizing $Q^{-}$. $\square$

Lemma 2.7

For all $E\in {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$

$$\begin{aligned} M_{z,E}(8,8) = 4 \gamma ^2 z\, M_{z,E}(7,7) . \end{aligned}$$

(2.38)

Proof

Using the Schur complement formula, the lower-right $2\times 2$ block of the inverse of $\varvec{H}^{\mathrm {(sc)}}_{E} - zJ_{8} \otimes \mathbb {1}_{{\mathcal {S}}}$ can be written as

$$\begin{aligned} \left( \begin{array}{cc} 4 \gamma ^2 c_{1} c_{1}^* &{}\quad E \mathbb {1}_{{\mathcal {S}}}-s+ \mathrm {i}\,\gamma (c_{1}c_{1}^* + c_{2}c_{2}^*) \\ E \mathbb {1}_{{\mathcal {S}}}-s- \mathrm {i}\,\gamma (c_{1}c_{1}^* + c_{2}c_{2}^*) &{}\quad \frac{1}{z} c_{2}c_{2}^* \end{array} \right) ^{-1} . \end{aligned}$$

(2.39)

For convenience, change the rows in the above matrix, so that

$$\begin{aligned} M_{3} \left( \begin{array}{cc} 0&{}\quad 1 \\ 1&{}\quad 0 \end{array} \right) = ({\mathrm {id}}_2\otimes \,\tau _{{\mathcal {S}}}) \left[ \left( \begin{array}{cc} E \mathbb {1}_{{\mathcal {S}}}-a &{}\quad \frac{1}{z} c_{2}c_{2}^* \\ 4 \gamma ^2 c_{1} c_{1}^* &{}\quad E \mathbb {1}_{{\mathcal {S}}}-a^* \end{array} \right) ^{-1} \right] , \end{aligned}$$

(2.40)

where we introduced

$$\begin{aligned} a := s+\mathrm {i}\,\gamma \big (c_{1}c_{1}^* + c_{2}c_{2}^*\big ) . \end{aligned}$$

(2.41)

Notice, that since $c_{1}c_{1}^* + c_{2}c_{2}^*$ has a free Poisson distribution of rate 2, $c_{1}c_{1}^* + c_{2}c_{2}^* \ge (1-\frac{1}{\sqrt{2}})^2 \mathbb {1}_{{\mathcal {S}}}$ and thus both diagonal elements of the matrix on the right-hand side of (2.40) are invertible. Rewrite the matrix in the square brackets in the following way: for the entries of the first row apply the Schur complement formula with respect to the (1, 1)-component, and for the second row apply the Schur complement formula with respect to the (2, 2)-component. This leads to the following expressions for $M_{z,E}(7,7)$ and $M_{z,E}(8,8)$

$$\begin{aligned}&M_{z,E}(7,7) = \frac{1}{z}\tau _{{\mathcal {S}}} \Bigg (-\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a} c_{2}c_{2}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a^*-\frac{4 \gamma ^2}{z} c_{1}c_{1}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a}c_{2}c_{2}^*} \,\Bigg ) , \end{aligned}$$

(2.42)

$$\begin{aligned}&M_{z,E}(8,8) = 4 \gamma ^2 \tau _{{\mathcal {S}}}\Bigg (-\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a^*} c_{1}c_{1}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a-\frac{4 \gamma ^2}{z} c_{2}c_{2}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}-a^*}c_{1}c_{1}^*} \, \Bigg ) . \end{aligned}$$

(2.43)

Under $\tau _{{\mathcal {S}}}$, we can swap the labels of $c_1$ and $c_2$ and replace a with $-\,a^*$ without changing the value in (2.43). After completing these operations, we obtain

$$\begin{aligned} M_{z,E}(8,8) = 4 \gamma ^2 \tau _{{\mathcal {S}}}\Bigg (-\frac{1}{E \mathbb {1}_{{\mathcal {S}}}+a} c_{2}c_{2}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}+a^*-\frac{4 \gamma ^2}{z} c_{1}c_{1}^*\frac{1}{E \mathbb {1}_{{\mathcal {S}}}+a}c_{2}c_{2}^*} \,\Bigg ) . \end{aligned}$$

(2.44)

Multiplying both fractions under $\tau _{{\mathcal {S}}}$ in (2.44) by $-\,1$, and swapping E to $-\,E$ by (2.34), a comparison with (2.42) yields (2.38). $\square$

Lemma 2.8

For all $E\in {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$

$$\begin{aligned} M_{z,E}(8,7)-M_{z,E}(7,8) = \frac{\mathrm {i}\,}{ \gamma } M_{z,E}(8,8) \end{aligned}$$

(2.45)

Proof

Denote

$$\begin{aligned} T_1 := \left( \begin{array}{cc} -\frac{1}{ \gamma ^2 z} &{}\quad \frac{\mathrm {i}\,}{ \gamma } \\ -\frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 \end{array} \right) + M_{3} ,\quad T_{2} := \left( \begin{array}{cc} 0 &{}\quad \frac{\mathrm {i}\,}{ \gamma } \\ -\frac{\mathrm {i}\,}{ \gamma } &{}\quad -4 \end{array} \right) + M_{3} , \end{aligned}$$

(2.46)

so that (2.32) can be rewritten as

$$\begin{aligned} \frac{1}{M_{3}}- E \sigma _{1} - \frac{1}{T_{1}} - \frac{1}{T_{2}} + \sigma _{1} M_{3} \sigma _{1} = 0 . \end{aligned}$$

(2.47)

Then from (2.38), we have that

$$\begin{aligned} \det T_{1} - \det T_{2} = 4M_{z,E}(7,7)- \frac{1}{ \gamma ^2 z} M_{z,E}(8,8) = 0 \end{aligned}$$

(2.48)

Rewrite (2.47) componentwise using (2.48)

$$\begin{aligned}&\Big (\frac{1}{\det M_{3}} -\frac{2}{\det T_{1}} + 1 \Big ) M_{z,E}(8,8) = -\frac{4}{\det T_1} , \end{aligned}$$

(2.49)

$$\begin{aligned}&\Big ( -\frac{1}{\det M_{3}}+\frac{2}{\det T_{1}} \Big ) M_{z,E}(7,8) + M_{z,E}(8,7) = E - \frac{2 \mathrm {i}\,}{ \gamma } \frac{1}{\det T_1 } , \end{aligned}$$

(2.50)

$$\begin{aligned}&\Big ( -\frac{1}{\det M_{3}}+\frac{2}{\det T_{1}} \Big ) M_{z,E}(8,7) + M_{z,E}(7,8) = E + \frac{2 \mathrm {i}\,}{ \gamma } \frac{1}{\det T_1 }, \end{aligned}$$

(2.51)

$$\begin{aligned}&\Big (\frac{1}{\det M_{3}} -\frac{2}{\det T_{1}} + 1 \Big ) M_{z,E}(7,7) = -\frac{1}{ \gamma ^2 z\det T_1} . \end{aligned}$$

(2.52)

Subtracting (2.51) from (2.50) gives

$$\begin{aligned} \Big ( \frac{1}{\det M_{3}} - \frac{2}{\det T_{1}} + 1\Big ) (M_{z,E}(8,7)-M_{z,E}(7,8)) = - \frac{ \mathrm {i}\,}{ \gamma } \frac{4}{\det T_1 } , \end{aligned}$$

(2.53)

which together with (2.49) implies (2.45). $\square$

2.3 Boundedness of $M_{z,E}$ Away from $z=0$ and $z=1$

Lemma 2.9

(Boundedness of $M_{z,E}$) For any small $\theta >0$, there exists $C_{\theta }>0$ such that

$$\begin{aligned} \sup \bigg \{\,\Vert M_{z,E}\Vert \, : \, | z |\ge \theta , |1- z | \ge \theta , \, \mathrm {Im}z > 0,\, |E|\le \frac{1}{\theta } \,\bigg \} \le C_{\theta }. \end{aligned}$$

(2.54)

Proof

Introduce the following notation for the entries of $M_{3}$

$$\begin{aligned} \left( \begin{array}{cc} m_{11}&{}\quad m_{12} \\ m_{21}&{}\quad m_{22} \end{array} \right) := \left( \begin{array}{cc} M_{z,E}(7,7) &{}\quad M_{z,E}(7,8) \\ M_{z,E}(8,7) &{}\quad M_{z,E}(8,8) \end{array} \right) , \end{aligned}$$

(2.55)

so that, in particular, (2.38) and (2.45) can be rewritten as

$$\begin{aligned} m_{22}&= 4 \gamma ^2 z m_{11} , \end{aligned}$$

(2.56)

$$\begin{aligned} m_{21} - m_{12}&= \frac{\mathrm {i}\,}{ \gamma } m_{22} . \end{aligned}$$

(2.57)

Our goal is to show that $M_{z,E}$ and in particular $M_{z,E}(1,1)$ is bounded everywhere if z is away from 0 or 1. From (2.31), we have that

$$\begin{aligned}&M_{1} = -\left( \begin{array}{ccc} m_{22} + z &{}\; m_{22} &{}\; m_{21} \\ m_{22} &{}\; m_{22} &{}\; m_{21} - \frac{\mathrm {i}\,}{ \gamma } \\ m_{12} &{}\; m_{12} + \frac{\mathrm {i}\,}{ \gamma } &{}\; m_{11} \end{array} \right) ^{-1} ,\nonumber \\&M_{2} = -\left( \begin{array}{ccc} m_{22} &{}\; m_{21} &{} \; m_{21} - \frac{\mathrm {i}\,}{ \gamma } \\ m_{12} &{}\; m_{11} + \frac{1}{4 \gamma ^2} &{}\; m_{11} \\ m_{12} + \frac{\mathrm {i}\,}{ \gamma } &{}\; m_{11} &{}\; m_{11} \end{array} \right) ^{-1} , \end{aligned}$$

(2.58)

which after some elementary computations yields

$$\begin{aligned} \det M_{1} = \frac{-1}{z \det T_1} ,\quad \det M_{2} = \frac{- \gamma ^2}{\det T_1} \end{aligned}$$

(2.59)

and

$$\begin{aligned} M_{z,E}(1,1) = -\frac{1}{z} - \frac{4m_{11}}{z\det T_{1}} , \end{aligned}$$

(2.60)

where $T_{1}$ was defined in (2.46). The functions $\{m_{ij}, 1\le i,j \le 2\}$, $\{M_{i}, 1\le i \le 3\}$ and $\{T_{i}, 1\le i \le 2\}$ defined above all depend on the variables z and E, but in order to make the exposition lighter we drop the explicit dependence on these variables from the notation. Using (2.58)–(2.60) it is enough to show that for any fixed $(z_{\infty },E_{\infty })$ with $z_{ \infty } \in \overline{{\mathbb {C}}_{+}}$, $z_{\infty } \notin \{0,1\}$ and $E_{\infty } \in {\mathbb {R}}$ we have that $\lim _{(z,E)\rightarrow (z_{\infty }, E_{\infty })} |m_{ ij }| <\infty$ for $i,j\in \{1,2\}$ and $\lim _{(z,E)\rightarrow (z_{\infty }, E_{\infty })}|\det T_{1}|>0$.

We now prove some additional relations that can be obtained from (2.56), (2.57) and (2.49)–(2.52). Plugging (2.56) and (2.57) into (2.50) (recall that we are using notation (2.55)) gives

$$\begin{aligned} \Big (-\frac{1}{\det M_{3}}+ \frac{2}{\det T_{1} }+1\Big ) m_{12} + 4 \gamma \mathrm {i}\,z m_{11} + \frac{2\mathrm {i}\,}{ \gamma \det T_{1}} - E = 0 , \end{aligned}$$

(2.61)

which, after applying (2.52) to the terms in the parenthesis, can be rewritten as

$$\begin{aligned} \Big ( \frac{1}{ \gamma ^2 z m_{11} \det T_{1}} +2\Big ) m_{12} = E - \frac{2 \mathrm {i}\,}{ \gamma \det T_{1}} - 4 \gamma \mathrm {i}\,z m_{11} . \end{aligned}$$

(2.62)

From the definitions of $T_{1}$ and $M_{3}$ we have

$$\begin{aligned} \det T_{1} = \det M_{3} + 4 (z-1) m_{11} - \frac{1}{ \gamma ^2} , \end{aligned}$$

(2.63)

while (2.52) gives

$$\begin{aligned} \frac{1}{\det T_{1}} \Big ( 2- \frac{1}{ \gamma ^2 z m_{11}} \Big ) = \frac{1}{\det M_{3}} + 1 . \end{aligned}$$

(2.64)

Combining (2.63) and (2.64), we get the following quadratic equation for $\det M_{3}$

$$\begin{aligned}&\big (\det M_{3}\big )^2 + \det M_{3}\bigg ( 4(z-1) m_{11} + \frac{1}{ \gamma ^2 z m_{11}} -\bigg (1+\frac{1}{ \gamma ^2}\bigg )\bigg )\nonumber \\&\quad + \bigg ( 4(z-1)m_{11} - \frac{1}{ \gamma ^2}\bigg ) = 0 . \end{aligned}$$

(2.65)

Note that (2.62)–(2.65) hold for all $E\in {\mathbb {R}}$ and all $z\in {\mathbb {C}}_{+}$.

Using the above relations, we proceed with a proof by contradiction. Assume that there exists a sequence $(z_{n},E_{n})_{n=1}^{\infty }\subset {\mathbb {C}}_{+}\times [-\theta ^{-1},\theta ^{-1}]$, such that $|m_{11}^{(n)}| \rightarrow \infty$ as $n\rightarrow \infty$ (here and below we denote the evaluations at $(z_{n},E_{n})$ by adding the superscript (n)). Solving (2.65) for $\det M_{3}$ allows us to express $\det M_{3}$ in terms of $m_{11}$

$$\begin{aligned} \det M_{3}= & {} \frac{1}{2} \bigg \{-4(z-1)m_{11}+\Big (1+\frac{1}{ \gamma ^2}\Big ) \nonumber \\&{\pm } 4(z-1)m_{11} \bigg [1 -\frac{1}{2}\bigg (\frac{1}{2(z-1)}\Big (1+\frac{1}{ \gamma ^2}\Big )+\frac{1}{z-1}\bigg )\frac{1}{m_{11}}\nonumber \\&+O\left( \frac{1}{|m_{11}|^2}\right) \bigg ] \bigg \} . \end{aligned}$$

(2.66)

By passing to a subsequence, we may assume that the choice of the ${\pm }$ sign in (2.66) is constant for all n.

If we take the − sign in (2.66), then

$$\begin{aligned} \det M_{3}^{(n)} = -4(z_{n}-1)m_{11}^{(n)} + O\left( 1\right) \end{aligned}$$

(2.67)

and by (2.64)

$$\begin{aligned} \det T_{1}^{(n)} \rightarrow 2 ,\quad n\rightarrow \infty . \end{aligned}$$

(2.68)

From (2.62), (2.56), (2.57) and (2.68),

$$\begin{aligned} m_{12}^{(n)} = -2 \gamma \mathrm {i}\,z_{n} m_{11}^{(n)} + O\left( 1\right) ,\quad m_{21}^{(n)} = 2 \gamma \mathrm {i}\,z_{n} m_{11}^{(n)} + O\left( 1\right) , \end{aligned}$$

(2.69)

and therefore

$$\begin{aligned} \det M_{3}^{(n)} = 4 \gamma ^2 z_{n} \left( m_{11}^{(n)}\right) ^2-4 \gamma ^2z_{n}^2 \left( m_{11}^{(n)}\right) ^2 = 4 \gamma ^2 z_{n}(1-z_{n})\left( m_{11}^{(n)}\right) ^2 {+} O\left( m_{11}^{(n)}\right) , \end{aligned}$$

(2.70)

which contradicts to (2.67) since $|z_{n}(1-z_{n})|$ is separated away from 0.

If we take $+$ sign in (2.66), then

$$\begin{aligned} \det M_{3}^{(n)} = -1 +O\left( \frac{1}{\big |m_{11}^{(n)}\big |}\right) , \end{aligned}$$

(2.71)

and from (2.63)

$$\begin{aligned} \det T_{1}^{(n)} = 4(z_{n}-1)m_{11}^{(n)} + O\left( 1\right) . \end{aligned}$$

(2.72)

But then again, from (2.62), (2.56), (2.57) and (2.72),

$$\begin{aligned} m_{12}^{(n)} = -2 \gamma \mathrm {i}\,z_{n} m_{11}^{(n)} + O\left( 1\right) ,\quad m_{21}^{(n)} = 2 \gamma \mathrm {i}\,z_{n} m_{11}^{(n)} + O\left( 1\right) , \end{aligned}$$

(2.73)

so that

$$\begin{aligned} \det M_{3}^{(n)} = 4 \gamma ^2 z_{n}(1-z_{n}) (m_{11}^{(n)})^2 + O\left( \big |m_{11}^{(n)}\big |\right) , \end{aligned}$$

(2.74)

which contradicts to (2.71). Therefore, we have proven that $m_{11}$ is bounded everywhere away from the points $z\in \{0,1\}$. It is clear from (2.56) that the boundedness of $m_{11}$ guarantees the boundedness of $m_{22}$. On the other hand, assuming that $m_{12}$ and $m_{21}$ [see (2.57)] are unbounded implies that $|\det M_{3}^{(n)}|\rightarrow \infty$ and $|\det T_1^{(n)}|\rightarrow \infty$ on some sequence $(z_n,E_n)_{n=1}^{\infty }$, which contradicts to the boundedness of $m_{11}$ in (2.61). We conclude that all entries of $M_{3}$ are bounded everywhere away from the points $z\in \{0,1\}$.

Assume now that there exists a sequence $(z_{n},E_{n})_{n=1}^{\infty }\subset {\mathbb {C}}_{+}\times [-\theta ^{-1},\theta ^{-1}]$ such that $\det T_{1}^{(n)} \rightarrow 0$ as $n\rightarrow \infty$. Then by (2.62)

$$\begin{aligned} \left( \frac{1}{ \gamma ^2 z_{n} m_{11}^{(n)} }+O\left( \det T_{1}^{(n)}\right) \right) m_{12}^{(n)} = -\frac{2\mathrm {i}\,}{ \gamma } + O\left( \det {T_{1}}^{(n)}\right) , \end{aligned}$$

(2.75)

which together with the fact that $m_{11}^{(n)}$ is bounded implies that

$$\begin{aligned} m_{12}^{(n)} = -2\mathrm {i}\,\gamma z_{n} m_{11}^{(n)} + O\left( \det T_{1}^{(n)}\right) . \end{aligned}$$

(2.76)

Then by (2.56) and (2.57) we find

$$\begin{aligned} \det M_{3}^{(n)} = 4 \gamma ^2 z_{n}(1-z_{n}) \big (m_{11}^{(n)}\big )^2 + O\left( \det T_{1}^{(n)}\right) , \end{aligned}$$

(2.77)

and conclude from (2.77) and (2.63) that $\det M_{3}^{(n)}$ does not vanish as $n\rightarrow \infty$. But then (2.64) implies

$$\begin{aligned} m_{11}^{(n)} = \frac{1}{2 \gamma ^2 z_{n} } + O\left( \det T_{1}^{(n)}\right) , \end{aligned}$$

(2.78)

and thus by (2.63) and (2.78) also

$$\begin{aligned} \det M_{3}^{(n)} = \frac{1}{ \gamma ^2} - 2(z_{n}-1) \frac{1}{ \gamma ^2 z_{n} } + O\left( \det T_{1}^{(n)}\right) , \end{aligned}$$

(2.79)

as well as

$$\begin{aligned} \det M_{3}^{(n)} = 4 \gamma ^2 z_{n}(1-z_{n}) \frac{1}{4 \gamma ^4 z_{n}^2 } + O\left( \det T_{1}^{(n)}\right) = (1-z_{n}) \frac{1}{ \gamma ^2 z_{n} } + O\left( \det T_{1}^{(n)}\right) , \end{aligned}$$

(2.80)

by (2.77) and (2.78), which contradicts to (2.79) since $z_{n}$ is away from 0. We conclude that $|\det T_{1}|$ is bounded away from 0, which together with (2.60) establishes (a non-effective version of) (2.54).

In order to keep the presentation simple, above we showed that $\Vert M_{z,E}\Vert$ is bounded for $z\in \overline{{\mathbb {C}}_{+}} {\setminus } \{0,1\}$ and $E\in {\mathbb {R}}$ without providing an explicit effective bound $C_{\theta }$ as formulated in (2.54). Note that the constants hidden in the $O\left( \cdot \right)$ terms (for example, in (2.67)–(2.70), (2.71)–(2.74) or (2.75)–(2.80)) depend only on $|z_n|$, $|1-z_n|$ and $E_n$. Therefore, using the assumptions $|m_{11}^{(n)}| \ge D_{\theta }$ and $|\det T_{1}^{(n)}|\le 1/D_{\theta }$ for n large enough and carefully chosen $D_{\theta }>0$ instead of $|m_{11}^{(n)}| \rightarrow \infty$ and $|\det T_{1}^{(n)}|\rightarrow 0$ as $n\rightarrow \infty$, together with the a priori bound (2.16), eventually leads to the uniform bound (2.54). $\square$

2.4 Singularities of $M_{z,E}(1,1)$

Lemma 2.10

(Behavior of $M_{z,E}(1,1)$ in the vicinities of $z=0$ and $z=1$) 

• (a) For all $E\in {\mathbb {R}}$

  $$\begin{aligned} M_{z,E}(1,1) = \root 3 \of {\frac{1+E^2}{4 \gamma ^{2}}}\,z^{-2/3} + O\left( \frac{1}{|z|^{1/3}}\right) \text{ as } z\rightarrow 0 ; \end{aligned}$$

  (2.81)

• (b) for all $|E| < \frac{1}{ \gamma } \sqrt{2 \sqrt{1 + 6 \gamma ^2 + \gamma ^4}-2-2 \gamma ^2}$

  $$\begin{aligned} M_{z,E}(1,1) = - \frac{16 \gamma ^2 \xi _{0}}{ \gamma ^2 E^2 + 4+16 \gamma ^4 \xi _{0}^{2}}\, (z-1)^{-1/2} + O\left( 1\right) \text{ as } z\rightarrow 1 , \end{aligned}$$

  (2.82)

  where the constant $\xi _{0}<0$ is as in part (ii) of Theorem 1.4 (see also (2.95)).

The choices of the branches for $(\cdot )^{1/3}$ and $(\cdot )^{1/2}$ are specified in the course of the proof below.

Proof

We multiply (2.32) from the left by $M_{3}$ and from the right by $(Z_1 + M_{3})(Z_1-Z_2)^{-1}(Z_2+M_{3})$ with the short hand notation

$$\begin{aligned} Z_1 := -\frac{1}{2 \gamma ^2 z}(I_2+\sigma _3) -\frac{1}{ \gamma }\sigma _2 , \qquad Z_2 := -2(I_2-\sigma _3) -\frac{1}{ \gamma }\sigma _2 . \end{aligned}$$

(2.83)

Subsequently, the equation for $M_{3}$ takes the form $\Delta =0$ with

$$\begin{aligned} \Delta := \Big (Z_1-M_{3} + M_{3} \sigma _1(M_{3}\sigma _1-E)(Z_1+M_{3})\Big )\frac{1}{Z_1-Z_2}(Z_2+M_{3}) -M_{3} . \end{aligned}$$

(2.84)

Using that $(Z_1-Z_2)^{-1} = - \frac{ \gamma ^2 z}{2}(I_2+\sigma _3) +\frac{1}{8}(I_2-\sigma _3)$ and performing the matrix products we see that the entries of $\Delta \in {\mathbb {C}}^{2 \times 2}$ are polynomials in the entries of $M_{3}$, E and z.

Step 1: Expansion around $z=0$. We will now first construct a solution to (2.32) in a vicinity of $z=0$ by asymptotic expansion. Later, in Step 3, we will show that the constructed solution coincides with $M_{3}$ defined in (2.15) and (2.30). For this purpose, we write $t = z^{1/3}$ with an analytic cubic root on ${\mathbb {C}}{\setminus } (\mathrm {i}\,(-\infty ,0]))$ such that $(-1)^{1/3} =-1$. Then, we make the ansatz

$$\begin{aligned} {\widetilde{M}}_3 = \left( \begin{array}{cc} \xi _{11} t^{-1}&{}\quad 4 \gamma ^2 \xi _{12} t \\ 4 \gamma ^2 \xi _{21} t &{}\quad 4 \gamma ^2 \xi _{22} t^2 \end{array} \right) , \end{aligned}$$

(2.85)

where we will determine the unknown functions $\Xi _t:=(\xi _{ij}(t))_{i,j=1}^2$. Plugging (2.85) into (2.84) reveals that

$$\begin{aligned} \Delta = \left( \begin{array}{cc} 4 \gamma ^2 ( q_{11} + t p_{11}) &{}\quad 4 \gamma ^2t(q_{12} + t p_{12}) \\ -4 \gamma ^2t(q_{21} + t p_{21}) &{}\quad -16 \gamma ^4 t^3 (q_{22} + t p_{22}) \end{array} \right) , \end{aligned}$$

(2.86)

where the entries of $P:=(p_{ij})_{i,j=1}^2$ are polynomials in $t,E,\xi _{ij}$ and where $Q:=(q_{ij})_{i,j=1}^2$ is given by

$$\begin{aligned} Q = \left( \begin{array}{cc} \frac{1}{16 \gamma ^4} + \xi _{11}^2 \xi _{22} - E \xi _{11} \xi _{12} &{}\quad \xi _{12} +E \xi _{11} \xi _{22} \\ \xi _{21} + E \xi _{11} \xi _{22} &{}\quad -\frac{1}{16 \gamma ^4}- \xi _{11} \xi _{22}^2+E \xi _{21} \xi _{22} \end{array} \right) . \end{aligned}$$

(2.87)

Thus, the equation $\Delta =0$ is equivalent to $Q+t P =0$. For $t=0$, the three possible solutions are

$$\begin{aligned} \begin{aligned} \Xi _0 = \left( \begin{array}{cc} \zeta &{}\quad -E \zeta ^2 \\ -E \zeta ^2&{}\quad \zeta \end{array} \right) , \quad \text {where} \quad (1+E^2) \zeta ^3 = -\frac{1}{16 \gamma ^4}. \end{aligned} \end{aligned}$$

(2.88)

Since $\det \nabla _{\Xi } Q|_{t=0} = -3 \zeta ^4(1+E^2)^2$ the equation $Q+tP = 0$ is linearly stable at $t=0$ and can thus be solved for $\Xi _t$ in a neighborhood of $\Xi _0$ when t is sufficiently small. Since the equation is polynomial, the solution $\Xi _t$ admits a power series expansion in t. Now we define the analytic function ${\widetilde{M}}_3(z)$ through (2.85) on a neighborhood of $z=0$ in ${\mathbb {C}}{\setminus } (\mathrm {i}\,(-\infty ,0]))$ with $\Xi$ the solution to $Q+tP = 0$ and the choice $\zeta <0$ in (2.88).

We will now check that for any phase $e^{\mathrm {i}\,\psi } \ne -1$ in the complex upper half-plane the imaginary part of ${\widetilde{M}}_3(\theta e^{\mathrm {i}\,\psi })$ is positive definite in the sense that for any fixed $\varepsilon >0$ we have

$$\begin{aligned} \inf _{\psi \in [-\pi +\varepsilon ,0]}\inf _{\Vert u \Vert _{2}=1}\mathrm {Im}\langle u, {\widetilde{M}}_3(\theta e^{\mathrm {i}\,\psi }) u \rangle > 0 , \end{aligned}$$

(2.89)

for sufficiently small $\theta >0$. Indeed, for any vector $u=(u_1,u_2) \in {\mathbb {C}}^2$ we find

$$\begin{aligned} \mathrm {Im}\langle u, {\widetilde{M}}_3 u \rangle\ge & {} \beta \mathrm {Im}(-t^{-1} |u_1|^2 - t^2 |u_2|^2 ) - C |u_1||u_2| |t| \nonumber \\\ge & {} \beta _\varepsilon \bigg (\frac{1}{|t|} |u_1|^2 + |t|^2 |u_2|^2\bigg ) - \frac{C}{R}\frac{1}{|t|}|u_1|^2 - C R|u_2| |t|^3\qquad \end{aligned}$$

(2.90)

for some constants $\beta ,C>0$, small t, an $\varepsilon$-dependent constant $\beta _\varepsilon$ and any $R>0$. For $R=2C/\beta _\varepsilon$ and sufficiently small t this is still positive.

Step 2: Expansion around $z=1$. For the expansion around $z=1$, we proceed similarly to the discussion at $z=0$. We set $t = \sqrt{z-1}$, where the square root has a branch cut at $\mathrm {i}\,(-\infty ,0]$ and $\sqrt{1}=1$. Here, we make an ansatz with a reduced number of unknown functions by exploiting the identities (2.38) and (2.45), namely

$$\begin{aligned} {\widetilde{M}}_3 = \left( \begin{array}{cc} \frac{\xi }{4 \gamma ^2 t} &{}\quad \frac{E}{2}- \frac{\mathrm {i}\,\xi }{2 \gamma } (t^{-1}+t) + \frac{\nu t}{4 \gamma } \\ \frac{E}{2} + \frac{\mathrm {i}\,\xi }{2 \gamma } (t^{-1}+t) + \frac{\nu t}{4 \gamma } &{}\quad \xi (t^{-1}+t) \end{array} \right) . \end{aligned}$$

(2.91)

We will determine the unknown functions $\xi$ and $\nu$. Plugging (2.91) into (2.84), multiplying out everything and simplifying afterwards reveals

$$\begin{aligned} \Delta = \left( \begin{array}{cc} -\frac{1}{64 \gamma ^4 }(q_1+t p_1) &{}\quad -\frac{1}{32 \gamma ^3} (q_2 + t p_2) \\ \frac{1}{32 \gamma ^3} (q_2 + t p_2) &{}\quad \frac{1+t^2}{16 \gamma ^2}(q_1+t p_1 -2 \mathrm {i}\,(q_2 + t p_2)) \end{array} \right) , \end{aligned}$$

(2.92)

where $p_1,p_2$ are polynomials in $t,E,\xi ,\nu$ and where

$$\begin{aligned} \begin{aligned} q_1 =\,&\xi ^4 + (2 E^2 \gamma ^2 + 4 + 4 \gamma ^2) \xi ^2 + \gamma ^2( \gamma ^2 E^4 + 4 E^2 (1 + \gamma ^2)- 16 ) \\&-\, 16 \mathrm {i}\,\gamma ^3 E + \mathrm {i}\,E^2 \gamma ^2 \xi \nu + 4 \mathrm {i}\,\xi \nu + \mathrm {i}\,\xi ^3 \nu , \\ q_2 = \,&\xi ^3 \nu + (4 + \gamma ^2 E^{2}) \xi \nu - 16 \gamma ^{3} E \, . \end{aligned} \end{aligned}$$

(2.93)

The solution to the system $(q_1,q_2)=0$ at $t=0$ has the form

$$\begin{aligned} \nu _0= & {} \frac{16 \gamma ^3 E}{\xi (4+ \gamma ^2 E^2+ \xi ^2 )}, \quad r:= \gamma ^2E^4 + \frac{1}{ \gamma ^2} \xi ^4 \nonumber \\&+ \frac{4}{ \gamma ^2} (1 + \gamma ^2) \xi ^2 + 4 E^2 \left( 1 + \gamma ^2 + \frac{1}{2} \xi ^2 \right) -16 =0 \end{aligned}$$

(2.94)

In $r=0$ from (2.94) we choose the unique negative solution (as long as the expression inside the square root is positive)

$$\begin{aligned} \xi _0 = -\sqrt{2 \sqrt{1 + 6 \gamma ^2 + \gamma ^4}-2-2 \gamma ^2 - \gamma ^2 E^2} . \end{aligned}$$

(2.95)

We compute the Jacobian

$$\begin{aligned} \begin{aligned} \det \nabla _{\xi ,\nu }(q_1,q_2) = 4 \xi ^2 \left( 8 + 2 \gamma ^2 (12 + E^2) + \gamma ^4\left( \frac{2}{ \gamma ^4} \xi ^2 - 2 E^2\right) - 2 \gamma ^2 \xi ^2 + \gamma ^2 r\right) , \end{aligned} \end{aligned}$$

(2.96)

and evaluate at $t=0$ with $\xi =\xi _0$ to find

$$\begin{aligned} \det \nabla _{\xi _0,\nu _0}(q_1,q_2) = 16 \xi _0^2 \left( 1 {+} \gamma ^4 {+} \sqrt{ 1 {+} 6 \gamma ^2 {+} \gamma ^4} - \gamma ^2 \left( -6 {+} \sqrt{ 1 {+} 6 \gamma ^2 {+} \gamma ^4}\right) \right) <0. \end{aligned}$$

(2.97)

In particular, $\xi$ and $\nu$ admit a power series expansion in t.

Step 3: Coincidence of ${\widetilde{M}}_3$ and $M_{3}$, asymptotic behavior of $M_{z,E}(1,1)$. Here, we show that the asymptotic expansion ${\widetilde{M}}_3$ around $z=0,1$ coincides with $M_{3}$ defined in (2.15) and (2.30), the solution to DEL (2.32) constructed from free probability. For this purpose, for any small $\delta >0$, we construct a modification $M_{z,E}^{\delta }$ of the explicit solution $M_{z,E}$ (given by (2.36)) as follows

$$\begin{aligned} M^{\delta }_{z,E} := ({\mathrm {id}}_{8} \otimes \,\tau _{{\mathcal {S}}})\big (\varvec{H}_E^{(\mathrm {sc}),\delta }- z J_{8} \otimes \mathbb {1}_{{\mathcal {S}}}\big )^{-1} , \end{aligned}$$

(2.98)

with

$$\begin{aligned} \qquad \varvec{H}_E^{(\mathrm {sc}),\delta } := \left( \begin{array}{ccc} \kappa _1 \otimes \mathbb {1}_{{\mathcal {S}}}&{}\quad 0 &{}\quad \kappa _5^t \otimes c_2^* \\ 0&{}\quad \kappa _2 \otimes \mathbb {1}_{{\mathcal {S}}}&{}\quad \kappa _4^t \otimes c_1^* \\ \kappa _5 \otimes c_2 &{}\quad \kappa _4 \otimes c_1 &{}\quad \sigma _1 \otimes (E \cdot \mathbb {1}_{{\mathcal {S}}}-(1-\delta )\cdot s) -\delta \sigma _1 {\widetilde{M}}_3 \sigma _1\otimes \mathbb {1}_{{\mathcal {S}}}\end{array} \right) , \end{aligned}$$

(2.99)

where $c_i$ are circular and $s$ semicircular elements. Since the original generalized resolvent has the bound $\Vert M_{z,E}\Vert \le C (1+\frac{1}{\mathrm {Im}z})$, we also get

$$\begin{aligned} \Vert M_{z,E}^{\delta }-M_{z,E}\Vert \le C_{E,z}\delta . \end{aligned}$$

(2.100)

Now, similarly as in (2.31)–(2.32), we derive the DEL corresponding to the generalized resolvent of $\varvec{H}_E^{(\mathrm {sc}),\delta }$ and find

$$\begin{aligned}&-\frac{1}{M_{1}^{\delta }} = z J_{3} - \kappa _{1} + (1-\delta ) \kappa _{5}^{t} M_{3}^{\delta } \kappa _{5} ,\quad -\frac{1}{M_{2}^{\delta }} = - \kappa _{2} + (1-\delta ) \kappa _{4}^{t} M_{3}^{\delta } \kappa _{4} , \end{aligned}$$

(2.101)

$$\begin{aligned}&-\frac{1}{M_{3}^{\delta }} = -E \sigma _1 - \frac{1}{Z_1 +M_{3}^{\delta }} - \frac{1}{Z_2 +M_{3}^{\delta }} + (1-\delta ) \sigma _1 M_{3}^{\delta }\sigma _1 + \delta \sigma _1 {\widetilde{M}}_{3}\sigma _1 , \end{aligned}$$

(2.102)

where we exploited the block structure of $M_{z,E}^{\delta }$ and denoted

$$\begin{aligned} M_{z,E}^{\delta } = \left( \begin{array}{ccc} M_{1}^{\delta }&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad M_{2}^{\delta } &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad M_{3}^{\delta } \end{array} \right) . \end{aligned}$$

(2.103)

Notice that $\mathrm {Im}\big (\delta \sigma _1 {\widetilde{M}}_{3}\sigma _1\big )>0$ and $R \mapsto -E \sigma _1 - (Z_1 +R)^{-1} - (Z_2 +R)^{-1} + (1-\delta ) \sigma _1 R \sigma _1$ is a positivity preserving analytic mapping. The existence and uniqueness of the solution to (2.102) now follow from [25, Theorem 2.1]. Inverting both sides of (2.102), we obtain a fixed point equation $M_{3}^{\delta }= \Phi _{\delta }(M_{3}^{\delta })$ for $M_{3}^{\delta }$, where the map $\Phi _{\delta }$ can be read off from the inverse of the right-hand side of (2.102). Clearly, $\Phi _{\delta }$ with any $\delta >0$ is a contraction with respect to the Carathéodory metric mapping the set of $2 \times 2$ matrices with (strictly) positive definite imaginary parts strictly into itself (for more details, see [25]). In particular, there is a unique solution with positive semidefinite imaginary part and thus $M_{3}^{\delta } = {\widetilde{M}}_{3}$ for $\delta >0$. Together with (2.100) and taking the limit $\delta \downarrow 0$, we conclude $M_{3} = \widetilde{M_{3}}$. In particular, $M_{3}$ has a power law expansion around the singularities at $z=0,1$.

Finally, plugging the expansions (2.85) and (2.91) into (2.60) yields the asymptotics (2.81)–(2.82). $\square$

2.5 Proof of (i) and (ii) of Theorem 1.4 for $\phi = 1$

We will need the following classical result from the theory of Herglotz functions (see, e.g., [22, Theorem 2.3])

Lemma 2.11

Let $m:{\mathbb {C}}_{+}\rightarrow {\mathbb {C}}_{+}$ be a Herglotz function with representation

$$\begin{aligned} m(z) = m^{\infty } + \int _{{\mathbb {R}}} \frac{1}{\lambda - z} v (d\lambda ) \end{aligned}$$

(2.104)

for $m^{\infty }\in {\mathbb {R}}$ and a Borel measure $v(d\lambda )$ on ${\mathbb {R}}$. If for some $p>1$ and interval $I\subset {\mathbb {R}}$

$$\begin{aligned} \sup _{0< \eta< 1}\int _{I}|\mathrm {Im}m(\lambda + \mathrm {i}\,\eta )|^{p} d\lambda < \infty , \end{aligned}$$

(2.105)

then the measure $v(d\lambda )$ is absolutely continuous on I.

Proof of (i) and (ii) of Theorem 1.4for $\phi = 1$. The weak limits in the part (i) of Theorem 1.4 have been established in Lemma 2.5.

We know from (2.17) that $M_{z,E}(1,1)$ is a Herglotz function admitting the representation (2.104) with $m^{\infty } = M_{E}^{\infty }(1,1)$ and $v(d\lambda ) = \langle e_1, V_{E}(d\lambda ) \, e_1\rangle = \rho _{E}(d\lambda )$. Therefore, the absolute continuity of $\rho _{E}(d\lambda ) = \rho _{E}(\lambda ) d\lambda$ is a direct consequence of Lemma 2.11 and Lemmas 2.9 and 2.10 establishing together the integrability of $\lambda \mapsto \mathrm {Im}M_{\lambda + \mathrm {i}\,\eta ,E}(1,1)$ in the form (2.105) on the whole real axis for any $p<3/2$.

Finally, the asymptotic behavior of $\rho _{E}(\lambda )$ (1.13)–(1.14) near its singularities at 0 and 1 follows from Lemma 2.10 and the inverse Stieltjes transform formula (2.27). This, together with the global law established in Lemma 2.5, finishes the proof of Theorem 1.4 for $\phi = 1$ . $\square$

3 Proof of Theorem 1.4 for General Rational $\phi \in (0,1)$

In this section, we explain how the techniques described in Sect. 2 can be used to prove Theorems 1.3 and 1.4 in the case when $\phi = k/l\in (0,1)$ for fixed $k,l\in {\mathbb {N}}$, i.e., when $M=ln$ and $N=kn$ for some integer n tending to infinity. In addition to the steps used in Sect. 2, we need to tensorize the setup to accommodate for rectangular matrices. For example, the $M\times N = ln\times kn$ matrices $W_1$ and $W_2$ will be viewed as $l\times k$ rectangular block matrices with blocks of dimension $n\times n$. As we will see later, this allows us to treat the linearization matrix as a Kronecker random matrix in independent Wigner and i.i.d. matrices, which in turn makes various probabilistic estimates of the error terms in the corresponding DEL readily accessible from [3]. The restriction $\phi = k/l$ makes the presentation conceptually easier.

Note that different values of $\phi = k/l \in (0,1)$ require slightly different approaches. For $\phi \in (1/2,1)$, the matrix $\varvec{T}_{E,\phi } \in {\mathbb {C}}^{kn \times kn}$ given by (1.10) is well defined and bounded with very high probability. Indeed, in this case the product $W W^*$ is a sample covariance matrix with concentration ratio $\frac{1}{2\phi } \in (0,1)$, therefore similarly as in Sect. 2.1, one can define a sequence of events $\Theta _{\phi ,N}$ holding a.w.o.p. [see (2.4) and (3.2)] such that for any small $\delta >0$ and big enough $n\ge n_0( \delta )$, the spectrum of $W W^{*}$ is contained inside the interval $[(1-\frac{1}{\sqrt{2 \phi }})^{2}(1- \delta ) ,(1+\frac{1}{\sqrt{2 \phi }})^{2}(1+ \delta ) ]$ when restricted to $\Theta _{\phi , N}$ (see, e.g., [5, Section 5]). Thus, as n tends to infinity, the matrices in the denominator of (1.10) have positive imaginary parts and therefore bounded inverses with very high probability.

On the other hand, for $\phi \in {(}0,1/2{]}$ we need to proceed via regularization by replacing $\mathrm {i}\,\gamma W W^*$ with $\mathrm {i}\,\gamma W W^{*} + \mathrm {i}\,\epsilon \phi I_{M}$ for some small $\epsilon > 0$ to ensure the invertibility of the denominators in (1.10). This requires a more careful analysis of the $\epsilon$-dependence of various bounds and identities before we can take $\epsilon \rightarrow 0$.

Note also that for $\phi > 1$, $\mathrm {Rank}(\varvec{T}_{E,\phi })\le M$ and the spectral measure of the matrix $\varvec{T}_{E,\phi }$ has an atom of mass $1-1/2\phi$ at zero, while, as we will show, for $\phi <1$ the spectral measure $\mu _{\varvec{T}_{E,\phi }}$ does not have the pure point component. The regime $\phi =1$ studied in Sect. 2 is borderline: the limiting spectral measure of $\varvec{T}_{E,\phi }$ does not have atom at 0, but its behavior near the origin is more singular than in the case $\phi <1$.

3.1 Linearization Trick and the Dyson Equation for Linearization

In order to apply the linearization trick for $\phi = k/l\in (0,1)$, we split H, $W_{1}$ and $W_{2}$ into blocks of size $n\times n$, so that

$$\begin{aligned} H = ({\widehat{H}}_{ij})_{\begin{array}{c} i=1 \ldots l \\ j=1 \ldots l \end{array}} ,\quad W_{1} = ({\widehat{W}}_{1,ij})_{\begin{array}{c} i=1 \ldots l \\ j=1 \ldots k \end{array}} ,\quad W_{2} = ({\widehat{W}}_{2,ij})_{\begin{array}{c} i=1 \ldots l \\ j=1 \ldots k \end{array}} , \end{aligned}$$

(3.1)

with ${\widehat{H}}_{ij}, {\widehat{W}}_{1,ij}, {\widehat{W}}_{2,ij} \in {\mathbb {C}}^{n\times n}$. Note that for any $i\in \{1,\ldots ,l\}$, $\sqrt{l}\cdot {\widehat{H}}_{ii}$ is a (normalized) Wigner matrix of size n, and for any $i\ne j$, $\sqrt{l} \cdot {\widehat{H}}_{ij}$ is a (normalized) i.i.d. matrix of size n. Similarly, all the matrices $\sqrt{l} \cdot {\widehat{W}}_{1,ij}$ and $\sqrt{l} \cdot {\widehat{W}}_{2,ij}$ are also i.i.d. matrices of size n. All these matrices are independent apart from the natural constraint ${\widehat{H}}_{ij} = {\widehat{H}}_{ji}^{*}$.

Define the events

$$\begin{aligned} \Theta _{\phi , N} := \left\{ \begin{array}{ll} \big \{ \Vert H\Vert \le 3,\, \Vert W_{1}\Vert \le 3, \, \Vert W_2\Vert \le 3, \,\Vert (W W^*)^{-1}\Vert \le \frac{1}{(1-\frac{1}{\sqrt{2 \phi }})^{2}(1-\delta )} \big \}, &{} \phi \in (1/2,1), \\ \{ \Vert H\Vert \le 3,\, \Vert W_{1}\Vert \le 3, \, \Vert W_2\Vert \le 3 \, \}, &{} \phi \in {(}0,1/2{]}, \end{array} \right. \end{aligned}$$

(3.2)

for some $\delta >0$. Similarly as in Sect. 2.1, one can show that the events $\Theta _{\phi ,N}$ hold a.w.o.p. and the random matrix models (1.9) and (1.10) for $\phi \in (1/2,1)$ and $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ restricted to $\Theta _{\phi ,N}$ are well defined (see Remark A.21 and Lemma B.1). At the same time, in order to deal with $\phi \in {(}0,1/2{]}$, we consider the regularized models (1.9) and (1.10) with $w\in {\mathbb {C}}_{+}$, i.e., $\mathrm {Im}w>0$ strictly positive, which guarantees the invertibility of $w-H + \mathrm {i}\,\gamma W W^{*}$ without any additional restrictions on H and W.

The linearization matrix $\varvec{H}_{w,\phi }$ for (1.10) is defined as in (2.1). This is a Kronecker random matrix consisting of $(6k+2l)\times (6k+2l)$ blocks of size n. We now introduce a tensorized version of the generalized resolvent that takes into account the additional structure coming from (3.1).

Definition 3.1

(Generalized resolvent). Let $(\phi , w) \in ({(}0,1/2{]}\times {\mathbb {C}}_{+})\cup ((1/2,1)\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$ with $\phi = k/l \in {\mathbb {Q}}$. We call the matrix-valued function ${\mathbb {C}}_{+}\ni z \mapsto (\varvec{H}_{w,\phi }-z\varvec{J}_{k}\otimes I_{n})^{-1}$ the generalized resolvent of $\varvec{H}_{w,\phi }$. Here, we denote $\varvec{J}_{k}:= \sum _{i=1}^{k} E_{ii} \in {\mathbb {C}}^{(6k+2l)\times (6k+2l)}$ with $\{E_{ij}\}$ being the standard basis of ${\mathbb {C}}^{(6k+2l)\times (6k+2l)}$.

Lemma 3.2

(Basic properties of the generalized resolvent). 

• (i) For any $\gamma >0$ and $\phi \in (1/2,1)\cap {\mathbb {Q}}$, there exists $C_{\gamma ,\phi }>0$ such that a.w.o.p.

  $$\begin{aligned} \big \Vert (\varvec{H}_{w,\phi } - z\varvec{J}_{k}\otimes I_{n})^{-1}\big \Vert \le C_{\gamma ,\phi }\bigg (1 + \frac{1}{\mathrm {Im}z}\bigg ) \end{aligned}$$

  (3.3)

  uniformly for all $z\in {\mathbb {C}}_{+}$ and $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$.

• (ii) For any $\gamma >0$, $\phi \in {(}0,1/2{]}\cap {\mathbb {Q}}$ and $w\in {\mathbb {C}}_{+}$, there exists $C_{\gamma ,\phi ,w}>0$ such that a.w.o.p.

  $$\begin{aligned} \big \Vert (\varvec{H}_{w,\phi } - z\varvec{J}_{k}\otimes I_{n})^{-1}\big \Vert \le C_{\gamma ,\phi ,w}\bigg (1 + \frac{1}{\mathrm {Im}z}\bigg ) \end{aligned}$$

  (3.4)

  for all $z\in {\mathbb {C}}_{+}$.

• (iii) For all $(\phi , w) \in \big (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+}\big )\cup \big (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}})\big )$ and $1\le i,j\le N$

  $$\begin{aligned} \Big [(\varvec{H}_{w,\phi }-z\varvec{J}_{k}\otimes I_{n})^{-1}\Big ]_{ij} = \Big [(\varvec{T}_{w,\phi }- zI_N)^{-1}\Big ]_{ij} ,\quad 1\le i,j \le N . \end{aligned}$$

  (3.5)

Proof

Denote by ${\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ the effective domain related to the noncommutative rational function $q_{1,w}(x,y_1,y_2,y_1^*,y_2^*)= w-x + \mathrm {i}\,\gamma (y_1y_1^* + y_2 y_2^*)$. More precisely [see (A.4)], the set ${\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ consists of all triples of elements $({\hat{x}},{\hat{y}}_1,{\hat{y}}_2)$ such that

$$\begin{aligned} \left\| \frac{1}{q_{1,w}({\hat{x}},{\hat{y}}_1,{\hat{y}}_2,{\hat{y}}_1^*,{\hat{y}}_2^*)}\right\| \le C . \end{aligned}$$

(3.6)

As explained at the beginning of this section, for $\phi = k/l \in (1/2,1)$ a.w.o.p. the random matrices H, $W_1$ and $W_2$ defined in (3.1) belong to the domain ${\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ with constant [see (3.2) with $\delta = 1/2$] $C=\frac{1}{2 \gamma (1-\frac{1}{\sqrt{2\phi }})}$ depending only on $\gamma$ and $\phi$.

For $\phi = k/l \in {(}0,1/2{]}$ on the other hand, the evaluation of the function $q_{1,w}$ on random matrices H, $W_1$ and $W_2$ is invertible only for $\mathrm {Im}w>0$, in which case the norm of the inverse of $q_{1,w}$ evaluated on any triple $({\hat{x}},{\hat{y}}_1,{\hat{y}}_2)$ is bounded by $\frac{1}{\gamma \mathrm {Im}w}$.

With the above choice of the constant C in ${\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ depending on the value of $\phi$, the proof of (i)–(iii) can be obtained from the same argument as in Lemma 2.3 by restricting $\varvec{H}_{w,\phi }$ to the events $\Theta _{\phi , N}$ defined in (3.2) and taking into account the dimensions of H, $W_{1}$ and $W_{2}$ and the relation $N=kn$ (see Remark A.21). $\square$

From the structure of the linearization, we derive the DEL corresponding to $\varvec{H}_{w,\phi }$

$$\begin{aligned} -\frac{1}{M} = z \varvec{J}_{k} - K_0(w) + \Gamma _{\phi }[M] \end{aligned}$$

(3.7)

for an unknown matrix-valued function M depending on z, $w$ and $\phi$, having the following components:

• (i) the expectation matrix is given by

  $$\begin{aligned} K_0(w) := \left( \begin{array}{ccc|ccc|ccc} &{}\quad \kappa _{1}\otimes I_{k}&{} &{}&{}&{} &{}&{}&{} \\ &{}&{}&{}&{}&{}&{}&{}&{} \\ \hline &{}&{}&{}&{}&{}&{}&{}&{} \\ &{}&{} &{} &{}\quad \kappa _{2} \otimes I_{k}&{} &{}&{}&{} \\ &{}&{}&{}&{}&{}&{}&{}&{} \\ \hline &{}&{}&{}&{}&{}&{}&{}&{} \\ &{}&{}&{}&{}&{} &{} &{}\quad \kappa _{3}(w) \otimes I_{l}&{} \end{array} \right) \end{aligned}$$

  (3.8)

  with matrices $\kappa _{1}, \kappa _{2},\kappa _{3}(w)$ defined as in (2.9)–(2.10);

• (ii) the operator $\Gamma _{\phi } : {\mathbb {C}}^{(6k+2l)\times (6k+2l)} \rightarrow {\mathbb {C}}^{(6k+2l)\times (6k+2l)}$ maps an arbitrary matrix

  $$\begin{aligned} R = \left( \begin{array}{ccc} R_{11}&{}\quad R_{12}&{}\quad R_{13} \\ R_{21}&{}\quad R_{22}&{}\quad R_{23} \\ R_{31}&{}\quad R_{32}&{}\quad R_{33} \end{array} \right) \in {\mathbb {C}}^{(6k+2l)\times (6k+2l)} \end{aligned}$$

  (3.9)

  with $R_{11}, R_{22} \in {\mathbb {C}}^{3k\times 3k}$, $R_{33} \in {\mathbb {C}}^{2l\times 2l}$ into a block diagonal matrix with the first $3k\times 3k$ block equal to

  $$\begin{aligned} \kappa _{5}^{t}\bigg (\Big ({\mathrm {id}}_{2}\otimes \frac{1}{l}{\mathrm {Tr}}_{l}\Big ) R_{33}\bigg )\kappa _{5}\otimes I_{k} , \end{aligned}$$

  (3.10)

  the second $3k\times 3k$ diagonal block equal to

  $$\begin{aligned} \kappa _{4}^{t}\bigg (\Big ({\mathrm {id}}_{2}\otimes \frac{1}{l}{\mathrm {Tr}}_{l}\Big ) R_{33}\bigg )\kappa _{4}\otimes I_{k} , \end{aligned}$$

  (3.11)

  and the lower-right $2l\times 2l$ block equal to

  $$\begin{aligned}&\kappa _{5}\bigg (\Big ({\mathrm {id}}_{3}\otimes \frac{1}{l}{\mathrm {Tr}}_{k}\Big ) R_{11}\bigg )\kappa _{5}^{t} \otimes I_{l} + \kappa _{4}\bigg (\Big ({\mathrm {id}}_{3}\otimes \frac{1}{l}{\mathrm {Tr}}_{k}\Big ) R_{22}\bigg )\kappa _{4}^{t} \otimes I_{l} \nonumber \\&\quad + \sigma _{1} \bigg ( \Big ({\mathrm {id}}_{2} \otimes \frac{1}{l}{\mathrm {Tr}}_{l}\Big ) R_{33}\bigg ) \sigma _{1} \otimes I_{l} . \end{aligned}$$

  (3.12)

  Here, for $n\in {\mathbb {N}}$, we denote by ${\mathrm {Tr}}_{n}$ the trace of an $n\times n$ matrix $\kappa _{4},\kappa _{5}$ are defined as in (2.10) and $\sigma _{1}$ is a standard Pauli matrix. The operator $\Gamma _{\phi }$ is the tensorized analogue of $\Gamma$ from (2.29).

Now we can proceed similarly as for $\phi =1$ in Sect. 2 just the $(3+3+2)$ $\times$ $(3+3+2)$ structure of the linearized matrices is replaced by larger block matrices structured as $(3k+3k+2l)\times (3k+3k+2l)$.

Lemma 3.3

(Existence and basic properties of the solution to the DEL (3.7)) For any $\gamma >0$, $(\phi , w) \in (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+})\cup (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$ and $z\in {\mathbb {C}}_{+}$ define $M_{z, w} \in {\mathbb {C}}^{\,(6k+2l)\times (6k+2l)}$ as

$$\begin{aligned} M_{z,w} :=\,&({\mathrm {id}}_{6k+2l} \otimes \,\tau _{{\mathcal {S}}}) \bigg [ \Big ((K_0(w) - z \varvec{J}_{k}) \otimes \mathbb {1}_{{\mathcal {S}}}+ K_{1}\otimes H^{\mathrm {(sc)}} + L_1\otimes W_{1}^{\mathrm {(sc)}} \nonumber \\&+\,L_1^{*}\otimes \left( W_{1}^{\mathrm {(sc)}}\right) ^* + L_2 \otimes W_{2}^{\mathrm {(sc)}} + L_2^{*} \otimes \left( W_{2}^{\mathrm {(sc)}}\right) ^{*}\Big )^{-1}\bigg ] , \end{aligned}$$

(3.13)

where $W_{1}^{(\mathrm {sc})}$ and $W_{2}^{(\mathrm {sc})}$ are $l\times k$ matrices consisting of freely independent circular elements multiplied by $1/\sqrt{l}$, and $H^{(\mathrm {sc})}$ is an $l\times l$ self-adjoint matrix with freely independent semicircular elements multiplied by $1/\sqrt{l}$ on the diagonal and freely independent circulars multiplied by $1/\sqrt{l}$ above the diagonal.

Then,

• (i) For any $\gamma >0$ and $\phi \in (1/2,1)\cap {\mathbb {Q}}$, there exists $C_{\gamma ,\phi }>0$ such that $M_{z,w}$ satisfies the a priori bound

  $$\begin{aligned} \Vert M_{z,w} \Vert \le C_{\gamma ,\phi } \Big (1 + \frac{1}{\mathrm {Im}z}\Big ) \end{aligned}$$

  (3.14)

  uniformly for all $w\in {\mathbb {C}}_{+}\cup {\mathbb {R}}$ and $z\in {\mathbb {C}}_{+}$.

• (ii) For any $\gamma >0$, $\phi \in {(}0,1/2{]}\cap {\mathbb {Q}}$ and $w\in {\mathbb {C}}_{+}$, there exists $C_{\gamma ,\phi ,w}>0$ such that function $M_{z,w}$ satisfies the a priori bound

  $$\begin{aligned} \Vert M_{z,w} \Vert \le C_{\gamma ,\phi ,w} \Big (1 + \frac{1}{\mathrm {Im}z}\Big ) . \end{aligned}$$

  (3.15)

• (iii) For any $\gamma >0$, $(\phi , w) \in (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+})\cup (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$ and $z\in {\mathbb {C}}_{+}$, matrix $M_{z,w}$ satisfies the DEL (3.7) and has positive semidefinite imaginary part, $\mathrm {Im}M_{z,w} \ge 0$. Moreover, for all $\gamma >0$, $(\phi , w) \in (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+})\cup (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$, the matrix-valued function $z\mapsto M_{z,w}$ is analytic on ${\mathbb {C}}_{+}$.

• (iv) For any $\gamma >0$ and $(\phi , w) \in (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+})\cup (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$ function $z\mapsto M_{z,w}$ admits the representation

  $$\begin{aligned} M_{z,w} = M^{\infty }_{w,\phi } + \int _{{\mathbb {R}}} \frac{V_{w,\phi }(d\lambda )}{\lambda - z} , \end{aligned}$$

  (3.16)

  where $M_{w,\phi }^{\infty } \in {\mathbb {C}}^{(6k + 2l)\times (6k+2l)}$ is a self-adjoint matrix, and $V_{w,\phi }(d\lambda )$ is a positive-semidefinite matrix-valued measure on ${\mathbb {R}}$ with compact support.

Proof

The proof follows from parts (i)–(v) of Lemma A.12 (see also Remark A.21). Similarly as in the proof of Lemma 3.2, notice that the noncommutative rational expression $q_{1,w}= w-x + \mathrm {i}\,\gamma (y_1y_1^* + y_2 y_2^*)$ evaluated on matrices $x= H^{\mathrm {(sc)}}$, $y_1 = W_1^{\mathrm {(sc)}}$ and $y_2 = W_2^{\mathrm {(sc)}}$ expresses different behavior in variable $w$ for $\phi \in (0,1/2)$ and $\phi \in [1/2,1)$.

In the first case, $\phi \in (1/2,1)$, the invertibility of $q_{1,w}$ evaluated on $(s,c_1,c_2)$ does not depend on $w$ and thus $(s,c_1,c_2) \in {\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ with $C=C(\gamma ,\phi )$. In the case $\phi \in {(}0,1/2{]}$, $q_{1,w}$ is invertible if and only if $\mathrm {Im}w>0$, and the norm of $(q_{1,w})^{-1}$ depends on $\mathrm {Im}w$. This, in particular, means that $(s,c_1,c_2) \in {\mathcal {D}}_{\varvec{q}_{0},\{q_{1,w}\};C}$ with a $w$-dependent constant $C(\gamma ,\phi ,w)$.

This leads to two different a priori estimates: a bound (3.14) uniform in $w$ for $\phi \in (1/2,1)$ and a $w$-dependent bound for $\phi \in {(}0,1/2{]}$. The rest of the proof follows directly from Lemma A.12. $\square$

We omit the dependence of $M_{z,w}$ on $\phi$ for brevity. With these notations, we have the following global law establishing Theorem 1.3 and partially (i) of Theorem 1.4 for $\phi \in (0,1)$. The proof of the weak limit (1.12) for $\phi \in (0,1/2)$ is postponed to Sect. 3.6.

Lemma 3.4

(Global law for $\varvec{T}_{w,\phi }$, $\phi \in (0,1)$). For $(\phi , w) \in ({(}0,1/2{]}\times {\mathbb {C}}_{+})\cup ((1/2,1)\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$, $\phi = k/l$, the empirical spectral measure $\mu _{\varvec{T}_{w,\phi }}(d\lambda )$ converges weakly in probability (and almost surely) to $\rho _{w, \phi }(d\lambda )$, where

$$\begin{aligned} \rho _{w,\phi }(d\lambda ) := \frac{1}{k}{\mathrm {Tr}}( \varvec{J}_{k} \, V_{w,\phi }(d\lambda ) ) \end{aligned}$$

(3.17)

is the normalized trace of the upper-left $k\times k$ submatrix of the matrix-valued measure $V_{w,\phi }(d\lambda )$ from (3.16). The support of the measure $\rho _{w, \phi }(d\lambda )$ is a subset of the interval [0, 1]. Moreover, for any $\phi \in (1/2,1)$ and $E\in {\mathbb {R}}$, the measure $\rho _{w, \phi }(d\lambda )$ converges weakly to $\rho _{ E , \phi }(d\lambda )$ as $w\in {\mathbb {C}}_+$ tends to $E\in {\mathbb {R}}$.

Proof

The proofs are similar to the case $\phi =1$ (see Lemma 2.5 and Remark A.21) after taking into account the dimensions of the matrices H, $W_1$ and $W_2$ and the additional structure (3.1). $\square$

Definition 3.5

(Self-consistent density of states). We call the function

$$\begin{aligned} \rho _{w,\phi }(\lambda ) := \lim _{\eta \downarrow 0} \frac{1}{\pi k} \mathrm {Im}{\mathrm {Tr}}(\varvec{J}_{k} M_{\lambda +\mathrm {i}\,\eta ,w}) \end{aligned}$$

(3.18)

that gives the absolutely continuous part of $\rho _{w,\phi }(d\lambda )$, the self-consistent density of states of the model (1.10).

Since $\mathrm {supp} (\rho _{w,\phi })\subset [0,1]$ by unitarity of $S(w)$, part (iii) of Theorem 1.4 can be established by proving the boundedness of the upper-left $k\times k$ minor of $M_{z,w}$ for the spectral parameter z bounded away from 0 and 1 (Sect. 3.3), and analyzing the asymptotic behavior of this upper-left submatrix in the vicinity of the special points $z=0$ and $z=1$ (Sect. 3.4). The study of $M_{z,w}$ is simplified by the particular form of $K_{0}(w)$ and $\Gamma _{\phi }$, which implies that

$$\begin{aligned} M_{z,w} = \left( \begin{array}{ccc} M_1\otimes I_{k}&{}&{} \\ &{}\quad M_{2} \otimes I_{k} &{} \\ &{}&{}\quad M_{3} \otimes I_{l} \end{array} \right) \end{aligned}$$

(3.19)

with $M_1,M_2\in {\mathbb {C}}^{\,3\times 3}$ and $M_{3}\in {\mathbb {C}}^{\,2\times 2}$ satisfying

$$\begin{aligned} -\frac{1}{M_1} = zJ_{3}-\kappa _{1} + \kappa _{5}^{t} M_{3} \kappa _{5} ,\quad -\frac{1}{M_{2}} = -\kappa _{2} + \kappa _{4}^{t} M_{3} \kappa _{4} \end{aligned}$$

(3.20)

and

$$\begin{aligned} -\frac{1}{M_{3}} = - \kappa _{3}(w) + \phi \kappa _{5} M_{1} \kappa _{5}^{t} + \phi \kappa _{4} M_{2} \kappa _{4}^{t} + \sigma _{1} M_{3} \sigma _{1} . \end{aligned}$$

(3.21)

Similarly as in the case $\phi =1$, plugging (3.20) into (3.21) leads to the following self-consistent equation for $M_{3}$

$$\begin{aligned} -\frac{1}{M_{3}}= & {} - \kappa _{3}(w) - \frac{ \phi }{-\frac{1}{2\gamma ^2 z}(I_{2}+ \sigma _{3})-\frac{1}{\gamma }\sigma _{2}+M_{3}} \nonumber \\&- \frac{\phi }{-2(I_{2}- \sigma _{3})-\frac{1}{\gamma }\sigma _{2}+M_{3}} + \sigma _{1} M_{3} \sigma _{1} , \end{aligned}$$

(3.22)

which is the analogue of (2.32) for $\phi \ne 1$.

3.2 Useful Identities

Below we prove that identities similar to (2.34), (2.38) and (2.45) hold for $\phi \in (0,1)$.

Lemma 3.6

For all $(\phi , w) \in (({(}0,1/2{]} \cap {\mathbb {Q}})\times {\mathbb {C}}_{+})\cup (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$, $\gamma >0$ and $z\in {\mathbb {C}}_{+}$

• (i) $M_{z,w}(i,i) = M_{z,-{\overline{w}}}(i,i)$ for all $1\le i \le 6k+2l$;

• (ii) $M_{z,w}(6k+l+1,6k+l+1) = 4 \gamma ^2 z M_{z,w}(6k+1,6k+1)$;

• (iii) $M_{z,w}(6k+l+1,6k+1) - M_{z,w}(6k+1,6k+l+1) = \frac{\mathrm {i}\,}{ \gamma }M_{z,w}(6k+l+1,6k+l+1)\Big (1-\frac{ \gamma \mathrm {Im}w}{2 \phi } \det T_1\Big )$, where, similarly as in (2.46), we denoted

  $$\begin{aligned} T_1 = \left( \begin{array}{cc} -\frac{1}{ \gamma ^2 z} &{}\quad \frac{\mathrm {i}\,}{ \gamma } \\ -\frac{\mathrm {i}\,}{ \gamma } &{}\quad 0 \end{array} \right) + M_{3} \end{aligned}$$

  (3.23)

Denote the entries of $M_{3}$ in (3.19) by $m_{ij}$, $1\le i,j\le 2$. Then, the parts (ii) and (iii) of the above lemma can be rewritten as

$$\begin{aligned} m_{22}&= 4 \gamma ^2 z m_{11} , \end{aligned}$$

(3.24)

$$\begin{aligned} m_{21} - m_{12}&= \frac{\mathrm {i}\,}{ \gamma }m_{22} \Big (1-\frac{ \gamma \mathrm {Im}w}{2 \phi } \det T_1\Big ) . \end{aligned}$$

(3.25)

Proof

In order to establish Lemma 3.6, we can follow the proofs of Lemmas 2.6–2.8 and apply them to the matrix

$$\begin{aligned}&(K_0(w) - z \varvec{J}_{k}) \otimes \mathbb {1}_{{\mathcal {S}}}+ K_{1}\otimes H^{\mathrm {(sc)}} + L_1\otimes W_{1}^{\mathrm {(sc)}} +L_1^{*}\otimes \left( W_{1}^{\mathrm {(sc)}}\right) ^* \nonumber \\&\quad + L_2 \otimes W_{2}^{\mathrm {(sc)}} + L_2^{*} \otimes \left( W_{2}^{\mathrm {(sc)}}\right) ^{*} . \end{aligned}$$

(3.26)

Note that the above matrix (3.26) is obtained from (2.35) by substituting $c_{1}$, $c_{2}$ and $s$ with matrices $W_{1}^{(\mathrm {sc})}$, $W_{2}^{(\mathrm {sc})}$ and $H^{(\mathrm {sc})}$ correspondingly, and taking into account the dimensions of these matrices. For example, if we replace each diagonal entry of the matrix $Q^{-}$ from the proof of Lemma 2.6 by the tensor product of this entry and a corresponding identity matrix ($I_{k}$ or $I_{l}$), we obtain that the diagonal blocks of $M_{z,w}$ and $M_{z, -{\overline{w}}}$ coincide.

In order to prove (3.24), similarly as in the proof of Lemma 2.7, use the Schur complement formula with respect to the invertible upper-left $6k\times 6k$ submatrix of (3.26) to write the $2l\times 2l$ lower-right submatrix of its inverse as

$$\begin{aligned} \left( \begin{array}{cc} 4 \gamma ^2 W_{1}^{(\mathrm {sc})} \left( W_{1}^{(\mathrm {sc})}\right) ^* &{} w{-}H^{(\mathrm {sc})}{+} \mathrm {i}\,\gamma \Big (W_{1}^{(\mathrm {sc})} \big (W_{1}^{(\mathrm {sc})}\big )^* {+} W_{2}^{(\mathrm {sc})} \big (W_{2}^{(\mathrm {sc})}\big )^*\Big ) \\ {\overline{w}}{-}H^{(\mathrm {sc})}{-} \mathrm {i}\,\gamma \left( W_{1}^{(\mathrm {sc})} \big (W_{1}^{(\mathrm {sc})}\big )^* {+} W_{2}^{(\mathrm {sc})} \big (W_{2}^{(\mathrm {sc})}\big )^*\right) &{} \frac{1}{z} W_{2}^{(\mathrm {sc})} \big (W_{2}^{(\mathrm {sc})}\big )^* \end{array} \right) ^{-1} . \end{aligned}$$

(3.27)

Then, we can switch the blocks of (3.27) as in (2.40) and apply the Schur complement formula with respect to

$$\begin{aligned} w- H^{(\mathrm {sc})} + \mathrm {i}\,\gamma \Big (W_{1}^{(\mathrm {sc})} \big (W_{1}^{(\mathrm {sc})}\big )^* + W_{2}^{(\mathrm {sc})} \big (W_{2}^{(\mathrm {sc})}\big )^*\Big ) . \end{aligned}$$

(3.28)

The expression in (3.28) is invertible: for $(\phi , w) \in (((1/2,1)\cap {\mathbb {Q}})\times ({\mathbb {C}}_{+}\cup {\mathbb {R}}))$ the spectrum of $W_{1}^{(\mathrm {sc})} (W_{1}^{(\mathrm {sc})})^* + W_{2}^{(\mathrm {sc})} (W_{2}^{(\mathrm {sc})})^*$ follows the free Poisson distribution with rate $2\phi \in (1,2)$ and is therefore bounded away from zero, and for $(\phi , w) \in (({(}0,1/2{]}\cap {\mathbb {Q}})\times {\mathbb {C}}_{+})$ the invertibility is guaranteed by $\mathrm {Im}w$ being strictly positive. Due to the properties of freely independent circular and semicircular elements, switching the labels of the pair $(W_{1}^{(\mathrm {sc})},W_{2}^{(\mathrm {sc})})$ or changing the sign of $H^{(\mathrm {sc})}$ does not change the value of an expression involving these matrices after applying ${\mathrm {id}}_{6k+2l}\otimes \,\tau _{{\mathcal {S}}}$. Therefore, by proceeding as in (2.42)–(2.44) with $M_{z,w}(7,7)$ and $M_{z, w}(8,8)$ replaced by the corresponding $l\times l$ blocks of $M_{z, w}$, and using the diagonal structure of these blocks (3.19) and the part (i) of this lemma, we obtain (3.24).

Now it is straightforward to check by plugging (3.24) into (3.22) and following the proof of Lemma 2.8, that (3.25) holds. This proves Lemma 3.6.

$\square$

3.3 Boundedness of $M_{z,w}$ away from $z=0$ and $z=1$ for $\phi \in (0,1)$

The goal of this section is to establish a uniform bound on $\Vert M_{z,w}\Vert$ for parameter $w$ close to the real line and parameter z bounded away from 0 and 1. This will be used later in the proof of Theorem 1.4, in particular to show the absolute continuity of the measure $\rho _{E,\phi }(d \lambda )$. In the case $\phi \in (1/2,1)$ we can set $w= E \in {\mathbb {R}}$ and work directly with $M_{z,E}$. For $\phi \in {(}0,1/2{]}$, we prove the uniform bound for $\Vert M_{z,w}\Vert$ with $w= E + \mathrm {i}\,\phi \, \epsilon$ and small $\epsilon >0$, which will allow taking the limit $\epsilon \rightarrow 0$ in Sect. 3.4.

Lemma 3.7

(Boundedness of $M_{z,w}$) 

• (i) Case $\phi \in {(}1/2,1{]}$: For any $\gamma >0$ and small $\theta >0$, there exists $C_{\theta , \gamma }>0$ such that

  $$\begin{aligned} \sup \Big \{\,\Vert M_{z,E}\Vert \, : \, \phi \in (1/2,1)\cap {\mathbb {Q}},\, | z |\ge \theta , \, |1- z | \ge \theta , \, \mathrm {Im}z > 0 ,\, |E|\le \frac{1}{\theta }\, \Big \} \le C_{\theta , \gamma }; \end{aligned}$$

  (3.29)

• (ii) Case $\phi \in (0, 1)$: Let $w= E + \mathrm {i}\,\phi \epsilon$, $\epsilon >0$. For any $\gamma >0$, small $\theta >0$, $\phi _0\in (0,1/2)$ and $\epsilon _0>0$ small enough there exists $C_{\theta , \gamma ,\phi _0,\epsilon _0}>0$ such that

  $$\begin{aligned} \Vert M_{z,w}\Vert \le C_{\theta , \gamma ,\phi _0,\epsilon _0}\end{aligned}$$

  (3.30)

  uniformly on the set

  $$\begin{aligned} \Big \{ \phi \in [\phi _0,1-\phi _0]\cap {\mathbb {Q}}, |z|\ge \theta , |1-z| \ge \theta , \mathrm {Im}z > 0, |E|\le \frac{1}{\theta }, \epsilon \in (0, \epsilon _0]\, \Big \} . \end{aligned}$$

  (3.31)

Proof

Consider first (3.29) for which $\phi \in (1/2,1)$. By setting $\epsilon = 0$ in Lemma 3.6, we can proceed by establishing (3.29) in the same manner as in the proof of Lemma 2.9. To guarantee a uniform bound in parameters z, E and $\phi$, instead of a sequence $(z_n, E_{n})_{n=1}^{\infty }$ as in the proof of Lemma 2.9, we assume the existence of a sequence $((z_n, E_n, \phi _n))_{n=1}^{\infty }$, $z_n \in {\mathbb {C}}_+$, $|E|\le 1/\theta$, $\phi _n \in (1/2,1)$, on which $|m_{11}^{(n)}|\rightarrow \infty$ or $|\det T_{1}^{(n)}|\rightarrow 0$. Note that for $\phi _n\in (1/2,1)$ the leading terms in the analogues of the mutually contradicting pairs of statements (2.67)/(2.70), (2.71)/(2.74) and (2.79)/(2.80) do not depend on $\phi _n$ (with the exception of (2.71) where the constant $-1$ is replaced by $\phi _n - 2$).

For (3.30), i.e., $\phi \in [\phi _0,1- \phi _0]$, the above argument has to be slightly adjusted to ensure a uniform bound for small $\epsilon >0$. Instead of a sequence $((z_n, E_n, \phi _n))_{n=1}^{\infty }$ as in the first part of the proof, we now assume the existence of a sequence $((z_n, E_n ,\phi _n, \epsilon _n))_{n=1}^{\infty }$, $z_n \in {\mathbb {C}}_+$, $|E|\le 1/\theta$, $\phi _n \in [\phi _0,1-\phi _0]$, $\epsilon _n>0$, on which $|m_{11}^{(n)}|\rightarrow \infty$ or $|\det T_{1}^{(n)}|\rightarrow 0$.

The analogues of (2.67)/(2.70) in this case are

$$\begin{aligned} \det M_{3}^{(n)}&= -4(z_{n}-1)m_{11}^{(n)} + O\left( 1\right) , \end{aligned}$$

(3.32)

$$\begin{aligned} \det M_{3}^{(n)}&= 4 \gamma ^2 z_n (1-z_n(1-\epsilon _n\phi _n \gamma )^2)\left( m_{11}^{(n)}\right) ^2 + O\left( m_{11}^{(n)}\right) , \end{aligned}$$

(3.33)

which contradict each other if $|m_{11}^{(n)}|\rightarrow \infty$, $\epsilon _n\le \epsilon _0$ is small enough and $z_n$ is bounded away from 0 and 1.

Instead of the pair (2.71)/(2.74), we take the analogues of (2.71) and (2.72) for $\phi _n \in [\phi _0, 1-\phi _0]$ and $\epsilon _n>0$

$$\begin{aligned} \det M_{3}^{(n)}&= \phi _n -2 +O\left( \frac{1}{\big |m_{11}^{(n)}\big |}\right) , \end{aligned}$$

(3.34)

$$\begin{aligned} \det T_{1}^{(n)}&= 4(z_{n}-1)m_{11}^{(n)} + O\left( 1\right) , \end{aligned}$$

(3.35)

and observe that since $1 + 1/(\phi _n-2)\ge \phi _0/(1+\phi _0)>0$ for $\phi _n \in [\phi _0,1-\phi _0]$, the above equations contradict to

$$\begin{aligned} \left( \frac{1}{\det M_{3}} -\frac{2 \phi _n}{\det T_{1}} + 1 \right) m_{11}^{(n)} = -\frac{\phi _n}{ \gamma ^2 z\det T_1}, \end{aligned}$$

(3.36)

the analogue of (2.52), in the regime $|m_{11}^{(n)}|\rightarrow \infty$.

For the last pair (2.79)/(2.80) note that in the analogue of (2.75)

$$\begin{aligned} \Big (\frac{\phi _n}{ \gamma ^2 z_{n} m_{11}^{(n)} }+O\left( \det T_{1}^{(n)}\right) \Big )m_{12}^{(n)} = -\frac{2 \phi _n \mathrm {i}\,}{ \gamma } + O\left( \det {T_{1}}^{(n)}\right) \end{aligned}$$

(3.37)

with $|\det {T_{1}}^{(n)}|\rightarrow 0$, the parameter $\phi _n$ disappears after dividing (3.37) by $\phi _n$; therefore, we can proceed exactly as in the proof of Lemma 2.9. We conclude, similarly as in Lemma 2.9, that $\Vert M_{z,w}\Vert$ is uniformly bounded provided that $\min \{|z|, |1-z|\} \ge \theta$, $|E|\le 1/\theta$, $\phi _0 \le \phi \le 1-\phi _0$ and $\epsilon \le \epsilon _0$ for some $\phi _0 >0$ and $\theta ,\epsilon _0>0$ small enough. $\square$

3.4 Singularities of $M_{z,E}(1,1)$

For $\phi \in (1/2,1)$, the solution matrix $M_{z,E}$ with $E\in {\mathbb {R}}$ is given directly via (3.13). For $\phi \in {(}0,1/2{]}$, this formula cannot be directly applied when $w= E \in {\mathbb {R}}$ is real; we need an additional regularization argument. Nevertheless, in the next lemma we show the existence of the solution to the Dyson equation (3.20)–(3.22) for $\phi \in {(}0,1/2{]}$ and $w= E \in {\mathbb {R}}$ and we establish the asymptotic behavior of $M_{z, E}(1,1)$ near $z=0$ and $z=1$ for $\phi \in (0,1)$. We start by constructing an expansion of the solution $M_{z, w}$ in the vicinity of $z=0$ for $\phi \in (0,1)$ and $w= E + \mathrm {i}\,\phi \epsilon$ with $\epsilon >0$ sufficiently small. We then use this expansion to extend $M_{z,w}$ to $w= E \in {\mathbb {R}}$ for $\phi \in {(}0,1/2{]}$ by taking $\epsilon \downarrow 0$ and to study the asymptotic behavior of $M_{z,E}$ at special points $z=0$ and $z=1$.

Recall that the solution to the Dyson equation has the block structure (3.19), where $M_{1},M_{2}$ are determined by $M_{3}$ through (3.20) and $M_{3}$ satisfies (3.22).

Lemma 3.8

(Existence of $M_{z,E}$ for $\phi \in {(}0,1/2{]}$ and singularities of $M_{z, E}(1,1)$ for $\phi \in (0,1)$). 

• (a) For $\gamma >0$, let $M_{z,w, \phi }$ denote the function $M_{z,w}$ defined in (3.13) evaluated at a point $(z,w,\phi )$ with

  $$\begin{aligned} z\in {\mathbb {C}}_{+} \quad \text{ and } \quad (w,\phi )\in \big ({\mathbb {C}}_{+}\times \big ({(}0,1/2{]}\cap {\mathbb {Q}}\big )\big )\cup \big (({\mathbb {C}}_{+}\cup {\mathbb {R}}) \times \big ((1/2,1)\cap {\mathbb {Q}}\big )\big ). \end{aligned}$$

  (3.38)

  Then $M_{z,w,\phi }$ can be continuously extended to the set

  $$\begin{aligned} z\in {\mathbb {C}}_{+} \quad \text{ and } \quad (w,\phi )\in {\mathbb {R}}\times (0,1) \end{aligned}$$

  (3.39)

  in the following sense: for any $(w,\phi ) \in {\mathbb {R}}\times (0,1)$ and any sequence $\{(w_{n},\phi _{n}) \}_{n\ge 1} \subset \big ({\mathbb {C}}_{+}\times \big ({(}0,1/2{]}\cap {\mathbb {Q}}\big )\big )\cup \big (({\mathbb {C}}_{+}\cup {\mathbb {R}}) \times \big ((1/2,1)\cap {\mathbb {Q}}\big )\big )$ with $(w_{n},\phi _{n})\rightarrow (w,\phi )$ as $n\rightarrow \infty$, there exists an analytic matrix-valued function $M_{z,w,\phi }:{\mathbb {C}}_+ \rightarrow {\mathbb {C}}^{(6k + 2l)\times (6k + 2l)}$, $z\mapsto M_{z,w,\phi }$, such that

  $$\begin{aligned} M_{z,w_{n},\phi _{n}} \rightarrow M_{z,w,\phi } \end{aligned}$$

  (3.40)

  uniformly on compact z-subsets of ${\mathbb {C}}_+$ as $n\rightarrow \infty$.

  For $\phi \in (0,1)$ and $w= E \in {\mathbb {R}}$, denote by $M_{z,E}:= M_{z,E,\phi }$ the function defined in (3.40) omitting explicitly the dependence on $\phi$.

• (b) For all $\phi \in (0,1)$, $\gamma >0$ and $E\in {\mathbb {R}}$

  $$\begin{aligned} M_{z,E}(1,1) = \mathrm {i}\,\frac{4+ \gamma ^2 \nu _0^2+ \gamma ^2 \xi _0^2}{4 \gamma \xi _0}\,z^{-1/2} + O(1)\, \text{ as } z\rightarrow 0 , \end{aligned}$$

  (3.41)

  with constants $\xi _{0} := \xi _{0} ( \phi , \gamma )$ and $\nu _{0} := \nu _{0} ( \phi , \gamma )$ given in (3.49) in the proof below.

• (c) For all $\phi \in (0,1)$, $\gamma >0$ and $|E| \le E_{0}:=E_{0}(\phi ,\gamma )$

  $$\begin{aligned} M_{z,E}(1,1) = \frac{4\xi _0}{(\xi _0^2 + \gamma ^2 E^2+4 )}(z-1)^{-1/2} + O\left( 1\right) \text{ as } z\rightarrow 1 , \end{aligned}$$

  (3.42)

  with constants $E_{0}$ and $\xi _{0}:= \xi _{0} ( \phi , \gamma ) <0$ given in (3.52) and (3.54) correspondingly.

The branch of the square root is chosen to be continuous on ${\mathbb {C}}{\setminus } (\mathrm {i}(-\infty ,0])$ such that $\sqrt{1}=1$.

Proof

The analysis of (3.22) for $\phi \in (0,1)$ will follow similar steps as the analysis of (2.32) for the $\phi =1$ case as performed in Sect. 2.4, and we will omit the details of some straightforward albeit tedious calculations. In the first step below, we analyze the solution to (3.22) for rational $\phi \in (0,1)$ and $w= E + \mathrm {i}\,\phi \, \epsilon$ with $E\in {\mathbb {R}}$ and $\epsilon >0$ small enough. Using the same procedure that led to (2.84), we rewrite this equation as $\Delta =0$ with

$$\begin{aligned}&\Delta := \Big (Z_1+(1-2\phi )M_{3} + M_{3} \sigma _1(M_{3}\sigma _1-E+\mathrm {i}\epsilon \phi \sigma _3 )(Z_1+M_{3})\Big )\nonumber \\&\qquad \frac{1}{Z_1-Z_2}(Z_2+M_{3}) -\phi M_{3} , \end{aligned}$$

(3.43)

and the two $2 \times 2$-matrices

$$\begin{aligned} Z_1 := -\frac{1}{2 \gamma ^2 z}(I_2+\sigma _3) -\frac{1}{ \gamma }\sigma _2 , \qquad Z_2 := -2(I_2-\sigma _3) -\frac{1}{ \gamma }\sigma _2 . \end{aligned}$$

(3.44)

Step 1: Expansion around $z=0$. We construct an expansion of $M_3$ as a power series in $t=\sqrt{z}$ in a neighborhood of $z=0$. We make an ansatz compatible with the symmetries (3.24) and (3.25), namely

$$\begin{aligned} {\widetilde{M}}_3(t) = \left( \begin{array}{cc} \frac{\mathrm {i}\xi }{4 \gamma t} &{}\quad \frac{\nu }{2} + \frac{\xi {t}}{2}(1+\frac{\epsilon }{ \gamma } f )+\epsilon \frac{\mathrm {i}\xi ^2}{4} \\ \frac{\nu }{2} - \frac{\xi {t}}{2} (1+\frac{\epsilon }{ \gamma } f)-\epsilon \frac{\mathrm {i}\xi ^2}{4} &{}\quad \mathrm {i}\gamma \xi {t}\end{array} \right) , \end{aligned}$$

(3.45)

for the two functions $\xi =\xi (t),\nu =\nu (t)$ of t to be determined and where the parameters $E,\gamma , \phi ,\epsilon$ are considered fixed. Later, we will show that with the right choice of functions $\xi , \nu$ this ansatz coincides with the solution of the Dyson equation, i.e., that ${\widetilde{M}}_3(t)= {M}_3(z)$. Here, f solves the equation $q_0=0$ for sufficiently small $\epsilon \ge 0$, where

$$\begin{aligned} q_0 = f+ \frac{ \gamma ^2 }{2}\det \Bigg [ \left( \begin{array}{cc} -\frac{1}{ \gamma ^2 t^2} &{}\quad \frac{\mathrm {i}}{ \gamma } \\ -\frac{\mathrm {i}}{ \gamma } &{}\quad 0 \end{array} \right) + {\widetilde{M}}_3\Bigg ]+\frac{\mathrm {i}\gamma \xi }{2{t}} \end{aligned}$$

(3.46)

is a polynomial in all variables $t,E, \gamma ,\xi ,\nu ,\epsilon$ and f, in which it is quadratic. The choice (3.46) for f ensures that the symmetry condition (3.25) is satisfied.

We plug (3.45) into (3.43) and, after a mechanical but very long calculation that uses $q_0=0$ in the (2, 1)- and (2, 2)-entries of $\Delta$, find

$$\begin{aligned} \Delta = \left( \begin{array}{cc} -\frac{\mathrm {i}}{16 \gamma t}q_{1,\epsilon ,t} &{}\quad \frac{1}{4}q_{2,\epsilon ,t} \\ -\frac{1}{4}q_{2,\epsilon ,t} &{}\quad \frac{ \mathrm {i}\gamma {t}}{4}q_{1,\epsilon ,t} + (\mathrm {i}\gamma t^2+\mathrm {i}t \epsilon f{t}- \frac{ \gamma \epsilon \xi {t}}{2})q_{2,\epsilon ,t} \end{array} \right) , \end{aligned}$$

(3.47)

where

$$\begin{aligned} q_{1,\epsilon ,t} =q_1+\epsilon {\widetilde{q}}_1 + \frac{t}{4 \gamma ^3 } p_{1}, \qquad q_{2,\epsilon ,t} =q_2 + \frac{t}{ \gamma } p_{2}, \end{aligned}$$

and $p_1,p_2$ are polynomials in $t,E, \gamma ,\phi ,\xi ,\nu , \epsilon$ and $q_1=q_{1,0,0},q_2=q_{2,0,0},{\widetilde{q}}_1$ are the following explicitly defined functions of the unknowns $(\xi ,\nu )$:

$$\begin{aligned} q_1= & {} \xi (\xi ^2 - \nu ^2+2E\nu +4(\phi -1)), \quad q_2= \xi ^2 (\nu -E)+2\nu \phi , \nonumber \\ {\widetilde{q}}_1= & {} -\mathrm {i}\,\xi q_2+\epsilon \, \Big (\frac{\xi ^{5}}{4}+\xi ^3\phi \Big )\, . \end{aligned}$$

(3.48)

In particular, the equation $\Delta =0$ is equivalent to $(q_{1,\epsilon ,t},q_{2,\epsilon ,t})=0$ and thus to $(q_1,q_2) = 0$ in the limit $t \rightarrow 0$ and $\epsilon \rightarrow 0$. This, in turn, fixes the values for $\nu _0=\nu |_{t=0}$ and $\xi _0=\xi |_{t=0}$ through

$$\begin{aligned} \nu _0 = \frac{E\xi _0^2 }{ \xi _0^2+2\phi }, \quad r := \xi _0^6 + (E^2+8\phi -4) \xi _0^4+ 4 \phi (E^2+ 5\phi -4) \xi _0^2+16(\phi -1)\phi ^2=0, \end{aligned}$$

(3.49)

where we choose the positive solution $\xi _0$ for $r=0$. The fact that $r=0$ has a unique positive solution $\xi _0>0$ is an explicit elementary calculation. The positivity of $\xi _0$ will ensure the positive definiteness of $\mathrm {Im}{\widetilde{M}}_3$ for $z \in {\mathbb {C}}_+$.

We compute the Jacobian of the function $(q_1, q_2)$ from (3.48) as

$$\begin{aligned} J(\xi ,\nu ):= & {} \det \nabla _{\xi ,\nu }(q_1,q_2) = 3\xi ^4+ ((3 \nu ^2-6 \nu E+4 E^2)+10\phi -4)\xi ^2\nonumber \\&\quad +2\phi (4\phi -4+2E\nu -\nu ^2). \end{aligned}$$

(3.50)

Using that at $t=0$ we have $q_1=0$ and $\xi _0\ne 0$, we can eliminate the quadratic terms in $\nu$ and obtain

$$\begin{aligned} J(\xi _0,\nu _0) = 2\xi _0^2(3 \xi _0^2 +2E^2 + 10\phi -8). \end{aligned}$$

Again an elementary calculation using the defining equation $r=0$ for $\xi _0$ shows that $J(\xi _0, \nu _0)$ never vanishes. Thus, the ansatz (3.45) solves the Dyson equation in a small neighborhood of $z=t^2=0$. Furthermore, for $t = u (1+\mathrm {i}u )$ with sufficiently small $u >0$ it is easy to see that its imaginary part is positive definite. By using a regularization argument analogous to the one from Step 3 of Sect. 2.4 and combining it with the uniqueness of solutions to the Dyson equation with positive definite imaginary part, this implies that ${\widetilde{M}}_3(t) = M_3(z)$ for all $\sqrt{z}=t = u (1+\mathrm {i}u )$, $\phi \in (0,1)$ and small enough $\epsilon >0$. Since both functions are analytic this also implies equality for $z=t^2$ in the complex upper half plane intersected with a neighborhood of $z=0$.

Step 2: Extending $M_{z,w}$ to $w= E \in {\mathbb {R}}$ for $\phi \in (0,1)$. Fix $\phi \in (0,1)$, $\gamma >0$ and $E\in {\mathbb {R}}$. For $\epsilon _0>0$ and a coordinate pair $(i,j)\in \{1,\ldots , 6k + 2l\}^2$, consider the family of functions $\{M_{z,E+\mathrm {i}\,\epsilon _n \phi _{n} }\,(i,j)\}_{n\ge 1}$ analytic in z with $\{\phi _{n},\epsilon _n\}_{n\ge 1}\subset ((0,1)\cap {\mathbb {Q}}) \times (0,\epsilon _0)$ and $(\phi _{n},\epsilon _{n})\rightarrow (\phi ,0)$ as $n\rightarrow \infty$. It follows from part (ii) of Lemma 3.7 that for small enough $\epsilon _0$ and any small $\theta >0$ the family of functions $\{M_{z,E+\mathrm {i}\,\epsilon _n \phi _{n}}\,(i,j)\}_{n\ge 1}$ is uniformly bounded on the set $\{z \in {\mathbb {C}}_+ \, : \, |z| \ge \theta , |1-z| \ge \theta \}$, and thus locally bounded on ${\mathbb {C}}_+$.

It was established in Step 1 above that for any $\gamma >0$, $E\in {\mathbb {R}}$, $\phi \in (0,1)$ and $\epsilon >0$ small enough the solution to Eq. (3.22) with positive semidefinite imaginary part can be explicitly given as ${\widetilde{M}}_3(\sqrt{z})$ [see (3.45)] in the neighborhood of the origin, i.e., on the set $\{|z|\le \delta , \mathrm {Im}z >0\}$ for $\delta > 0$ sufficiently small. Moreover, Step 1 shows that for any $z\in \{|z|\le \delta , \mathrm {Im}z >0\}$, there exists a well-defined limit $\lim _{(\phi _{n},\epsilon _{n})\rightarrow (\phi , 0)} {\widetilde{M}}_3(\sqrt{z})$. Together with (3.19) and (3.20), this implies that on the set $\{|z|\le \delta , \mathrm {Im}z >0\}$ the limit $M_{z,E}:= \lim _{(\phi _{n},\epsilon _{n})\rightarrow (\phi , 0)} M_{z,E+\mathrm {i}\,\epsilon _n \phi _{n}}$ exists as well.

Combining the above information, we see that for any index pair (i, j), $\{M_{z,E+\mathrm {i}\,\epsilon _n \phi _{n}}\,(i,j)\}_{n\ge 1}$ is a family of analytic functions, locally bounded on ${\mathbb {C}}_+$ that converges on $\{|z|\le \delta , \mathrm {Im}z >0\}$ to $M_{z,E}(i,j)$. Applying the Vitali–Porter theorem (see, e.g., Section 2.4 in [34]), we conclude that for any $z\in {\mathbb {C}}_+$ the limit $M_{z,E}\,(i,j):=\lim _{(\phi _{n},\epsilon _{n})\rightarrow (\phi , 0) }M_{z,E+\mathrm {i}\,\epsilon _n \phi _{ n }}\,(i,j)$ exists, the convergence holds uniformly on the compact subsets of ${\mathbb {C}}_+$ and, as a result, the function $z\mapsto M_{z,E}\,(i,j)$ is analytic on ${\mathbb {C}}_+$. Taking $M_{z,E}\,(i,j)$ as the entries of the matrix-valued function $M_{z,E}$ defines the solution to the Dyson equation (3.21) for $\phi \in (0,1)$ and $w= E \in {\mathbb {R}}$.

To compute the asymptotic behavior of $M_{z,E}(1,1)$, we use (3.45) with $\epsilon =0$ [see (3.48)], (3.20) and $t = \sqrt{z}$ to find $M_1$ and its upper left corner element $M_{z,E}(1,1)$ in the neighborhood of $z=0$

$$\begin{aligned} M_{z,E}(1,1) = \mathrm {i}\frac{4+ \gamma ^2 \nu _0^2+ \gamma ^2 \xi _0^2}{4 \gamma \xi _0} \, z^{-1/2} + O(1). \end{aligned}$$

Step 3: Expansion around $z=1$. We apply exactly the same procedure as in Step 2 of Lemma 2.10, just we insert the parameters $\phi$ and $\gamma$ into the identities (2.83) and (2.84) and follow them through the analysis. Here, we record the final result of this elementary calculation. Our ansatz is

$$\begin{aligned} {\widetilde{M}}_3 = \left( \begin{array}{cc} \frac{\xi }{4 \gamma ^2t}&{}\quad \frac{E}{2}+\frac{\nu t}{4 \gamma } - \frac{\mathrm {i}\xi }{2 \gamma }(t +t^{-1}) \\ \frac{E}{2}+\frac{\nu t}{4 \gamma } + \frac{\mathrm {i}\xi }{2 \gamma }(t +t^{-1}) &{} \quad \xi (t +t^{-1}) \end{array} \right) . \end{aligned}$$

(3.51)

The expansion in $t=\sqrt{z-1}$ gives that for all $|E| \le E_0$ with

$$\begin{aligned} E_0 := \frac{\sqrt{2}}{ \gamma }\Big ( \gamma ^2 (1-2\phi ) -1+\sqrt{ 1 + \gamma ^4 (1 - 2 \phi )^2 + 2 \gamma ^2 (1+2\phi )}\Big )^{1/2} \end{aligned}$$

(3.52)

the upper-left component of $M_{z,E}$ is given by

$$\begin{aligned} M(1,1) = \frac{4\xi _0}{(\xi _0^2 + \gamma ^2 E^2+4 )t} + O(1), \end{aligned}$$

(3.53)

where $\xi _{0}$ is defined by

$$\begin{aligned} \xi _0 = - \gamma \sqrt{E_0^2-E^2} . \end{aligned}$$

(3.54)

This finishes the proof of the lemma. $\square$

3.5 Explicit Solution for $\phi \rightarrow 0$

Lemma 3.9

Let $\gamma >0$, $w= E \in {\mathbb {R}}$ and $\phi = 0$. Then, the Dyson equation (3.21)–(3.22) admits a solution ${\mathbb {C}}_{+}\ni z\mapsto M_{z,E} \in {\mathbb {C}}^{8\times 8}$ with $\mathrm {Im}M_{z,E}\ge 0$ and the upper-left entry is explicitly given by

$$\begin{aligned} M_{z,E}(1,1) = \frac{ \gamma ^2 (4-E^2)-(1+ \gamma ^2 )^2+\mathrm {i}\gamma (1+ \gamma ^2 )}{ \gamma ^2 (4-E^2)+((1+ \gamma ^2 )^2-(4-E^2) \gamma ^2 )z} \sqrt{\frac{4-E^2}{z(1-z)}}. \end{aligned}$$

(3.55)

Moreover, this solution can be continuously extended to the set $\phi \in [0,\phi _{0}]$, $|1-z|\ge \theta$, $\theta \le |z| \le \theta ^{-1}$ for $E\in {\mathbb {R}}$ and sufficiently small $\theta >0$ and $\phi _0 = \phi _0(\theta ) >0$.

Proof

At $\epsilon =0$ and setting $\phi = 0$, the Eq. (3.22) simplifies to a quadratic matrix equation for $M_3 \sigma _1$, where $M_3$ satisfies the symmetry constraints (3.24) and (3.25), i.e.,

$$\begin{aligned} -\frac{1}{M_{3}} = -E\,\sigma _{1} + \sigma _{1} M_{3} \sigma _{1} , \qquad M_3 = \left( \begin{array}{cc} \frac{\mathrm {i}\xi }{4 \gamma } &{}\quad \frac{\nu }{2}+\frac{ z \xi }{2} \\ \frac{\nu }{2}-\frac{ z \xi }{2} &{}\quad z \gamma \mathrm {i}\xi \end{array} \right) , \end{aligned}$$

(3.56)

for two functions $\xi$ and $\nu$ that are easily computed to be $\nu =E$ and

$$\begin{aligned} \xi = \sqrt{\frac{4-E^2}{z(1-z)}}. \end{aligned}$$

(3.57)

The choice of root is determined by $\mathrm {Re}\xi >0$. Inserting $M_{3}$ into (3.20) leads to

$$\begin{aligned} M_{z,E}(1,1) = \frac{ \gamma ^2 (4-E^2)-(1+ \gamma ^2 )^2+\mathrm {i}\gamma (1+ \gamma ^2 ) \xi }{ \gamma ^2 (4-E^2)+((1+ \gamma ^2 )^2-(4-E^2) \gamma ^2 )z}. \end{aligned}$$

(3.58)

We now construct the solution of (3.22) perturbatively for $\phi \in (0,\phi _0)$ with some sufficiently small $\phi _0=\phi _0(\theta )>0$ with $|1-z|\ge \theta$ and $\theta \le |z| \le \theta ^{-1}$. Recall that (3.22) is equivalent to $\Delta =0$ with $\Delta$ defined as in (3.43). In particular,

$$\begin{aligned} \Delta ={\widetilde{\Delta }} (Z_1+M_{3})\frac{1}{Z_1-Z_2}(Z_2+M_{3}) , \end{aligned}$$

(3.59)

implicitly defining ${\widetilde{\Delta }}$ that satisfies

$$\begin{aligned} {\widetilde{\Delta }}|_{\phi =0}=1+ M_{3}( \sigma _1M_{3}\sigma _1- \sigma _1E ). \end{aligned}$$

(3.60)

Similarly as we did in the proof of Lemma 3.8 instead of considering the equation $\Delta =0$ for a solution $M_{3} \in {\mathbb {C}}^{2\times 2}$, we can equivalently consider it as an equation for the two unknown functions $\xi$ and $\nu$ from the ansatz (3.56) for $M_{3}$. Since clearly the factors $Z_1+M_{3}$ and $Z_2+M_{3}$ in (3.59) have bounded inverses when $M_{3}$ is the explicit solution at $\phi =0$ from (3.56) with (3.57), we can equivalently consider ${\widetilde{\Delta }}=0$ as the equation for $\xi$ and $\nu$. Plugging the ansatz (3.56) into (3.60), we see that ${\widetilde{\Delta }}_{11}=0$ and ${\widetilde{\Delta }}_{12}=0$ already imply ${\widetilde{\Delta }}=0$ and that

$$\begin{aligned} {\widetilde{\Delta }}_{11}|_{\phi =0}= & {} \frac{1}{4} (4 + \nu ^2 + 2 z \nu \xi - z \xi ^2 + z^2 \xi ^2 - 2 E (\nu + z \xi )),\\ {\widetilde{\Delta }}_{12}|_{\phi =0}= & {} -\frac{\mathrm {i}(E - \nu ) \xi }{4 \gamma }. \end{aligned}$$

We compute the determinant of the Jacobian of the function $(\xi ,\nu )\rightarrow ( {\widetilde{\Delta }}_{11},{\widetilde{\Delta }}_{12})$ to be

$$\begin{aligned} \det \nabla _{\xi ,\nu }({\widetilde{\Delta }}_{11},{\widetilde{\Delta }}_{12})|_{\phi =0,\nu =E} = \frac{\mathrm {i}(z-1)z\xi ^2}{8\gamma }. \end{aligned}$$

Since $\xi$ from (3.57) does not vanish we infer that (3.22) is linearly stable as an equation for $\xi$, $\nu$ for small enough parameters $\phi$ in a vicinity of the explicit solution $M_{3}$ from (3.56) with (3.57) for $\phi =0$. $\square$

From (3.55), we read off the density of transmission eigenvalues $\rho (\lambda )= \rho _{E, \phi = 0}(\lambda ) :=\frac{1}{\pi }\lim _{\eta \downarrow 0}\mathrm {Im}M_{\lambda + \mathrm {i}\eta ,E}(1,1)$. The corresponding Fano factor for $\phi = 0$ is now computable as

$$\begin{aligned} F(E,\gamma ) = 1-\frac{\int \lambda ^2 \rho (\lambda ) \mathrm {d} \lambda }{\int \lambda \rho (\lambda ) \mathrm {d} \lambda } = \frac{1+ \gamma ^2 }{2(1+ \gamma ^2 + \gamma \sqrt{4-E^2})}. \end{aligned}$$

(3.61)

For $\gamma =1$ and $E=0$, we recover the density (1.5) and the Fano factor $F=\frac{1}{4}$ obtained in [7], see Sect. 4 for more details.

3.6 Proof of Parts (i), (iii) and (iv) of Theorem 1.4

In this section, we collect the results established in Sects. 3.1–3.5 and complete the proof of Theorem 1.4. Recall that Theorem 1.3 was proven in Lemma 2.5 for $\phi = 1$ and Lemma 3.4 for $\phi \in (0,1)$.

Proof of part (i) of Theorem 1.4. The extension of $\rho _{w,\phi }(d\lambda )$ to $w= E \in {\mathbb {R}}$ for $\phi \in {(}0,1/2{]}\cap {\mathbb {Q}}$ as well as the limit (1.12) and the extension of $\rho _{E,\phi }(d\lambda )$ to irrational $\phi \in (0,1)$ follows from the equivalence between the weak convergence of measures defined by (3.16) and the pointwise convergence of $M_{z,w}$ established in Lemma 3.4 and part (a) of Lemma 3.8 for the corresponding limits.

The weak limit $\lim _{\phi \downarrow 0}\rho _{E,\phi }(d\lambda ) = \rho _{E,\phi =0}(\lambda )d\lambda$ follows from the continuity of $\phi \mapsto M_{z,E}$ at $\phi = 0$ for all $z\in {\mathbb {C}}_{+}$, which was established in Lemma 3.8 in the regimes $|z|\le \theta$, $|1-z|\le \theta$ and in Lemma 3.9 in the complementary regimes $|z|\ge \theta$, $\theta \le |1-z|\le \theta ^{-1}$. Since the Stieltjes transform of $\rho _{E,\phi }$ is given by $M_{z,E}(1,1)$, the exact expression for $\rho _{E,0}$ can be derived as an inverse Stieltjes transform of $M_{z,E}$ from (3.55).

Similarly as for the case $\phi =1$ in Sect. 2.5, Lemma 2.11 and the integrability of $M_{z,E}$ that can be deduced from Lemmas 3.7 and 3.8 yield the absolute continuity of $\rho _{w,\phi }(d\lambda )=\rho _{w,\phi }(\lambda ) d\lambda$. $\square$

Proof of part (iii) of Theorem 1.4. Follows from the asymptotic behavior of $M_{z,E}$ near $z=0$ and $z=1$ (3.41)–(3.42) established in Lemma 3.8, block structure of $M_{z,E}$ (3.19) and the definition of the self-consistent density of states (3.18). $\square$

Proof of part (iv) of Theorem 1.4. Follows from the explicit formula for $M_{z,E}$ in the regime $\phi \rightarrow 0$ (3.58) established in Sect. 3.5, the block structure of $M_{z,E}$ (3.19) and the definition of the self-consistent density of states (3.18). $\square$

4 Comparison with the Results of Beenakker and Brouwer

Consider the scattering matrix (1.9)

$$\begin{aligned} S(E) := I - 2 \gamma \mathrm {i}\,W^* (E\cdot I - H + \mathrm {i}\,\gamma W W^*)^{-1} W \in {\mathbb {C}}^{N_0\times N_0} \end{aligned}$$

(4.1)

with $w= E \in {\mathbb {R}}$ in the regime $\phi = N/M \rightarrow 0$ as $M\rightarrow \infty$ with $N_0 = 2N$. This model was studied in the case of the Gaussian entries by Beenakker and Brouwer in [7, 10], and one of the remarkable results of their theory is that in the experimentally relevant setting of the ideal coupling the limiting transmission eigenvalue density is given by the arcsine law (1.5) (see [8, Eq. (3.12)]. The ideal coupling assumption is formulated in terms of the matrix S(E) having zero mean [8, Eq. (3.8)]. Below we show that in the regime $\phi \rightarrow 0$ the assumption ${\mathbb {E}}[S(E)] = 0$ is equivalent to $E=0$ and $\gamma = 1$. By plugging these values into (1.19) and (3.61), we recover the arcsine distribution and the corresponding Fano factor $F(0,1) = 1/4$.

Since the results of the current section do not affect the main outcomes of this paper and are meant to be of expository nature, we will keep the presentation rather informal, focusing only on the crucial steps and omitting the technical details.

For simplicity, we will assume in this section that H is Gaussian. Note that $W\in {\mathbb {C}}^{M\times N_{0}}$, so

$$\begin{aligned} W = U \Gamma V^* \end{aligned}$$

(4.2)

with unitary matrices $U\in {\mathbb {C}}^{M\times M}$ and $V\in {\mathbb {C}}^{N_{0}\times N_{0}}$ and

$$\begin{aligned} \Gamma = \left( \begin{array}{c} {\widetilde{\Gamma }} \\ 0 \end{array} \right) , \quad {\widetilde{\Gamma }} = \mathrm {diag}(\gamma _{1},\ldots , \gamma _{N_0}) , \end{aligned}$$

(4.3)

where $\gamma _{i}$ are the singular values of W and $N_0 \ll M$.

Note that $N_{0} = 2 N = 2 \phi M$, and thus, the eigenvalues of $W^*W$ have Marchenko–Pastur distribution with parameter $2\phi$. For $\phi \rightarrow 0$, regime that we are interested in, the eigenvalues of $W^*W$ will be concentrated around point 1, in the neighborhood of size $O(\sqrt{\phi })$, so for simplicity of presentation we will omit the asymptotically small term $O(\sqrt{\phi })$ and assume throughout these computations that all $\gamma _i=1$, i.e., ${\widetilde{\Gamma }} = I_{N_0}$.

After applying the singular value decomposition to W and factoring out matrices U and $U^*$ from the inverse, we get

$$\begin{aligned} S(E)&= I - 2 \gamma \mathrm {i}\,V \Gamma ^t U^* (E\cdot I - H + \mathrm {i}\,\gamma U \Gamma \Gamma ^{t} U^*)^{-1} U \Gamma V^* \end{aligned}$$

(4.4)

$$\begin{aligned}&= I - 2 \gamma \mathrm {i}\,V ({\widetilde{\Gamma }}^t, 0) \left( E\cdot I - {\widetilde{H}} + \mathrm {i}\,\gamma \left( \begin{array}{cc} {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t}&{}\quad 0 \\ 0 &{}\quad 0_{(M-N_{0})\times (M-N_0) } \end{array} \right) \right) ^{-1} \left( \begin{array}{c} {\widetilde{\Gamma }} \\ 0 \end{array}\right) V^* \end{aligned}$$

(4.5)

$$\begin{aligned}&\!=\! I \!+\! 2 \gamma \mathrm {i}\,V ({\widetilde{\Gamma }}^t, 0) \left( {\widetilde{H}} -E\cdot I - \mathrm {i}\,\gamma \left( \begin{array}{cc} {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t}&{}\quad 0 \\ 0 &{}\quad 0_{(M-N_{0})\times (M-N_0) } \end{array} \right) \right) ^{-1} \left( \begin{array}{c} {\widetilde{\Gamma }} \\ 0 \end{array}\right) V^* , \end{aligned}$$

(4.6)

where ${\widetilde{H}}$ remains a GUE matrix. Separate the upper-left $N_0 \times N_0$ block of ${\widetilde{H}}$

$$\begin{aligned} {\widetilde{H}} = \left( \begin{array}{cc} {\widetilde{H}}_{1}&{}\quad {\widetilde{H}}_{2} \\ {\widetilde{H}}_{2}^{*} &{}\quad {\widetilde{H}}_{3} \end{array} \right) \end{aligned}$$

(4.7)

and note that $\frac{1}{2\phi }{\widetilde{H}}_{1}$ and $\frac{1}{1-2\phi }{\widetilde{H}}_{3}$ are both independent GUE matrices. Now the inverse matrix in (4.6) can be rewritten as

$$\begin{aligned} \left( \begin{array}{cc} {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t}&{}\quad {\widetilde{H}}_{2} \\ {\widetilde{H}}_{2}^* &{}\quad {\widetilde{H}}_{3} -E \end{array} \right) ^{-1} . \end{aligned}$$

(4.8)

Using the Schur complement formula we have that the upper-left $N_{0}\times N_{0}$ block of (4.8), the only part that does not vanish after sandwiching (4.8) by $({\widetilde{\Gamma }}^t, 0)$ and its transpose, is given by

$$\begin{aligned} \left( {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t} - {\widetilde{H}}_{2} \Big ({\widetilde{H}}_{3} -E\Big )^{-1} {\widetilde{H}}_{2}^* \right) ^{-1} . \end{aligned}$$

(4.9)

The semicircular law for the Hermitian (GUE) matrix $\frac{1}{1-2\phi }{\widetilde{H}}_{3}$ implies that

$$\begin{aligned} ({\widetilde{H}}_{3} -E)^{-1} \!=\! \frac{1}{1-2\phi }\Big (\frac{1}{1-2\phi }{\widetilde{H}}_{3} \!-\!\frac{1}{1-2\phi }E\Big )^{-1} \!\approx \! \frac{1}{1\!-\!2\phi }m_{sc}\Big (\frac{1}{1\!-\!2\phi }E\Big ) I_{M-N_{0}} , \end{aligned}$$

(4.10)

as $M\rightarrow \infty$, where $m_{sc}(z)$ denotes the Stieltjes transform of the semicircular distribution and “$\approx$” denotes that the corresponding equality holds asymptotically with a vanishing additive term and with high probability. Note that random matrices ${\widetilde{H}}_{1}$, ${\widetilde{H}}_{2}$ and ${\widetilde{H}}_{3}$ are independent. Therefore, by the concentration for quadratic forms (see, e.g., [15, Theorem C.1]) can be approximated by

$$\begin{aligned} {\widetilde{H}}_{2} \Big ({\widetilde{H}}_{3} -E \Big )^{-1} {\widetilde{H}}_{2}^* \approx m_{sc}\Big (\frac{1}{1-2\phi }E\Big ) I_{N_{0}} . \end{aligned}$$

(4.11)

From (4.9) and (4.11), it remains to check the limiting behavior of

$$\begin{aligned} \left( {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t} - m_{sc}\Big (\frac{1}{1-2\phi }E\Big ) \right) ^{-1} . \end{aligned}$$

(4.12)

Recall that since $\phi$ is small, we assumed that ${\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t} = I_{N_0}$. In this case, using again the semicircular law for the Hermitian matrix $\frac{1}{2\phi }{\widetilde{H}}_{1}$, we have

$$\begin{aligned}&\left( {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t} - m_{sc}\Big (\frac{1}{1-2\phi }E\Big ) \right) ^{-1} \nonumber \\&\quad \approx \left( {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma - m_{sc}\Big (\frac{1}{1-2\phi }E\Big ) \right) ^{-1} \end{aligned}$$

(4.13)

$$\begin{aligned}&\quad = \frac{1}{2\phi } \left( \frac{1}{2\phi }{\widetilde{H}}_{1} -\frac{1}{2\phi }\Big (E + \mathrm {i}\,\gamma + m_{sc}\Big (\frac{1}{1-2\phi }E\Big )\Big ) \right) ^{-1} \end{aligned}$$

(4.14)

$$\begin{aligned}&\quad \approx \frac{1}{2\phi } m_{sc}\left( \frac{1}{2\phi }\Big (E + \mathrm {i}\,\gamma + m_{sc}\Big (\frac{1}{1-2\phi }E\Big )\Big ) \right) . \end{aligned}$$

(4.15)

From the asymptotics $z m_{sc}(z) \rightarrow -1, |z|\rightarrow \infty$, we get that

$$\begin{aligned} \left( {\widetilde{H}}_{1} -E - \mathrm {i}\,\gamma {\widetilde{\Gamma }}{\widetilde{\Gamma }}^{t} - m_{sc}\Big (\frac{1}{1-2\phi }E\Big ) \right) ^{-1} \approx -\frac{1}{E + \mathrm {i}\,\gamma + m_{sc}(E)} ,\quad \phi \rightarrow 0 . \end{aligned}$$

(4.16)

We conclude that as $N_{0}, M \rightarrow \infty$, $\phi \rightarrow 0$ (see (4.6) and (4.16))

$$\begin{aligned} {\mathbb {E}}[S(E)] \approx 1+ 2 \gamma \mathrm {i}\,\left( -\frac{1}{E + \mathrm {i}\,\gamma + m_{sc}(E)} \right) . \end{aligned}$$

(4.17)

Now, from $m_{sc}(0) = \mathrm {i}\,$ we have that

$$\begin{aligned} {\mathbb {E}}[S(0)] \approx 1- \frac{2 \gamma \mathrm {i}\,}{\mathrm {i}\,\gamma + \mathrm {i}\,} . \end{aligned}$$

(4.18)

Finally, taking $\gamma = 1$ gives ${\mathbb {E}}[S(0)] \approx 0$.

Note that from (4.17) we have that in the limit $\phi \rightarrow 0$

$$\begin{aligned} {\mathbb {E}}[S(0)] = 0 \quad \Leftrightarrow \quad E + m_{sc}(E) = \mathrm {i}\,\gamma , \end{aligned}$$

(4.19)

and for $E\in {\mathbb {R}}$, the expression $E + m_{sc}(E)$ is purely imaginary if and only if $E=0$

$$\begin{aligned} E + m_{sc}(E) = E+\frac{-E + \sqrt{E^2-4}}{2} = \frac{E + \sqrt{E^2-4}}{2} . \end{aligned}$$

(4.20)

Therefore, for $\phi \rightarrow 0$, ${\mathbb {E}}[S(E)]=0$ if and only if $E=0$ and $\gamma = 1$.

Appendices

Spectral Properties for a General Class of Rational Expressions in Random Matrices

1.1 Noncommutative (NC) Rational Expressions and Their Linearizations

NC rational expressions are formally defined as expressions obtained by applying the four algebraic operations (including taking inverses) to a tuple of NC variables. A systematic overview of the abstract theory of NC rational expression and functions (equivalence classes of rational expressions) can be found in [9]. Note that unlike polynomials, rational expressions do not have a canonical representation, which may lead to a situation when, after evaluating on some algebra, two rational expressions represent the same function. In this paper, we leave aside the question of identification of rational functions, and will work instead directly with rational expressions and their evaluations, specifying each time on which domain the evaluation is taking place. Below we introduce a standard set-up in which we will work and define recursively the classes of rational expressions, denoted by letter ${\mathcal {Q}}$, together with corresponding domains of evaluation denoted by ${\mathcal {D}}$.

Let ${\mathcal {H}}$ be a Hilbert space, ${\mathcal {A}}\subseteq {\mathcal {B}}({\mathcal {H}})$ be a $C^*$-algebra (of bounded operators on ${\mathcal {H}}$) with norm $\Vert \cdot \Vert _{{\mathcal {A}}}$ and let $x_{1},\ldots ,x_{\alpha _*},y_{1},\ldots ,y_{\beta _*}$ be the NC variables taking values in ${\mathcal {A}}$ with $x_{\alpha }=x_{\alpha }^*$ for $1\le \alpha \le \alpha _*$. Denote by ${\mathcal {A}}_{sa}\subset {\mathcal {A}}$ the set of self-adjoint elements of ${\mathcal {A}}$ and let ${\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^* \rangle$ be the set of polynomials in $\varvec{x}:=(x_{1},\ldots ,x_{\alpha _*})$, $\varvec{y}:=(y_{1},\ldots ,y_{\beta _*})$ and $\varvec{y}^*:=(y_1^*,\ldots ,y_{\beta _*}^*)$. We define the NC rational expressions recursively on their height using the following procedure:

• (i) Let $\varvec{q}_{0}:= \mathbb {1}_{{\mathcal {A}}}$. The set of rational expressions of height 0 is defined to be the set of polynomials in $\varvec{x},\varvec{y},\varvec{y}^*$ with the domain of definition ${\mathcal {A}}_{sa}^{\alpha _*} \times {\mathcal {A}}^{\beta _*}$

  $$\begin{aligned} ({\mathcal {Q}}_{\varvec{q}_0},{\mathcal {D}}_{\varvec{q}_0}):=({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle , {\mathcal {A}}_{sa}^{\alpha _*}\times {\mathcal {A}}^{\beta _*}) . \end{aligned}$$

  (A.1)

• (ii) Let $\varvec{q}_{1}:=(q_{1,1},\ldots ,q_{1,\ell _{1}}) \in ({\mathcal {Q}}_{\varvec{q}_0})^{\ell _{1}}$, $\varvec{q}_{2}:=(q_{2,1},\ldots ,q_{2,\ell _{2}}) \in ({\mathcal {Q}}_{\varvec{q}_0,\varvec{q}_{1}})^{\ell _{2}}$, $\ldots$, $\varvec{q}_{n}:=(q_{n,1},\ldots , q_{n,\ell {n}})\in ({\mathcal {Q}}_{\varvec{q}_0,\ldots ,\varvec{q}_{n-1}})^{\ell _{n}}$, assuming that $({\mathcal {Q}}_{\varvec{q}_{0}},{\mathcal {D}}_{\varvec{q}_{0}})$, $\ldots$, $({\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1}},{\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1}})$ are defined. Then, we define

  $$\begin{aligned}&{\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1},\varvec{q}_{n}} := {\mathbb {C}}\Big \langle \varvec{x},\varvec{y},\varvec{y}^*,\frac{1}{\varvec{q}_{1}},\frac{1}{\varvec{q}_1^*},\ldots ,\frac{1}{\varvec{q}_{n}},\frac{1}{\varvec{q}_n^*} \Big \rangle , \end{aligned}$$

  (A.2)
  $$\begin{aligned}&{\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1},\varvec{q}_{n}} \!:=\! \Big \{(\varvec{x},\varvec{y})\!\in \! {\mathcal {D}}_{\varvec{q}_{0},\varvec{q}_{1},\ldots ,\varvec{q}_{n-1}} \!:\! \Big \Vert \frac{1}{q_{n,j}(\varvec{x},\varvec{y},\varvec{y}^*)}\Big \Vert _{{\mathcal {A}}} \!<\! \infty \text{ for } 1\!\le \! j \!\le \! \ell _{n}\Big \} , \end{aligned}$$

  (A.3)

  where $\frac{1}{\varvec{q}_{i}}:=(q_{i,1}^{-1},\ldots ,q_{i,\ell _{i}}^{-1})$ and $\frac{1}{\varvec{q}_{i}^*}:=((q_{i,1}^*)^{-1},\ldots ,(q_{i,\ell _{i}}^*)^{-1})$.

We say that the rational expression $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1},\varvec{q}_{n}}$ defined on ${\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1},\varvec{q}_{n}}$ has height n.

For any $C\!>\!0$ and $\varvec{q}_{1}\!\in \! ({\mathcal {Q}}_{\varvec{q}_0})^{\ell _{1}}$, $\varvec{q}_{2}\!\in \! ({\mathcal {Q}}_{\varvec{q}_0,\varvec{q}_{1}})^{\ell _{2}}$, $\ldots$, $\varvec{q}_{n}\in ({\mathcal {Q}}_{\varvec{q}_0,\ldots ,\varvec{q}_{n-1}})^{\ell _{n}}$, define the effective domain

$$\begin{aligned} {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}:= & {} \Big \{(\varvec{x},\varvec{y})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1};C} : \Big \Vert \frac{1}{q_{n,j}(\varvec{x},\varvec{y},\varvec{y}^*)}\Big \Vert _{{\mathcal {A}}} \le C \text{ for } 1\le j \le \ell _{n}\Big \} \nonumber \\\subset & {} {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}} . \end{aligned}$$

(A.4)

This domain will allow an effective control of the norm of rational expressions of height n.

Remark A.1

Similar constructions of rational functions/expressions involving their height have been exploited actively in the literature (see, e.g., [9, Chapter 4], [32, 39]). Note that here we allow the “denominators” without constant terms, so they cannot be automatically expanded into geometric series for small $\varvec{x}, \varvec{y}$. Hence, we need to introduce and follow explicit domains.

Remark A.2

In the sequel, the statement “q is a rational expression of height n” will implicitly mean that there (uniquely) exist $\varvec{q}_{1}\!\in \! ({\mathcal {Q}}_{\varvec{q}_0})^{\ell _{1}}$, $\varvec{q}_{2}\!\in \! ({\mathcal {Q}}_{\varvec{q}_0,\varvec{q}_{1}})^{\ell _{2}}$, $\ldots$, $\varvec{q}_{n}\!\in \! ({\mathcal {Q}}_{\varvec{q}_0,\ldots ,\varvec{q}_{n-1}})^{\ell _{n}}$ and $C>0$ such that $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$ and q is evaluated on the effective domain ${\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$. Note that many of the basic results, in particular about constructing the linearizations, can be formulated in a completely abstract form or without restriction to effective domains.

Remark A.3

When evaluating a rational expression of height n on different $C^*$-algebras ${\mathcal {A}}_{1},\ldots ,{\mathcal {A}}_{k}$, we will use the notation ${\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}({\mathcal {A}}_{i})$ and ${\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {A}}_{i})$ correspondingly.

Definition A.4

(Self-adjoint rational expression). We say that a rational expression $q=q(\varvec{x},\varvec{y}, \varvec{y}^*)$ is self-adjoint if $q(\varvec{x},\varvec{y}, \varvec{y}^*) = [q(\varvec{x},\varvec{y}, \varvec{y}^*)]^{*}$ for all $(\varvec{x},\varvec{y})\in {\mathcal {D}}$.

1.2 Linearizations and Linearization Algorithm

Definition A.5

Let q be a self-adjoint rational expression of height n in NC variables $\varvec{x}$, $\varvec{y}$ and $\varvec{y}^*$. We say that the self-adjoint matrix

$$\begin{aligned} \varvec{L}= \left( \begin{array}{c|ccc} \lambda &{} &{}\quad \varvec{\ell }^*&{} \\ \hline &{}&{}&{} \\ \varvec{\ell } &{} &{}\quad {\widehat{\varvec{L}}} &{} \\ &{}&{}&{} \end{array} \right) \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m} \end{aligned}$$

(A.5)

with $\lambda \in {\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle$, $\varvec{\ell } \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{(m-1)\times 1}$, ${\widehat{\varvec{L}}}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{(m-1)\times (m-1)}$, whose entries are polynomials of degree at most 1, is a (self-adjoint) linearization of q if

• (i) the submatrix ${\widehat{\varvec{L}}}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^* \rangle )^{(m-1)\times (m-1)}$ is invertible, and

• (ii) $\lambda - \varvec{\ell }^* {{\widehat{\varvec{L}}}}^{-1} \varvec{\ell } = q$

for all $(\varvec{x},\varvec{y})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$.

The linearization of q can be written as

$$\begin{aligned} \varvec{L}= K_0 \otimes \mathbb {1}_{{\mathcal {A}}}- \sum _{\alpha =1}^{\alpha _*} K_{\alpha } \otimes x_{\alpha } - \sum _{\beta =1}^{\beta _*}\big ( L_{\beta } \otimes y_{\beta } + L_{\beta }^* \otimes y_{\beta }^*\big ) , \end{aligned}$$

(A.6)

where $K_0,K_{\alpha },L_{\beta } \in {\mathbb {C}}^{m\times m}$.

The idea of studying noncommutative rational functions/expressions via linearizations goes back to Kleene [26]. Since the publication of this work, various approaches and algorithms have been developed for constructing linearizations of general classes of rational functions/expressions (see, e.g., [9] or [24] for a pedagogical presentation of the subject). For reader’s convenience, we provide below a simple linearization algorithm based on the method described in [13, Section A.1]. We use the following observation: for matrices $\varvec{A}_{i}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_i\times m_i}$ and $\varvec{B}_{j} \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_{j}\times m_{j+1}}$, $m_{j}\in {\mathbb {N}}$, the lower right $m_1 \times m_{k}$ submatrix of the inverse of the block matrix

(A.7)

is equal to

$$\begin{aligned} -\varvec{A}_{1}^{-1}\varvec{B}_{1}\varvec{A}_{2}^{-1}\varvec{B}_{2}\cdots \varvec{A}_{k-1}^{-1} \varvec{B}_{k-1} \varvec{A}_{k}^{-1} \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_1\times m_k} . \end{aligned}$$

(A.8)

Now, if $q_{1},\ldots ,q_{\ell }$ are rational expressions having known linearizations $\varvec{A}_{1}, \ldots , \varvec{A}_{\ell }$, then one can easily check that the linearization of the product $w_0 \frac{1}{q_{1}}w_{1}\frac{1}{q_{2}}$ $\cdots$ $\frac{1}{q_{k-1}}w_{k-1}\frac{1}{q_{k}}w_{k}$ with $w_{j}\in \{\mathbb {1}_{{\mathcal {A}}}, x_{\alpha }, y_{\beta }, y_{\beta }^{*} \, | \,1\le \alpha \le \alpha _*, 1\le \beta \le \beta _{*}\}$ can be given by

(A.9)

with $\varvec{b}_0=(w_0,0,\ldots ,0)\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{1\times m_{1}}$, $\varvec{b}_k=(w_{k},0,\ldots ,0)^{t} \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_{k}\times 1}$ and $\varvec{B}_{j}$ being matrices of the form

$$\begin{aligned} \varvec{B}_{j} = \left( \begin{array}{cccc} w_{j}&{}\quad 0&{}\quad \cdots &{}\quad 0 \\ 0&{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \end{array} \right) \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_{j}\times m_{j+1}} . \end{aligned}$$

(A.10)

Using (A.9), we can construct a linearization of rational expressions using the induction on the height. In the case of a rational expression of height 0 (polynomial function), linearization can be constructed using, e.g., the algorithm from [13, Section A.1]. If $\varvec{q}_{1}\in ({\mathcal {Q}}_{\varvec{q}_{0}})^{\ell _{1}}$ and $q\in {\mathcal {Q}}_{\varvec{q}_0,\varvec{q}_1}$ is a rational expression of height 1, the linearization can be obtain using the following algorithm:

• (B0) write q as a sum of monomials in $\varvec{x}$, $\varvec{y}$, $\varvec{y}^*$, $\frac{1}{\varvec{q}_{1}}$, $\frac{1}{\varvec{q}_{1}^*}$ of the form $w_{1}\frac{1}{q_{1}}w_{2}\frac{1}{q_{2}}\cdots w_{k-1}\frac{1}{q_{k-1}} w_{k}$ with $w_{j} \in \{\mathbb {1}_{{\mathcal {A}}},x_{\alpha },y_{\beta },y_{\beta }^* \, | \, 1\le \alpha \le \alpha _*,1\le \beta \le \beta _*\}$ and $q_{i}\in \{\mathbb {1}_{{\mathcal {A}}}, q_{1,\gamma }, q_{1,\gamma }^* \, | \, 1\le \gamma \le \ell _{1}\}$ using $\frac{1}{\mathbb {1}_{{\mathcal {A}}}}=\mathbb {1}_{{\mathcal {A}}}$ if necessary;

• (B1) for each polynomial $q_{1,\gamma _{1}}, q_{1,\gamma _{1}}^*, 1\le \gamma _{1}\le \ell _{1}$ construct a linearization (not necessarily self-adjoint) using the algorithm from [13, Section A.1];

• (B2) linearization of a monomial in $\varvec{x}$, $\varvec{y}$, $\varvec{y}^*$, $\frac{1}{\varvec{q}_{1}}$, $\frac{1}{\varvec{q}_{1}^*}$ of the form $w_{1}\frac{1}{q_{1}}w_{2}\frac{1}{q_{2}}\cdots w_{k-1}\frac{1}{q_{k-1}} w_{k}$ is given by (A.9) with $A_{i}$ being either linearizations of polynomials $q_{i}\in \{q_{1,\gamma _{1}}, q_{1,\gamma _{1}}^*\, | \, 1\le \gamma _{1} \le \ell _{1} \}$, or $\mathbb {1}_{{\mathcal {A}}}$ (linearization of $\frac{1}{\mathbb {1}_{{\mathcal {A}}}} = \mathbb {1}_{{\mathcal {A}}}$), and $w_{j} \in \{\mathbb {1}_{{\mathcal {A}}},x_{\alpha },y_{\beta },y_{\beta }^* \, | \, 1\le \alpha \le \alpha _*,1\le \beta \le \beta _*\}$;

• (B3) the (possibly not self-adjoint) linearization of a linear combination of monomials (and thus q) is constructed by putting the linearizations of monomials obtained at (B2) into a block-diagonal form using a procedure similar to (R1)–(R2) from [13, Section A.1];

• (B4) if after step (B3) the resulting linearization is not self-adjoint, the symmetrized linearization of $q = (q+q^*)/2$ can be obtained by putting the linearization obtained at step (B3) and its conjugate transpose into a block-skew-diagonal form similarly as in (R3) from [13, Section A.1].

Suppose that we know how to construct linearizations for rational expressions of height $\le n-1$ and suppose that $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$ is a rational expression of height n. Then, the linearization of q can be constructed using the following algorithm, that is an adaptation of (B0)–(B4):

• (S0) write q as a sum of monomials in $\varvec{x}$, $\varvec{y}$, $\varvec{y}^*$, $\frac{1}{\varvec{q}_{1}}$, $\frac{1}{\varvec{q}_{1}^*},\ldots ,\frac{1}{\varvec{q}_{n}},\frac{1}{\varvec{q}_{n}^*}$ of the form $w_{1}\frac{1}{q_{1}}w_{2}\frac{1}{q_{2}}\cdots w_{k-1}\frac{1}{q_{k-1}} w_{k}$ with $w_{j} \in \{\mathbb {1}_{{\mathcal {A}}},x_{\alpha },y_{\beta },y_{\beta }^* \, | \, 1\le \alpha \le \alpha _*,1\le \beta \le \beta _*\}$ and $q_{i}\in \{\mathbb {1}_{{\mathcal {A}}}, q_{t,\gamma _{t}}, q_{t,\gamma _{t}}^* \, | \, 1\le t \le n, 1\le \gamma _{t} \le \ell _{n}\}$ using $\frac{1}{\mathbb {1}_{{\mathcal {A}}}}=\mathbb {1}_{{\mathcal {A}}}$ if necessary;

• (S1) construct linearizations (not necessarily self-adjoint) of each rational expression $q_{t,\gamma _{t}}$ for $1\le t \le n$, $1\le \gamma _{t}\le \ell _{t}$ of height $\le n-1$ (rational expressions of height $\le n-1$);

• (S2) linearization of a monomial in $\varvec{x}$, $\varvec{y}$, $\varvec{y}^*$, $\frac{1}{\varvec{q}_{1}}, \frac{1}{\varvec{q}_{1}^*}, \ldots , \frac{1}{\varvec{q}_{n}}, \frac{1}{\varvec{q}_{n}^*}$ of the form $w_{1}\frac{1}{q_{1}}w_{2}\frac{1}{q_{2}}\cdots w_{k-1}\frac{1}{q_{k-1}} w_{k}$ is given by (A.9) with $A_{i}$ being either linearizations of polynomials $q_{i}\in \{q_{t,\gamma _{t}}, q_{t,\gamma _{t}}^* \, | \, 1\le t \le n, 1\le \gamma _{t} \le \ell _{n}\}$, or $\mathbb {1}_{{\mathcal {A}}}$, and $w_{j} \in \{\mathbb {1}_{{\mathcal {A}}},x_{\alpha },y_{\beta },y_{\beta }^* \, | \, 1\le \alpha \le \alpha _*,1\le \beta \le \beta _*\}$.

The steps (S3)–(S4) are identical to (B3)–(B4).

Remark A.6

It is always possible to obtain the specific form of monomials required in steps (B0) and (S0) by adding factors $\mathbb {1}_{{\mathcal {A}}}$ or $\frac{1}{\mathbb {1}_{{\mathcal {A}}}}$, as, for example, in

$$\begin{aligned} \frac{1}{(\mathbb {1}_{{\mathcal {A}}}-x)^{2}}x^2 = \mathbb {1}_{{\mathcal {A}}}\frac{1}{\mathbb {1}_{{\mathcal {A}}}- x} \mathbb {1}_{{\mathcal {A}}}\frac{1}{\mathbb {1}_{{\mathcal {A}}}- x} x \frac{1}{\mathbb {1}_{{\mathcal {A}}}} x . \end{aligned}$$

(A.11)

Note that there is some ambiguity at steps (B0) and (S0), representing the fact that we have freedom to choose the order in which the monomials appear in the sum, as well as freedom to put the product of the constant terms $\mathbb {1}_{{\mathcal {A}}}$ and $\frac{1}{\mathbb {1}_{{\mathcal {A}}}}$ between the terms $w_j$ and $\frac{1}{q_i}$. All the results of Sect. A hold for any choice of the representation of q as a sum of monomials; therefore, we do not fix any particular order or other rules to guarantee the uniqueness of the representation in (B0) and (S0).

The condition (ii) in the definition of the linearization is satisfied by construction. The condition (i) follows from the following lemma, which gives a bound on $\Vert ( {\widehat{\varvec{L}}}(\varvec{x},\varvec{y},\varvec{y}^*))^{-1}\Vert _{{\mathbb {C}}^{(m-1)\times (m-1)}\otimes \,{\mathcal {A}}}$, where for any $n\in {\mathbb {N}}$ and $\varvec{R}= (\varvec{R}_{ij})_{i,j=1}^{n} \in {\mathbb {C}}^{n\times n}\otimes {\mathcal {A}}$ we denote

$$\begin{aligned} \Vert \varvec{R}\Vert _{{\mathbb {C}}^{n\times n}\otimes \,{\mathcal {A}}} := \max _{1\le i,j\le n} \Vert \varvec{R}_{ij}\Vert _{{\mathcal {A}}} . \end{aligned}$$

(A.12)

Lemma A.7

(Invertibility of ${\widehat{\varvec{L}}}$) Let q be a self-adjoint rational expression of height n and let $\varvec{L}= \varvec{L}_{q} \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m}$ be the linearization of q constructed via the above algorithm. Let ${\widehat{\varvec{L}}}$ be the submatrix of $\varvec{L}$ defined using the decomposition (A.5). Then, there exist ${\widehat{C}}_{\varvec{L}_{q}}>0$ and ${\widehat{n}}_{\varvec{L}_{q}}\in {\mathbb {N}}$ such that for any $(\varvec{x},\varvec{y})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$

$$\begin{aligned} \Vert ({\widehat{\varvec{L}}}(\varvec{x},\varvec{y},\varvec{y}^*))^{-1} \Vert _{{\mathbb {C}}^{(m-1)\times (m-1)}\otimes \,{\mathcal {A}}} \le {\widehat{C}}_{\varvec{L}_{q}}(1 + C + \max _{\alpha }\Vert x_{\alpha }\Vert _{{\mathcal {A}}} + \max _{\beta }\Vert y_{\beta }\Vert _{{\mathcal {A}}})^{{\widehat{n}}_{\varvec{L}_{q}}} . \end{aligned}$$

(A.13)

Proof

We prove (A.13) by induction on n. For $n=0$ (the special case of polynomial functions) (A.13) follows from, for example, [13, (3.16)]. Suppose (A.13) holds for all rational expressions of height $k\le n-1$. Consider q of height n with linearization obtained via (S0)–(S4). Steps (S3) and (S4) of the linearization algorithm endow ${\widehat{\varvec{L}}}$ with block-diagonal (S3) or block-skew-diagonal (S4) structure with blocks being the linearizations of monomials obtained at step (S2). Therefore, in order to obtain the bound (A.13) it is enough to consider only the inverses of the blocks of the form

(A.14)

with $\varvec{A}_{i}$ being the linearizations of the rational expressions $q_{i}\in \{\frac{1}{q_{t,\gamma _{t}}},\frac{1}{q_{t,\gamma _{t}}^*}\, | \, 0 \le t \le n, 1\le \gamma _{t} \le \ell _{t}\}$, and $\varvec{B}_{j}$ being of the form (A.10). One can easily check that the inverse of (A.14) consists of the blocks of the type $\varvec{A}_{i}^{-1} \varvec{B}_{i} \varvec{A}_{i + 1}^{-1} \varvec{B}_{i + 1}\ldots$. The induction step, together with the Schur complement formula for $\varvec{A}_{i}^{-1}$ and the condition that for $(\varvec{x},\varvec{y}) \in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$

$$\begin{aligned} \Big \Vert \frac{1}{q_{i}(\varvec{x},\varvec{y},\varvec{y}^*)}\Big \Vert _{{\mathcal {A}}} \le C \end{aligned}$$

(A.15)

implies that for each block of type (A.14) there exist $C'>0$ and $n'\in {\mathbb {N}}$ such that

(A.16)

where $\varvec{A}_{i}$’s and $\varvec{B}_{j}$’s in the left-hand side are evaluated at $(\varvec{x},\varvec{y})\in {\mathcal {D}}$. Taking ${\widehat{C}}_{\varvec{L}_{q}}$ and ${\widehat{n}}_{\varvec{L}_{q}}$, respectively, the maximum over all $C'$’s and the maximum over all $n'$’s in the bounds (A.16) running through all monomials in the representation of q, leads to (A.13) $\square$

Remark A.8

Suppose that $P\in {\mathbb {C}}^{m\times m}$ is of the form

$$\begin{aligned} P = \left( \begin{array}{c|ccc} 1&{}\quad 0 &{}\quad \cdots &{} 0 \\ \hline 0 &{} &{} &{} \\ \vdots &{}&{}\quad Q &{} \\ 0 &{} &{} &{} \end{array} \right) \end{aligned}$$

(A.17)

with $Q\in {\mathbb {C}}^{(m-1)\times (m-1)}$ invertible. It is easy to see that if $\varvec{L}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m}$ is a linearization of a rational expression of height q, then so is $(P\otimes \mathbb {1}_{{\mathcal {A}}}) \varvec{L}(P^{-1} \otimes \mathbb {1}_{{\mathcal {A}}})$. We will use this freedom to bring linearizations to more convenient form.

1.3 A Priori Bound on Generalized Resolvents

Definition A.9

(Generalized resolvent). Let $\varvec{L}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m}$. We call the matrix-valued function $z \mapsto (\varvec{L}- z J_{m}\otimes \mathbb {1}_{{\mathcal {A}}})^{-1}$ defined for $z\in {\mathbb {C}}_+$ the generalized resolvent of $\varvec{L}$.

Lemma A.10

Let q be a self-adjoint rational expression of height n and let $\varvec{L}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m}$ be the linearization of q constructed using the algorithm from Sect. A.2. Then, there exist $C_{\varvec{L}_{q}}>0$ and $n_{\varvec{L}_{q}}\in {\mathbb {N}}$ such that for all $C>1$, $(\varvec{x},\varvec{y})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$ and $z\in {\mathbb {C}}_+$

$$\begin{aligned}&\Vert (\varvec{L}(\varvec{x},\varvec{y},\varvec{y}^*)-zJ_{m}\otimes \mathbb {1}_{{\mathcal {A}}})^{-1}\Vert _{{\mathbb {C}}^{m\times m}\otimes \,{\mathcal {A}}} \nonumber \\&\quad \le C_{\varvec{L}_{q}}(1+C + \max _{\alpha }\Vert x_{\alpha }\Vert _{{\mathcal {A}}} + \max _{\beta } \Vert y_{\beta }\Vert _{{\mathcal {A}}})^{n_{\varvec{L}_{q}}}\Big (1+\frac{1}{\mathrm {Im}z}\Big ) . \end{aligned}$$

(A.18)

Proof

Rewrite $\varvec{L}- zJ_{m}\otimes \mathbb {1}_{{\mathcal {A}}}$ using the block decomposition from (A.5)

$$\begin{aligned} \varvec{L}- zJ_{m}\otimes \mathbb {1}_{{\mathcal {A}}}= \left( \begin{array}{c|ccc} \lambda - z \mathbb {1}_{{\mathcal {A}}}&{}&{}\quad \varvec{\ell }^*&{} \\ \hline &{}&{}&{} \\ \varvec{\ell } &{} &{}\quad {\widehat{\varvec{L}}} &{} \\ &{}&{}&{} \end{array} \right) \end{aligned}$$

(A.19)

with ${\widehat{\varvec{L}}}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{(m-1)\times (m-1)}$. By (ii) in the definition of the linearization and the Schur complement formula, we have

$$\begin{aligned} (\varvec{L}- zJ_{m}\otimes \mathbb {1}_{{\mathcal {A}}})^{-1} = \left( \begin{array}{c|ccc} (q-z\mathbb {1}_{{\mathcal {A}}})^{-1}&{} &{}\quad -(q-z\mathbb {1}_{{\mathcal {A}}})^{-1} \varvec{\ell }^* {\widehat{\varvec{L}}}^{-1} \\ \hline &{}&{}&{} \\ -{\widehat{\varvec{L}}}^{-1} \varvec{\ell } (q-z\mathbb {1}_{{\mathcal {A}}})^{-1} &{} &{}\quad {\widehat{\varvec{L}}}^{-1}+{\widehat{\varvec{L}}}^{-1} \varvec{\ell } (q-z\mathbb {1}_{{\mathcal {A}}})^{-1} \varvec{\ell }^* {\widehat{\varvec{L}}}^{-1} &{} \\ &{}&{}&{} \end{array} \right) . \end{aligned}$$

(A.20)

Now (A.18) follows from Lemma A.7 and the trivial bound for resolvents of self-adjoint elements

$$\begin{aligned} \Big \Vert \frac{1}{q - z\mathbb {1}_{{\mathcal {A}}}}\Big \Vert _{{\mathcal {A}}} \le \frac{1}{\mathrm {Im}z} \quad \text{ uniformly } \text{ for } z\in {\mathbb {C}}_{+} . \end{aligned}$$

(A.21)

$\square$

1.4 Dyson Equation for Linearizations of NC Rational Expressions

Let q be a self-adjoint rational expression of height n and let $\varvec{L}\in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m\times m}$ be its linearization constructed using the algorithm from Sect. A.2. Write $\varvec{L}$ as

$$\begin{aligned} \varvec{L}= K_{0} \otimes \mathbb {1}_{{\mathcal {A}}}- \sum _{\alpha = 1}^{\alpha _*} K_{\alpha } \otimes x_{\alpha }- \sum _{\beta = 1}^{\beta ^*} \big ( L_{\beta } \otimes y_{\beta } + L_{\beta }^* \otimes y_{\beta }^* \big ) \end{aligned}$$

(A.22)

with $K_0,K_{\alpha },L_{\beta } \in {\mathbb {C}}^{\,m\times m}$ and $K_{0}$, $K_{\alpha }$ self-adjoint. Define the completely positive map $\Gamma : {\mathbb {C}}^{\,m \times m} \rightarrow {\mathbb {C}}^{\,m \times m}$ by

$$\begin{aligned} \Gamma \big [R\,\big ] = \sum _{\alpha = 1}^{\alpha _*} K_{\alpha } R K_{\alpha } + \sum _{\beta =1}^{\beta _*} \big ( L_{\beta } R L_{\beta }^* + L_{\beta }^* R L_{\beta }\big ) ,\quad R\in {\mathbb {C}}^{m \times m} . \end{aligned}$$

(A.23)

Definition A.11

(Dyson equation for linearizations). We call the equation

$$\begin{aligned} -\frac{1}{M} = z J_{m} - K_{0} + \Gamma [ M] \end{aligned}$$

(A.24)

the Dyson equation for the linearization (DEL) (of a rational expression).

Lemma A.12

(Solution of DEL: existence and basic properties). Let ${\mathcal {H}}$ be a Hilbert space and let ${\mathcal {S}}\subset {\mathcal {B}}({\mathcal {H}})$ be a $C^*$-algebra containing a freely independent family $\{s_1,\ldots ,s_{\alpha _*}, c_{1},\ldots ,c_{\beta _*}\}$ of $\alpha _*$ semicircular and $\beta _*$ circular elements in a NC probability space $({\mathcal {S}},\tau _{{\mathcal {S}}})$. Let $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$ and assume that $(\varvec{s},\varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$ for $\varvec{s}= (s_{1},\ldots ,s_{\alpha _*})$, $\varvec{c}:=(c_1,\ldots ,c_{\beta _*})$ and some $C>0$. Define

$$\begin{aligned} M^{(\mathrm {sc})}_{z} := ({\mathrm {id}}_{m}\otimes \,\tau _{{\mathcal {S}}})\bigg ( \Big [(K_0-zJ)\otimes \mathbb {1}_{{\mathcal {S}}}- \sum _{\alpha =1}^{\alpha _*} K_{\alpha } \otimes s_{\alpha } - \sum _{\beta = 1}^{\beta _*}\big (L_{\beta } \otimes c_{\beta } + L_{\beta }^* \otimes c_{\beta }^*\big )\Big ]^{-1}\bigg ) . \end{aligned}$$

(A.25)

Then

• (i) there exists $C^{(\mathrm {sc})}_{\varvec{L}_{q}}>0$ such that

  $$\begin{aligned} \Vert M^{(\mathrm {sc})}_{z}\Vert _{ {\mathbb {C}}^{m\times m}} \le C^{(\mathrm {sc})}_{\varvec{L}_{q}}\Big (1 + \frac{1}{\mathrm {Im}z}\Big ) ; \end{aligned}$$

  (A.26)

• (ii) $M^{(\mathrm {sc})}_{z}$ satisfies the DEL (A.24);

• (iii) $M^{(\mathrm {sc})}_{z}$ depends analytically on z;

• (iv) $\mathrm {Im}M^{(\mathrm {sc})}_{z} \ge 0$;

• (v) $M^{(\mathrm {sc})}_{z}$ admits the representation

  $$\begin{aligned} M^{(\mathrm {sc})}_{z} = M^{\infty } + \int _{{\mathbb {R}}} \frac{V(d\lambda )}{\lambda -z} , \end{aligned}$$

  (A.27)

  where $M^{\infty }\in {\mathbb {C}}^{m \times m }$ is a self-adjoint matrix, and $V(d\lambda )$ is a positive semidefinite matrix-valued measure on ${\mathbb {R}}$ with compact support;

• (vi) for almost every $\lambda \in R$ the limit $\lim _{\eta \rightarrow 0} \pi ^{-1} \mathrm {Im}M_{\lambda + \mathrm {i}\,\eta } = V(\lambda ) \in {\mathbb {C}}^{m\times m}$ exists; if the limit is finite on some interval $I\subset {\mathbb {R}}$ everywhere, then $V(d \lambda )$ is absolutely continuous on I and $V(d\lambda ) = V(\lambda ) d\lambda$;

• (vii) $\mathrm {supp}(V_{11}) = \mathrm {supp}({\mathrm {Tr}}V)$.

Proof

Proof of (i). It follows from Lemma A.10 and the norm bounds for semicircular and circular operators

$$\begin{aligned} \Vert s_{\alpha }\Vert _{{\mathcal {S}}} = 2,\quad \Vert c_{\beta }\Vert _{{\mathcal {S}}} = 2,\quad 1\le \alpha \le \alpha _* ,\quad 1\le \beta \le \beta _* \end{aligned}$$

(A.28)

that

$$\begin{aligned}&\Big \Vert \Big ((K_{0}-zJ)\otimes \mathbb {1}_{{\mathcal {S}}}-\sum _{\alpha =1}^{\alpha _*} K_{\alpha } - \sum _{\beta = 1}^{\beta _*} \big (L_{\beta } \otimes c_{\beta } + L_{\beta }^* \otimes c_{\beta }^* \big )\otimes s_{\alpha }\Big )^{-1}\Big \Vert _{{\mathbb {C}}^{m\times m}\otimes \,{\mathcal {S}}} \nonumber \\&\quad \le C^{(\mathrm {sc})}_{\varvec{L}_{q}}\Big (1+\frac{1}{\mathrm {Im}z}\Big ) \end{aligned}$$

(A.29)

for $C^{(\mathrm {sc})}_{\varvec{L}_{q}}:= C_{\varvec{L}_{q}}(1+C + 4)^{n_{\varvec{L}_{q}}}$.

Proof of (ii). First note that the real and imaginary parts of free circular elements form a freely independent family of semicirculars. Therefore, by defining for each $1\le \beta \le \beta _*$

$$\begin{aligned}&s_{\alpha _*+\beta } := \sqrt{2}\mathrm {Re}c_{\beta } ,\quad s_{\alpha _*+\beta _*+\beta } := \sqrt{2}\mathrm {Im}c_{\beta } , \quad K_{\alpha _* + \beta } := \sqrt{2} \mathrm {Re}L_{\beta } ,\nonumber \\&K_{\alpha _* + \beta _* + \beta } := -\sqrt{2} \mathrm {Im}L_{\beta } , \end{aligned}$$

(A.30)

$M^{(\mathrm {sc})}_{z}$ can be rewritten as

$$\begin{aligned} M^{(\mathrm {sc})}_{z} = ({\mathrm {id}}_{m}\otimes \,\tau _{{\mathcal {S}}})\bigg ( \Big [(K_0-zJ)\otimes \mathbb {1}_{{\mathcal {S}}}- \sum _{\alpha =1}^{\alpha _*+2\beta _*} K_{\alpha } \otimes s_{\alpha } \Big ]^{-1}\bigg ) \end{aligned}$$

(A.31)

and it will be enough to show that $M^{(\mathrm {sc})}_{z}$ satisfies the DEL (A.24) with $\Gamma [\,\cdot \,] = \sum _{\alpha =1}^{\alpha + 2\beta _*} K_{\alpha } \,\cdot \, K_{\alpha }$. This last fact can be established via the argument similar to the proof of the existence of the solution to the DEL for the linearizations of polynomials in [13, Lemma 2.6 (iv)]. This proof relies on the results of [23, Lemma 5.4] and [27, Proposition 4.1] establishing the existence of the solution to a particular class of matrix Dyson equations (MDE), as well as regularization technique which allows to extend these results from MDE to DEL. The trivial bound from [13, Lemma 2.5], which justifies the application of the Schur complement formula and relies on the nilpotency structure of the linearizations, in the setting of the current paper can be replaced by the bound (A.29) coming from the specific choice of the domain of evaluation.

Proof of (iii)–(vii). The analyticity of $M^{(\mathrm {sc})}_{z}$ follows from (A.29) and the positive semidefiniteness of $\mathrm {Im}M^{(\mathrm {sc})}_{z}$ is a direct consequence of the representation

$$\begin{aligned} \mathrm {Im}M^{(\mathrm {sc})}_{z} = \eta \, ({\mathrm {id}}_{m} \otimes \,\tau _{{\mathcal {S}}})\Big ( \big ( \varvec{L}^{(\mathrm {sc})} - {\overline{z}}J_{m}\otimes \mathbb {1}_{{\mathcal {S}}}\big )^{-1} (J_{m} \otimes \mathbb {1}_{{\mathcal {S}}})\big ( \varvec{L}^{(\mathrm {sc})} - z J_{m}\otimes \mathbb {1}_{{\mathcal {S}}}\big )^{-1}\Big ) \end{aligned}$$

(A.32)

with $\varvec{L}^{(\mathrm {sc})}:= K_0\otimes \mathbb {1}_{{\mathcal {S}}}- \sum _{\alpha =1}^{\alpha _*+2\beta _*} K_{\alpha } \otimes s_{\alpha }$. Properties $(v){-}(vii)$ follow from the general properties of matrix-valued Herglotz functions, Schur formula (A.20) applied to $(\varvec{L}^{(\mathrm {sc})} - zJ_{m}\otimes \mathbb {1}_{{\mathcal {S}}})^{-1}$ and the bound (A.29) using the similar argument as in the proof of [13, Lemma 2.7]. $\square$

1.5 Convergence of Spectrum for the Rational Expressions in Random Matrices and the a Priori Bound for the Generalized Resolvent in Random Matrices

The next two sections are devoted to the study of the eigenvalues of a general class of rational expressions evaluated on random Wigner and iid matrices.

Assumption A.13

(Wigner and iid matrices). Let $X_{1},\ldots , X_{\alpha _*}\in {\mathbb {C}}^{N\times N}$ and $Y_{1},\ldots ,Y_{\beta _*} \in {\mathbb {C}}^{N\times N}$ be two independent families of independent random matrices satisfying the following assumptions

(H1):

$X_{\alpha }=(X_{\alpha }(i,j))_{i,j=1}^{N}$, $1\le \alpha \le \alpha _*$, are Hermitian random matrices having independent (up to symmetry constraints) centered entries of variance 1/N;

(H2):

$Y_{\beta }=(Y_{\beta }(i,j))_{i,j=1}^{N}$, $1\le \beta \le \beta _*$, are (non-Hermitian) random matrices having independent centered entries of variance 1/N;

(H3):

there exist $\varphi _{n}>0$, $n \in {\mathbb {N}}$, such that

$$\begin{aligned} \max _{1\le i,j \le N} \Big ( \max _{1\le \alpha \le \alpha _*} {\mathbb {E}}\big [|\sqrt{N}X_{\alpha }(i,j)|^{n}\big ] + \max _{1\le \beta \le \beta _*} {\mathbb {E}}\big [|\sqrt{N}Y_{\beta }(i,j)|^{n}\big ]\Big ) \le \varphi _{n} . \end{aligned}$$

(A.33)

We call $X_{\alpha }$ Wigner matrices and $Y_{\beta }$ iid matrices.

Denote $\varvec{X}:=(X_{1},\ldots ,X_{\alpha _*})$, $\varvec{Y}:=(Y_{1},\ldots ,Y_{\beta _*})$, $\varvec{Y}^*:=(Y_{1}^*,\ldots ,Y^*_{\beta _*})$ and let q be a (self-adjoint) rational expression in $\alpha _*$ self-adjoint and $\beta _*$ non self-adjoint noncommutative variables. In order to prove the local law for $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$, we will need to show that the spectrum of $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ converges to the spectrum of $q(\varvec{s},\varvec{c},\varvec{c}^*)$. To this end, for any $\varepsilon >0$, $m\in {\mathbb {N}}$ and operator $\varvec{R}\in {\mathbb {C}}^{mN\times mN}$, denote by $\mathrm {Spec}_{\varepsilon }(\varvec{R})$ the $\varepsilon$-pseudospectrum of $\varvec{R}$ defined by

$$\begin{aligned} \mathrm {Spec}_{\varepsilon }(\varvec{R}) = \mathrm {Spec}(\varvec{R})\cup \{z\in {\mathbb {C}}\, : \Vert (\varvec{R}- z I_{m}\otimes I_{N})^{-1}\Vert _{{\mathbb {C}}^{mN\times mN}} \ge \varepsilon ^{-1}\} . \end{aligned}$$

(A.34)

It is easy to check that for any $\varvec{R}\in {\mathbb {C}}^{mN\times mN} \cong {\mathbb {C}}^{m\times m} \otimes {\mathbb {C}}^{N\times N}$

$$\begin{aligned} \Vert \varvec{R}\Vert _{{\mathbb {C}}^{mN\times mN}} \le m \, \Vert \varvec{R}\Vert _{{\mathbb {C}}^{m\times m}\otimes \,{\mathbb {C}}^{N\times N}} , \end{aligned}$$

(A.35)

where $\Vert \, \cdot \, \Vert _{{\mathbb {C}}^{mN\times mN}}$ and $\Vert \, \cdot \, \Vert _{{\mathbb {C}}^{N\times N}}$ denote the operator norms on ${\mathbb {C}}^{mN \times mN}$ and ${\mathbb {C}}^{N\times N}$ correspondingly.

For any (not necessarily self-adjoint) rational expression $r(\varvec{x},\varvec{y},\varvec{y}^*)$ in NC variables $\varvec{x},\varvec{y},\varvec{y}^*$, denote by $\varvec{L}_{r}:=\varvec{L}_{r}(\varvec{x},\varvec{y},\varvec{y}^*) \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{m_{r}\times m_{r}}$ its (not necessarily self-adjoint) linearization, which can be constructed using, for example, the algorithm from Sect. A.2 omitting steps (B4) and (S4) if r is not self-adjoint. Define the corresponding Hermitized linearization by

$$\begin{aligned} \varvec{L}^{r,z}(\varvec{x},\varvec{y},\varvec{y}^*) := \left( \begin{array}{cc} 0 &{} \quad \varvec{L}_{r}(\varvec{x},\varvec{y},\varvec{y}^*)-zJ_{m_{r}}\otimes \mathbb {1}_{{\mathcal {A}}}\\ \big (\varvec{L}_{r}(\varvec{x},\varvec{y},\varvec{y}^*)\big )^{*}-{\overline{z}}J_{m_{r}}\otimes \mathbb {1}_{{\mathcal {A}}}&{}\quad 0 \end{array} \right) \end{aligned}$$

(A.36)

with $\varvec{L}^{r,z}(\varvec{x},\varvec{y},\varvec{y}^*) \in ({\mathbb {C}}\langle \varvec{x},\varvec{y},\varvec{y}^*\rangle )^{2m_{r}\times 2m_{r}}$. For the Hermitized linearization (A.36), we define [similarly as in (A.23)] the self-energy operator $\Gamma ^{r,z}:{\mathbb {C}}^{2m_{r}\times 2m_{r}} \rightarrow {\mathbb {C}}^{2m_{r}\times 2m_{r}}$, given by the completely positive map

$$\begin{aligned} \Gamma ^{r,z}[R] = \sum _{\alpha =1}^{\alpha _*} K_{\alpha }^{r,z} R K_{\alpha }^{r,z} + \sum _{\beta =1}^{\beta _*} \left( L_{\beta }^{r,z} R \left( L_{\beta }^{r,z}\right) ^{*} + \left( L_{\beta }^{r,z}\right) ^* R L_{\beta }^{r,z}\right) , \end{aligned}$$

(A.37)

where $K_{\alpha }^{r,z}$ and $L_{\beta }^{r,z}$ are the coefficient matrices of $\varvec{L}^{r,z}$ [see, e.g., (A.6)]. Note that if we evaluate $\varvec{L}^{r,z}$ on the tuple of random matrices $(\varvec{X},\varvec{Y},\varvec{Y}^*)$, then $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$ belongs to the class of Kronecker random matrices, which were studied in [3]. Therefore, from [3, Lemma 2.2], we have that the corresponding Matrix Dyson equation

$$\begin{aligned} -\frac{1}{M^{r,z}_{\omega }} = \omega I_{2m_{r}} - K^{r,z}_{0} + \Gamma ^{r,z}[M^{r,z}_{\omega }] ,\quad z\in {\mathbb {C}},\quad \omega \in {\mathbb {C}}_{+} \end{aligned}$$

(A.38)

has a unique solution with positive semidefinite imaginary part $\mathrm {Im}M^{r,z}_{\omega } \ge 0$. Moreover, for each $z\in {\mathbb {C}}$, the solution matrix $M^{r,z}_{\omega }$ admits the Stieltjes transform representation

$$\begin{aligned} M^{r,z}_{\omega } = \int _{{\mathbb {R}}} \frac{V^{r,z}(d\lambda )}{\lambda -\omega } , \end{aligned}$$

(A.39)

where $\{V^{r,z}\}_{z\in {\mathbb {C}}}$ is a family of measures taking values in the set of positive definite matrices. In the limit $N \rightarrow \infty$ the solution $M^{r,z}_{\omega }\otimes I_N$ well approximates $(\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)-\omega I)^{-1}$ in the entrywise maximum norm (see [3, Lemma B.1]).

Remark A.14

The statement of [3, Lemma 2.2] is formulated in the more general setting of Wigner-type matrices allowing independent but not necessarily identically distributed entries. This model, in general, leads to a system of N matrix equations. In our case, the matrices in $\varvec{X}$ and $\varvec{Y}$ have i.i.d. entries (up to symmetry constraints), which reduces the system of N possibly different matrix equations (see, e.g., [3, Eq. (2.6)]) to N identical matrix equations of the form (A.38).

For any rational expression r with linearization $\varvec{L}_r$ define the set ${\mathbb {D}}_{\varepsilon }^{\varvec{L}_{r}}\subset {\mathbb {C}}$ by

$$\begin{aligned} {\mathbb {D}}_{\varepsilon }^{\varvec{L}_{r}} := \Big \{z \, : \, \mathrm {dist}(0,\mathrm {supp}\, \rho ^{r,z}) \le \varepsilon \Big \} , \end{aligned}$$

(A.40)

where $\rho ^{r,z}(d\lambda ):=\frac{1}{2m_r} {\mathrm {Tr}}V^{r,z}(d\lambda )$ and the family of measures $V^{r,z}(d\lambda )$ were defined in (A.39). The set ${\mathbb {D}}_{\varepsilon }^{\varvec{L}_{r}}$ is called the self-consistent $\varepsilon$-pseudospectrum of r related to its linearization $\varvec{L}_{r}$, and $\rho ^{r,z}$ is called the self-consistent density of states of $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$.

The next lemma contains the main result of this section.

Lemma A.15

(Convergence of the (pseudo)spectrum). Suppose that $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$ is a (not necessarily self-adjoint) rational expression of height n, and $(\varvec{s}, \varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {S}})$ with some constant $C>0$. Then, there exists ${\widehat{C}}^{\mathrm {w}}_{\varvec{L}_{q}}>0$ such that

$$\begin{aligned} \Big \Vert \big ({\widehat{\varvec{L}}}_{q}(\varvec{X},\varvec{Y},\varvec{Y}^*)\big )^{-1}\big \Vert _{{\mathbb {C}}^{(m-1)\times (m-1)}\otimes \,{\mathbb {C}}^{N\times N}} \le {\widehat{C}}^{\mathrm {w}}_{\varvec{L}_{q}}\quad \text{ a.w.o.p. }, \end{aligned}$$

(A.41)

where ${\widehat{\varvec{L}}}_{q}$ is defined as in (A.5). There exists also a constant ${\widetilde{C}}>0$ depending only on the linearization $\varvec{L}_q$ and the constant C such that

$$\begin{aligned} (\varvec{X},\varvec{Y}) \in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};{\widetilde{C}}}\big ({\mathbb {C}}^{N \times N}\big )\qquad a.w.o.p. \end{aligned}$$

(A.42)

Moreover, for any $\varepsilon \in (0,1)$ the $\varepsilon$-pseudospectrum of $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ satisfies

$$\begin{aligned} \mathrm {Spec}_{\varepsilon }(q(\varvec{X},\varvec{Y},\varvec{Y}^*)) \subset {\mathbb {D}}_{2\varepsilon }^{\varvec{L}_{q}} \quad \text{ a.w.o.p. } . \end{aligned}$$

(A.43)

Proof

We split the proof of this lemma in two parts. First, we show that the condition (A.41) together with (A.42) implies (A.43) for any, not necessarily self-adjoint, rational expression and its linearization. After that we prove that (A.41) and (A.42) are satisfied for q if $(\varvec{s}, \varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {S}})$ using induction on the height n.

Suppose that we have an arbitrary rational expression r and its linearization $\varvec{L}_{r}$ of size $m_{r}$, and suppose that there exists ${\widehat{C}}^{\mathrm {w}}_{\varvec{L}_{r}}>0$ such that a.w.o.p.

$$\begin{aligned} \Vert ({\widehat{\varvec{L}}}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*))^{-1}\Vert _{{\mathbb {C}}^{(m_{r}-1)\times (m_{r}-1)}\otimes \,{\mathbb {C}}^{N \times N}}\le {\widehat{C}}^{\mathrm {w}}_{\varvec{L}_{r}}, \end{aligned}$$

(A.44)

where ${\widehat{\varvec{L}}}_{r}$ is defined similarly as in (A.5). Then from the definition of the linearization (Definition A.5) and the Schur complement formula (A.20), we can choose $C_{3},C_{4}>0$ such that

$$\begin{aligned} \Vert r(\varvec{X},\varvec{Y},\varvec{Y}^*) \Vert _{{\mathbb {C}}^{N\times N}} \le C_{3} \end{aligned}$$

(A.45)

and the sequence of inequalities

$$\begin{aligned}&\Vert (r(\varvec{X},\varvec{Y},\varvec{Y}^*) - z I_{N})^{-1} \Vert _{{\mathbb {C}}^{N \times N}} \nonumber \\&\quad \le \Vert (\varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m_{r}}\otimes I_{N})^{-1}\Vert _{{\mathbb {C}}^{m_{r}N\times m_{r}N}} \end{aligned}$$

(A.46)

$$\begin{aligned}&\quad \le m_{r} \Vert (\varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m_{r}}\otimes I_{N})^{-1}\Vert _{{\mathbb {C}}^{m_{r}\times m_{r}}\otimes \,{\mathbb {C}}^{N\times N}} \end{aligned}$$

(A.47)

$$\begin{aligned}&\quad \le C_{4}\Vert (r(\varvec{X},\varvec{Y},\varvec{Y}^*) - z I_{N})^{-1} \Vert _{{\mathbb {C}}^{N\times N}} \end{aligned}$$

(A.48)

hold a.w.o.p. for all $z\in {\mathbb {C}}$. Here the first and third inequalities follow from the Schur complement formula (A.20) and the norm bounds $\max _{1\le \alpha \le \alpha _{*}}\Vert X_{\alpha }\Vert _{{\mathbb {C}}^{N\times N}}$ $\le 3$, $\max _{1\le \beta \le \beta _{*}}\Vert Y_{\beta }\Vert _{{\mathbb {C}}^{N\times N}} \le 3$ holding a.w.o.p., the second inequality holds deterministically for all realizations of $\varvec{X}$ and $\varvec{Y}$ [see (A.35)], $C_{3}>0$ depends on ${\widehat{C}}_{r}$, $m_{r}$ and the norms of $X_{\alpha }$, $Y_{\beta }$, which we bound by 3, and $C_{4}>1$ additionally depends on $C_{3}$.

Note again from (A.20) that if ${\widehat{\varvec{L}}}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)$ is invertible, then

$$\begin{aligned} \mathrm {Spec}(r(\varvec{X},\varvec{Y},\varvec{Y}^*)) = \{z\in {\mathbb {C}}\, : \varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m_{r}}\otimes I_N \text{ is } \text{ not } \text{ invertible }\} . \end{aligned}$$

(A.49)

On the other hand, using the definition of $\varvec{L}^{r,z}$ from (A.36), the set on the right-hand side of (A.49) can be described via the spectrum of $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$ as

$$\begin{aligned}&\{z\in {\mathbb {C}}\, : \varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m_{r}}\otimes I_N \text{ is } \text{ not } \text{ invertible }\} \nonumber \\&\quad = \{z\in {\mathbb {C}}\, : 0 \in \mathrm {Spec}(\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*))\} \end{aligned}$$

(A.50)

with the identity (A.50) holding deterministically for all realizations of $\varvec{X}$ and $\varvec{Y}$.

Under the condition (A.44), the equality (A.49) can be rewritten in terms of the pseudospectrum using (A.46) as

$$\begin{aligned} \mathrm {Spec}_{\varepsilon }(r(\varvec{X},\varvec{Y},\varvec{Y}^*))&\subset \Big \{z\in {\mathbb {C}}: \Vert ( \varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*) - zJ_{m_{r}} \otimes I_{N})^{-1}\Vert _{{\mathbb {C}}^{m_{r}N\times m_{r}N}} \ge \frac{1}{\varepsilon } \Big \} \end{aligned}$$

(A.51)

$$\begin{aligned}&\subset \mathrm {Spec}_{C_{4} \varepsilon }(r(\varvec{X},\varvec{Y},\varvec{Y}^*)) \end{aligned}$$

(A.52)

holding a.w.o.p. At the same time, from the definition (A.36) we have that the set of the singular values of the Hermitian matrix $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$ coincides with the set of the singular values of $\varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*) - zJ_{m_{r}} \otimes I_{N}$, so that

$$\begin{aligned}&\Big \{z\in {\mathbb {C}}\, : \Vert ( \varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*) - zJ_{m_{r}} \otimes I_{N})^{-1}\Vert _{{\mathbb {C}}^{m_{r} N\times m_{r} N}} \ge \frac{1}{\varepsilon } \Big \} \nonumber \\&\quad = \{ z\in {\mathbb {C}}\, : \mathrm {dist}(0, \mathrm {Spec}(\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*))\le \varepsilon \} . \end{aligned}$$

(A.53)

In order to study the spectrum of $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$, we will exploit the fact that $\varvec{L}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m_{r}}\otimes I_{N}$ belongs to the class of Kronecker random matrices and thus we can use the results from [3] about the location of spectrum for this class of random matrix ensembles. By applying part (i) of [3, Theorem 4.7] to $\varvec{L}^{r,z}$, we have (similarly as in the proof of [3, Lemma 6.1] for bounded $\zeta$) that for any z satisfying $\mathrm {dist}\,(0,\mathrm {supp}\,\rho ^{r,z}) \ge 2\varepsilon$, a.w.o.p.

$$\begin{aligned} \mathrm {Spec}(\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)) \cap \Big [-\frac{3\varepsilon }{2} ,\frac{3\varepsilon }{2}\Big ] = \emptyset . \end{aligned}$$

(A.54)

Now we claim that (A.54) holds simultaneously a.w.o.p. for all $\{z \, : |z|\le 2C_{3},\, z\notin {\mathbb {D}}^{\varvec{L}_{r}}_{2\varepsilon } \}$. To prove this strengthening, we apply the standard grid argument (again analogously as in the proof of [3, Lemma 6.1]) together with the Lipschitz continuity of the eigenvalues of $\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*)$ in z. Together with (A.51) and (A.53) this implies that a.w.o.p.

$$\begin{aligned}&\mathrm {Spec}_{\varepsilon }(r(\varvec{X},\varvec{Y},\varvec{Y}^*)) \nonumber \\&\quad \subset \{ z\in {\mathbb {C}}\, : \mathrm {dist}(0, \mathrm {Spec}(\varvec{L}^{r,z}(\varvec{X},\varvec{Y},\varvec{Y}^*))\le \varepsilon \} \subset \Big ({\mathbb {D}}^{\varvec{L}_{r}}_{2\varepsilon } \cup \{z \, : |z|\ge 2C_{3} \} \Big ) .\nonumber \\ \end{aligned}$$

(A.55)

We conclude from (A.45) that a.w.o.p.

$$\begin{aligned} \mathrm {Spec}_{\varepsilon }(r(\varvec{X},\varvec{Y},\varvec{Y}^*)) \subset {\mathbb {D}}^{\varvec{L}_{r}}_{2\varepsilon } . \end{aligned}$$

(A.56)

This finishes the first part of the proof by establishing that for any rational expression the conditions (A.41) and (A.42) imply (A.43).

In order to prove (A.41) and (A.42), we proceed with a proof by induction on the height of q. If q has height 0 (i.e., if q is a polynomial), then (A.42) is trivially true and (A.41) holds by nilpotency [13, Lemma 2.5] and the norm bounds

$$\begin{aligned} \Vert X_{\alpha }\Vert _{{\mathbb {C}}^{N\times N}} \le 3 ,\quad \Vert Y_{\beta }\Vert _{{\mathbb {C}}^{N\times N}} \le 3 \quad \text{ a.w.o.p. } \end{aligned}$$

(A.57)

Suppose that the statement of the theorem is true for all rational expressions of height $\le n-1$. Together with $(\varvec{s}, \varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {S}})$ this, in particular, means that a.w.o.p.

$$\begin{aligned} (\varvec{X},\varvec{Y}) \in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n-1};{\widetilde{C}}_1}\big ({\mathbb {C}}^{N \times N}\big ) \end{aligned}$$

(A.58)

for some ${\widetilde{C}}_1>0$, and

$$\begin{aligned} \Vert ({\widehat{\varvec{L}}}_{r}(\varvec{X},\varvec{Y},\varvec{Y}^*))^{-1}\Vert _{{\mathbb {C}}^{(m_r-1)\times (m_r-1)}\otimes \,{\mathbb {C}}^{N\times N}} \le {\widehat{C}}^{\mathrm {w}}_{\varvec{L}_{r}} \end{aligned}$$

(A.59)

for all $r\in {\mathcal {T}}_{q}:= \{q_{i,\gamma _{i}} \,: 0\le i\le n, 1\le \gamma _{i} \le \ell _{i}\}$ with $q_{i,\gamma _{i}}$ as in the definition of ${\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$. We now show that there exists ${\widetilde{C}}>0$ such that $(\varvec{X},\varvec{Y}) \in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};{\widetilde{C}}}\big ({\mathbb {C}}^{N \times N}\big )$ a.w.o.p., or equivalently, that for all $\gamma _{n} \in \{1,\ldots ,\ell _n\}$ a.w.o.p.

$$\begin{aligned} \Big \Vert \frac{1}{q_{n,\gamma _n}(\varvec{X},\varvec{Y},\varvec{Y}^*)}\Big \Vert _{{\mathbb {C}}^{N\times N}} \le {\widetilde{C}} . \end{aligned}$$

(A.60)

Using the result established in the first part of the proof, (A.58) and (A.59) with $r = q_{n, \gamma _n}$ imply that a.w.o.p.

$$\begin{aligned} \mathrm {Spec}_{\varepsilon }(q_{n,\gamma _n}(\varvec{X},\varvec{Y},\varvec{Y}^*)) \subset {\mathbb {D}}^{\varvec{L}_{q_{n,\gamma _n}}}_{2\varepsilon } . \end{aligned}$$

(A.61)

Therefore, it is enough to show that there exists ${\widetilde{C}}>0$ such that for all $\gamma _n \in \{1,\ldots ,\ell _n\}$ the point $z=0$ does not belong to ${\mathbb {D}}^{\varvec{L}_{q_{n,\gamma _n}}}_{2/{\widetilde{C}}}$. By the definition (A.40), the last condition is equivalent to

$$\begin{aligned} \mathrm {dist}(0,\mathrm {supp}\, \rho ^{q_{n,\gamma _n},0}) \ge \frac{2}{{\widetilde{C}}} . \end{aligned}$$

(A.62)

We now show that (A.62) holds for one fixed $\gamma _n$. The desired bound (A.60) can then be obtained by taking the maximum over ${\widetilde{C}}$ for all $\gamma _n\in \{1,\ldots , \ell _n\}$.

Fix $\gamma _{n}\in \{1,\ldots , \ell _n\}$, and denote by $\varvec{L}_{q_{n,\gamma _n}}(\varvec{s},\varvec{c},\varvec{c}^*) \in {\mathbb {C}}^{m\times m}\otimes {\mathcal {S}}$ the linearization of $q_{n,\gamma _n}$ evaluated at $(\varvec{s},\varvec{c},\varvec{c}^*)$. By examining carefully the derivation of the a priori bound (A.18), we observe that there exists a constant $t>0$ depending only on the linearization and the constant C, such that

$$\begin{aligned} \Big \Vert \Big (\varvec{L}_{q_{n,\gamma _n}}(\varvec{s},\varvec{c},\varvec{c}^*)\Big )^{-1}\Big \Vert _{{\mathbb {C}}^{m \times m} \otimes \, {\mathcal {S}}} \le t. \end{aligned}$$

(A.63)

Indeed, each entry of $(\varvec{L}_{q_{n,\gamma _n}}(\varvec{s},\varvec{c},\varvec{c}^*))^{-1}$ is a polynomial in $\varvec{s}$, $\varvec{c}$, $\varvec{c}^*$ and $(r(\varvec{s},\varvec{c},\varvec{c}^*))^{-1}$ with $r\in {\mathcal {T}}_{q}$. Therefore, the bounds $\Vert (r(\varvec{s},\varvec{c},\varvec{c}^*))^{-1}\Vert _{{\mathcal {S}}}\le C$ that follow from the assumption $(\varvec{s},\varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}$ imply that (A.63) holds for some $t>0$.

Next notice, that from the definition of $\varvec{L}^{r,z}$ [see (A.36)], we have that

$$\begin{aligned} \Big \Vert \Big (\varvec{L}^{q_{n,\gamma _n},0 }(\varvec{s},\varvec{c},\varvec{c}^*)\Big )^{-1}\Big \Vert _{{\mathbb {C}}^{2m \times 2m} \otimes \, {\mathcal {S}}} = \Big \Vert \Big (\varvec{L}_{q_{n,\gamma _n}}(\varvec{s},\varvec{c},\varvec{c}^*)\Big )^{-1}\Big \Vert _{{\mathbb {C}}^{m \times m} \otimes \, {\mathcal {S}}} . \end{aligned}$$

(A.64)

The resolvent identity

$$\begin{aligned}&\Big (\varvec{L}^{q_{n,\gamma _n},z }(\varvec{s},\varvec{c},\varvec{c}^*)- \omega I \otimes \mathbb {1}_{{\mathcal {S}}}\Big )^{-1} \Big (I\otimes \mathbb {1}_{{\mathcal {S}}}- \omega \, \Big (\varvec{L}^{q_{n,\gamma _n},z }(\varvec{s},\varvec{c},\varvec{c}^*)\Big )^{-1} \Big ) \nonumber \\&\quad = \Big (\varvec{L}^{q_{n,\gamma _n},z }(\varvec{s},\varvec{c},\varvec{c}^*)\Big )^{-1} \end{aligned}$$

(A.65)

together with (A.63) and (A.64) implies that

$$\begin{aligned} \Big \Vert \Big (\varvec{L}^{q_{n,\gamma _n},0 }(\varvec{s},\varvec{c},\varvec{c}^*)-\omega I\otimes \mathbb {1}_{{\mathcal {S}}}\Big )^{-1} \Big \Vert _{{\mathbb {C}}^{2m \times 2m} \otimes \, {\mathcal {S}}} \le 2t\end{aligned}$$

(A.66)

for all $\omega \in {\mathbb {C}}$ satisfying $|\omega | \le 1/(4m t)$, where we used the relation between the operator and max norms as in (A.47) to estimate the operator norm of $(\varvec{L}^{q_{n,\gamma _n},z }(\varvec{s},\varvec{c},\varvec{c}^*))^{-1}$. The bound (A.66) means, in particular, that

$$\begin{aligned} \mathrm {Spec}\Big (\varvec{L}^{q_{n,\gamma _n},0 }(\varvec{s},\varvec{c},\varvec{c}^*)\Big ) \cap \Big [-\frac{1}{4mt}, \frac{1}{4mt}\Big ] = \emptyset . \end{aligned}$$

(A.67)

The spectrum of the self-adjoint operator $\varvec{L}^{q_{n,\gamma _n},0 }(\varvec{s},\varvec{c},\varvec{c}^*)$ is characterized by the solution to the self-consistent Eq. A.38 (see, e.g., [36, Theorem 4.1.12]) via

$$\begin{aligned} \mathrm {Spec}\Big (\varvec{L}^{q_{n,\gamma _n},0 }(\varvec{s},\varvec{c},\varvec{c}^*) \Big ) = \mathrm {supp} \, \rho ^{q_{n,\gamma _n},0 } , \end{aligned}$$

(A.68)

which establishes (A.62) with ${\widetilde{C}} = 8 m t$.

Finally, (A.42) together with the Schur complement formula, (A.15)–(A.16), and the special form of the linearization blocks (A.9)–(A.10), yields (A.41) for $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$. This finishes the proof of the lemma. $\square$

The bound (A.41) together with the Schur complement formula (A.20) applied to $(\varvec{L}_{q}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m}\otimes I_{N})^{-1}$, the trivial bound (A.21) applied to $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ and the norm bounds $\Vert X_{\alpha }\Vert _{{\mathbb {C}}^{N\times N}}\le 3$ and $\Vert Y_{\beta }\Vert _{{\mathbb {C}}^{N\times N}}\le 3$ holding [similarly as in (2.3)–(2.4)] a.w.o.p., imply the trivial bound for the generalized resolvent in random matrices.

Corollary A.16

There exists $C^{\mathrm {w}}_{\varvec{L}_{q}}>0$ depending only on the linearization $\varvec{L}_{q}$, such that a.w.o.p.

$$\begin{aligned} \Vert (\varvec{L}_{q}(\varvec{X},\varvec{Y},\varvec{Y}^*)-zJ_{m}\otimes I_{N})^{-1} \Vert _{{\mathbb {C}}^{m\times m} \otimes {\mathbb {C}}^{N\times N}} \le C^{\mathrm {w}}_{\varvec{L}_{q}}\Big (1+\frac{1}{\mathrm {Im}z} \Big ) . \end{aligned}$$

(A.69)

1.6 Global and Local Laws for Rational Expressions in Random Matrices

Denote, as before, by ${\mathcal {S}}$ a $C^*$-algebra containing a freely independent family $\{s_1,\ldots ,s_{\alpha _*}, c_{1},\ldots ,c_{\beta _*}\}$ of $\alpha _*$ semicircular and $\beta _*$ circular elements in a NC probability space $({\mathcal {S}},\tau _{{\mathcal {S}}})$. Let $q\in {\mathcal {Q}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n}}$ be a rational expression of height n and assume that $(\varvec{s},\varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {S}})$ for $\varvec{s}= (s_{1},\ldots ,s_{\alpha _*})$, $\varvec{c}:=(c_1,\ldots ,c_{\beta _*})$ and some $C>0$. Let

$$\begin{aligned} \varvec{L}= \varvec{L}(\varvec{x},\varvec{y},\varvec{y}^*) := K_0\otimes \mathbb {1}_{{\mathcal {A}}}- \sum _{\alpha =1}^{\alpha _*} K_{\alpha } \otimes x_{\alpha } - \sum _{\beta =1}^{\beta _*}\big ( L_{\beta } \otimes y_{\beta } + L_{\beta }^* \otimes y_{\beta }^*\big ) \end{aligned}$$

(A.70)

be the linearization of q constructed via the algorithm from Sect. A.2.

In order to formulate the local law, we need to introduce the notion of the stochastic domination.

Definition A.17

(Stochastic domination). Let $\Phi := (\Phi _{N})_{N\in {\mathbb {N}}}$ and $\Psi := (\Psi _{N})_{N\in {\mathbb {N}}}$ be two sequences of nonnegative random variables. We say that $\Phi$ is stochastically dominated by $\Psi$ (denoted $\Phi \prec \Psi$), if for any $\varepsilon , D > 0$ there exists $C(\varepsilon ,D)>0$ such that for all $N\in {\mathbb {N}}$

$$\begin{aligned} {\mathbb {P}}[\Phi _{N} \ge N^{\varepsilon } \Psi _{N}] \le \frac{C(\varepsilon ,D)}{N^{D}} . \end{aligned}$$

(A.71)

Let $q(\varvec{X},\varvec{Y},\varvec{Y}^*)\in {\mathbb {C}}^{N\times N}$ be the evaluation of q on the $\alpha _*$-tuple of Wigner and $\beta _*$-tuple of iid random matrices satisfying (H1)–(H3), and define the linearization matrix

$$\begin{aligned} \varvec{H}:= \varvec{L}(\varvec{X},\varvec{Y},\varvec{Y}^*) = K_0\otimes I_{N} - \sum _{\alpha =1}^{\alpha _*} K_{\alpha } \otimes X_{\alpha } - \sum _{\beta =1}^{\beta _*}\big ( L_{\beta } \otimes Y_{\beta } + L_{\beta }^* \otimes Y_{\beta }^*\big ) . \end{aligned}$$

(A.72)

Let $\varvec{G}_{z}: = (\varvec{H}- zJ\otimes I_N)^{-1} \in {\mathbb {C}}^{mN \times mN}$ be the generalized resolvent of $\varvec{H}$. Note that the generalized resolvent $\varvec{G}_{z}$, when viewed as taking values in ${\mathbb {C}}^{m\times m} \otimes {\mathbb {C}}^{N \times N}$, can be written as

$$\begin{aligned} \varvec{G}_{z} = \sum _{i,j=1}^{N} G_{z, ij} \otimes E_{ij}, \end{aligned}$$

(A.73)

where the collection of matrices $E_{ij} := (\delta _{ki} \delta _{jl})_{1\le k,l \le N}$ form a standard basis of ${\mathbb {C}}^{N \times N}$ and $G_{z,ij} \in {\mathbb {C}}^{m \times m}$ is an $m\times m$ matrix for each (i, j) pair. In general, we will follow the convention that for any $\varvec{A}\in {\mathbb {C}}^{m\times m} \otimes {\mathbb {C}}^{N \times N}$, we denote by $\varvec{A}_{kl} \in {\mathbb {C}}^{N \times N}$, $k,l=1,2,\ldots , m$ the (k, l)-th block according to the ${\mathbb {C}}^{m\times m}$ factor in the tensor product, while $A_{ij} \in {\mathbb {C}}^{m\times m}$, $i,j =1,2,\ldots , N$ denotes the (i, j)-th block in the second factor, i.e.,

$$\begin{aligned} \varvec{A}= \sum _{k,l=1}^{m} E_{kl} \otimes \varvec{A}_{kl}=\sum _{i,j=1}^{N} A_{ij} \otimes E_{ij}, \end{aligned}$$

(A.74)

in particular $A_{ij}(k,l) = \varvec{A}_{kl}(i,j)$.

Let $M^{(\mathrm {sc})}_{z} : {\mathbb {C}}_{+} \rightarrow {\mathbb {C}}^{m \times m}$ be a matrix valued function given by (A.25). For each $z\in {\mathbb {C}}_{+}$ we define the stability operator ${\mathscr {L}}_{z}: {\mathbb {C}}^{m\times m} \rightarrow {\mathbb {C}}^{m \times m}$ corresponding to $M^{(\mathrm {sc})}_{z}$ by

$$\begin{aligned} {\mathscr {L}}_{z}\big [R\,\big ] = R - M^{(\mathrm {sc})}_{z} \Gamma \big [R\,\big ] M^{(\mathrm {sc})}_{z} ,\quad R \in {\mathbb {C}}^{m \times m} . \end{aligned}$$

(A.75)

For $n \in {\mathbb {N}}$ and an operator ${\mathscr {R}}: {\mathbb {C}}^{n\times n} \rightarrow {\mathbb {C}}^{n \times n}$, we denote by $\Vert {\mathscr {R}}\Vert _{{\mathbb {C}}^{n \times n} \rightarrow {\mathbb {C}}^{n \times n}}$ the operator norm of ${\mathscr {R}}$ generated by the the operator norm on ${\mathbb {C}}^{n\times n}$. Then, the following holds.

Theorem A.18

(Local law for rational expressions). Let $M^{(\mathrm {sc})}_{z}$ be defined as in (A.25) and let ${\mathscr {L}}_{z}$ be the stability operator corresponding to $M^{(\mathrm {sc})}_{z}$. If there exist $C_{0}>0$ and ${\mathcal {I}}\subset {\mathbb {R}}$ such that for all z with $\mathrm {Re}z \in {\mathcal {I}}$ and $0\le \mathrm {Im}z < \infty$

• (M1) $\Vert M^{(\mathrm {sc})}_{z}\Vert _{{\mathbb {C}}^{m \times m} } \le C_{0}$;

• (M2) $\Vert {\mathscr {L}}_{z}^{-1} \Vert _{{\mathbb {C}}^{m \times m} \rightarrow {\mathbb {C}}^{m \times m} } \le C_{0}$,

then the optimal local law holds for $\varvec{G}(z)$ on the set ${\mathcal {I}}$, i.e., uniformly for $\mathrm {Re}z \in {\mathcal {I}}$

$$\begin{aligned} \max _{1\le i,j \le N} \Big \Vert G_{z,ij}- M^{(\mathrm {sc})}_{z} \delta _{ij} \Big \Vert _{{\mathbb {C}}^{m\times m}} \quad&\prec \sqrt{\frac{1}{N\mathrm {Im}z}},\nonumber \\ \Big \Vert \frac{1}{N} \sum _{i=1}^{N} G_{z,ii} - M^{(\mathrm {sc})}_{z} \Big \Vert _{{\mathbb {C}}^{m\times m} }&\prec \frac{1}{N\mathrm {Im}z} . \end{aligned}$$

(A.76)

In particular, this implies that on the set ${\mathcal {I}}$ the optimal local law holds for the rational expression in random matrices $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$, i.e., uniformly for $\mathrm {Re}z \in {\mathcal {I}}$

$$\begin{aligned} \max _{1 \le i,j\le N}\Big |g_{z,ij} - M^{(\mathrm {sc})}_{z}(1,1) \delta _{ij} \Big |&\prec \sqrt{\frac{1}{N\mathrm {Im}z}} ,\nonumber \\ \Big |\frac{1}{N}\sum _{i=1}^{N} g_{z,ii} -M^{(\mathrm {sc})}_{z}(1,1) \Big |&\prec \frac{1}{N\mathrm {Im}z} , \end{aligned}$$

(A.77)

where $\varvec{g}_{z}=(g_{z,ij})_{i,j=1}^{N} :=(q(\varvec{X},\varvec{Y},\varvec{Y}^*)-zI_N)^{-1}\in {\mathbb {C}}^{N\times N}$.

Note that by the definition of the generalized resolvent and (A.20) and the notational convention (A.74), we have that $g_{z,ij} = G_{z,ij}(1,1)$ for all $1\le i,j \le N$.

Proof

Our proof of the local law for linearizations of rational expressions (A.76) is analogous to the proof of the corresponding result for linearizations of polynomials in Wigner and iid matrices [13, Theorem 5.1]. Below we provide a summary of the important steps of that proof and show how these steps are adjusted to the current setting of rational expressions.

1. Restricting analysis to the set where $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ is well defined. In contrast to the case when q is a polynomial, the evaluation $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ may not always be well defined and the generalized resolvent $\varvec{G}_{z}$ may not always be bounded, even when $z\in {\mathbb {C}}_{+}$, i.e., the a priori bound analogous to (2.5) in [13, Lemma 2.5] may not hold. But according to Lemma A.15 and Corollary A.16, this bound can be replaced by (A.69) and the existence of $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ can be guaranteed on an event $\Theta =\Theta _{N}$ of asymptotically overwhelming probability. The entire analysis is then restricted to this set. In particular, the indicator sets $\chi (\cdot )$ in the proof of [13, Theorem 5.1] should be replaced by $\chi (\cdot )\cap \Theta$.

2. Exploiting the Kronecker structure of $\varvec{H}$ and regularization of the DEL. Similarly as in the polynomial case, any linearization of a rational expression evaluated on Wigner and iid ensembles belongs to the class of Kronecker random matrices. Therefore, in order to obtain the initial estimates on the error term, we use the results of [3, Lemma 4.4]. As in the polynomial setup, these results require introducing a small regularization $\omega = i u$ with $u>0$ in order to use the stability theory of the MDE (A.38). The bounds in [13, Lemma 5.2] are uniform in this regularization and are a consequence of the a priori estimate [13, Lemma 2.5]. In our current setting, they remain true when this a priori estimate is replaced by (A.18).

3. Effective replacement of the Ward identity. Using again the a priori bound (A.18) instead of [13, Lemma 2.5], one can obtain the Ward identity type estimates (see, e.g., [13, formula (5.13)]) for the error terms involving the generalized resolvent.

4. Finishing the proof. With the above modifications, the proof of Theorem A.18 can be obtained by following the proof of [13, Theorem 5.1] line by line. $\square$

In the same spirit, we can follow line by line the proof of [13, Proposition 2.17], replacing the use of [13, Lemma 2.5] with the trivial bound (A.18), to obtain the following global law for (the linearizations of) rational expressions. Note that in the proof of Proposition A.19, we do not have to assume that the conditions (M1)–(M2) hold as stated in Theorem A.18. For the global law the boundedness of $M^{(sc)}_z$ and ${\mathscr {L}}_z^{-1}$ is needed only for z away from the real axis, in which case it follows from (A.18) and the representation [13, equation (5.22)] of ${\mathscr {L}}_z^{-1}$.

Proposition A.19

(Global law). For any $\theta >0$, uniformly on $\{\,z\, : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$

$$\begin{aligned}&\max _{1\le i,j \le N} \Big \Vert G_{z,ij}- M^{(\mathrm {sc})}_{z} \delta _{ij} \Big \Vert _{{\mathbb {C}}^{m\times m}} \prec _{\theta } \frac{1}{\sqrt{N}} ,\nonumber \\&\Big \Vert \frac{1}{N} \sum _{i=1}^{N} G_{z,ii} - M^{(\mathrm {sc})}_{z} \Big \Vert _{{\mathbb {C}}^{m\times m} } \prec _{\theta } \frac{1}{N} , \end{aligned}$$

(A.78)

where $\prec _{\theta }$ indicates that the constant in the definition of the stochastic domination (see Definition A.17) may depend on $\theta$. In particular, uniformly on $\{\,z\, : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$

$$\begin{aligned} \Big |\frac{1}{N} {\mathrm {Tr}}\varvec{g}_{z} - M^{(\mathrm {sc})}_{z}(1,1) \Big | \prec _{\theta } \frac{1}{N} . \end{aligned}$$

(A.79)

The global law for the trace of the resolvent of a rational expression, ${\mathrm {Tr}}\varvec{g}_{z}$, has already been proven in [39] with a somewhat different method; we comment on this point in Remark A.20.

Proof

Firstly we show that for any rational expression q as defined at the beginning of this section and any $\theta >0$, the bounds (M1)–(M2) from Theorem A.18 hold uniformly on $\{\,z\, : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$, namely that for any $\theta >0$ there exist $C_{\theta }>0$ such that for $\{\,z\, : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$

$$\begin{aligned} \Vert M^{(\mathrm {sc})}_{z}\Vert _{{\mathbb {C}}^{m \times m} } \le C_{\theta }, \qquad \Vert {\mathscr {L}}_{z}^{-1} \Vert _{{\mathbb {C}}^{m \times m} \rightarrow {\mathbb {C}}^{m \times m} } \le C_{\theta } . \end{aligned}$$

(A.80)

The first estimate in (A.80) follows directly from (A.26) and $(\mathrm {Im}z)^{-1}\le \theta$. In order to obtain the second estimate, we use the identity

$$\begin{aligned} {\mathscr {L}}_{z}[\,R\,]= & {} ({\mathrm {id}}\otimes \,\tau _{{\mathcal {S}}}) \bigg ( \Big (\varvec{L}^{\mathrm {sc}} -z J_m \otimes \mathbb {1}_{{\mathcal {S}}}\Big )^{-1} \Big ( (M_z^{\mathrm {sc}})^{-1}\, R\, (M_{z}^{\mathrm {sc}})^{-1} \otimes \mathbb {1}_{{\mathcal {S}}}\Big )\nonumber \\&\Big (\varvec{L}^{\mathrm {sc}} -z J_m \otimes \mathbb {1}_{{\mathcal {S}}}\Big )^{-1}\bigg ) \end{aligned}$$

(A.81)

for all ${\mathbb {R}}\in {\mathbb {C}}^{m\times m}$, where $\varvec{L}^{\mathrm {sc}}:=\varvec{L}(\varvec{s},\varvec{c}, \varvec{c}^*)$ is the linearization of the rational expression q evaluated on the freely independent semicircular and circular elements (see the proof of [13, Proposition 2.17] for the derivation of this identity). The boundedness of $(M_{z}^{\mathrm {sc}})^{-1}$ follows from the boundedness of $M_{z}^{\mathrm {sc} }$, the Dyson equation (A.24) and $|z|<\theta$, while the trivial bound in Lemma A.10 implies the boundedness of $\Vert (\varvec{L}^{\mathrm {sc}} -z J_m \otimes \mathbb {1}_{{\mathcal {S}}})^{-1}\Vert _{{\mathbb {C}}^{m\times m} \otimes {\mathcal {S}}}$ holding polynomially in $\theta$. Combining this with (A.81) and $(\mathrm {Im}z)^{-1} \le \theta$, we obtain the second inequality in (A.80). Note that in the above argument, we rely on the fact that $(\varvec{s},\varvec{c})\in {\mathcal {D}}_{\varvec{q}_{0},\ldots ,\varvec{q}_{n};C}({\mathcal {S}})$.

Now we can proceed with the proof (A.78) by following the argument used in Theorem A.18 but restricting the analysis to a subset $\{\,z\, : \, \mathrm {Im}z \ge \theta ^{-1}, \, |z| \le \theta \,\}$ bounded away from the real line. The concentration inequalities (A.78) then follow from (A.77) with $(\mathrm {Im}z)^{-1} \le \theta$. Finally, taking the (1,1)-component in the averaged global law for the linearization in (A.78) yields (A.79). $\square$

From (A.27), the function $M^{(\mathrm {sc})}_{z}(1,1)$ is a Stieltjes transform of a probability measure $\langle e_1, V(d\lambda ) e_1 \rangle$, which together with Proposition A.19 implies that in probability (and almost surely) the empirical spectral measure of $q(\varvec{X},\varvec{Y},\varvec{Y}^*)$ converges weakly to $\langle e_1, V(d\lambda ) e_1 \rangle$ as $N\rightarrow \infty$.

Remark A.20

In [39] an induction argument on the height of rational functions (similar as in the proof of Lemma A.15) was used to show, that the trace and the norm of a rational expression in GUE (or Wigner) matrices converges almost surely to the trace and norm of the same rational expression in semicircular elements. This result implies in particular the global law for self-adjoint rational expressions in Wigner matrices as in (A.79) (without giving the optimal convergence rate) and the convergence of the (pseudo)spectrum. We use a slightly different approach compared to [39], in particular in the proof of Lemma A.15, by working with the linearization matrix and the generalized resolvent rather than directly with the rational expressions. This allows us to prove not only the global law but also the local laws for the linearized models in Sect. A.6, leading to the global and local laws for the rational expressions (A.77) and (A.79) with a precise control of the convergence speed. This also allows us to link the main outcomes of the present paper with the results about the convergence of the (pseudo)spectrum and the global and local laws established in [3, 13].

Remark A.21

The results of Sects. A.3–A.6 can be extended to the setting when the NC variables $x_{1},\ldots ,x_{\alpha _*},y_1,\ldots ,y_{\beta _*}$ are replaced by the matrices of NC variables

$$\begin{aligned} x_{\alpha } = \big (x_{\alpha }(i,j)\big )_{1\le i,j\le l_{\alpha }}\in {\mathcal {A}}^{l_{\alpha }\times l_{\alpha }} , \quad y_{\beta } = \big (y_{\beta }(i,j)\big )_{\begin{array}{c} 1\le i \le l_{\beta } \\ 1 \le j \le k_{\beta } \end{array}}\in {\mathcal {A}}^{l_{\beta }\times k_{\beta }}\nonumber \\ \end{aligned}$$

(A.82)

and $q(\varvec{x},\varvec{y},\varvec{y}^*)\in {\mathcal {A}}^{l_{q}\times l_{q}}$ with properly chosen dimensions that make the corresponding matrix operations well defined. The linearization of such model has the same structure (A.6) and we can analyze it by considering the matrix entries $x_{\alpha }(i,j)$, $1\le i,j\le k_{\alpha }$, and $y_{\beta }(i,j)$, $1\le i \le l_{\beta }, 1 \le j \le k_{\beta }$, as the new NC variables. Note that in this case the diagonal entries of $x_{\alpha }$ are self-adjoint.

In order to relate the spectrum of the rational expression $q(\varvec{x},\varvec{y},\varvec{y}^*)$ to its linearization, the generalized resolvent should be replaced by

$$\begin{aligned} z\mapsto (\varvec{L}- z \varvec{J}_{l_q})^{-1} . \end{aligned}$$

(A.83)

where $\varvec{J}_{l_{q}} = \sum _{i=1}^{l_{q}} E_{ii} \in {\mathbb {C}}^{m\times m}$ with $E_{ij}$ being the standard basis of ${\mathbb {C}}^{m\times m}$ and m the dimension of the linearization. With the above definition the upper-left $l_{q}\times l_{q}$ corner of $(\varvec{L}- z \varvec{J}_{l_q})^{-1}$ is equal to $(q(\varvec{x},\varvec{y},\varvec{y}^*) - I_{l_{q}}\otimes \mathbb {1}_{{\mathcal {A}}})^{-1}$, the resolvent of the rational expression $q(\varvec{x},\varvec{y},\varvec{y}^*)$.

We can then proceed with the study of the corresponding random matrix model, with $x_{\alpha }(i,i)$ independent Wigner matrices and all other elements being independent i.i.d. matrices. The linearization of this model is again a Kronecker random matrix; therefore, all the probabilistic estimates in the above analysis remain valid. The proof of the main results of Sects. A.3–A.6 follows the same lines as in the $l_{q}=l_{\alpha } = l_{\beta } = k_{\beta } = 1$ setting; the main changes are notational and are reduced to incorporating the additional structure (A.82) and (A.83).

Norm Bounds for Random Sample Covariance Matrices

In this section, we prove the norm bounds for random sample covariance ensembles that are used frequently in the paper. Although various forms of these bounds are well known in the literature (see, e.g., [6, 17] or [33]), we provide a short alternative proof to make sure that the assumptions about the random matrix ensembles and the probabilistic estimates match the setting of the present paper.

Lemma B.1

Let $l,m \in {\mathbb {N}}$ and let $W\in {\mathbb {C}}^{\,l n\times m n}$ be a (non-Hermitian) random matrix with independent centered entries of variance 1/(ln). Suppose that the entries of W have finite moments of all orders. Then for any $m<l$ and $\delta >0$, as $n\rightarrow \infty$, a.w.o.p.

$$\begin{aligned} \big \Vert (WW^*)^{-1} \big \Vert _{{\mathbb {C}}^{ln\times ln}} \le \frac{1}{\Big (1-\frac{1}{\sqrt{m/l}}\Big )^{2}(1-\delta )} . \end{aligned}$$

(B.1)

Proof

Let ${\widehat{W}}_{ij}\in {\mathbb {C}}^{n\times n}$, $1\le i \le l$, $1 \le j \le m$, be independent $n\times n$ random matrices having i.i.d. centered entries with variance 1/n, so that W can be written as a block matrix

$$\begin{aligned} W=\left( \frac{1}{\sqrt{l}}{\widehat{W}}_{ij}\right) _{\begin{array}{c} i = 1\ldots l \\ j=1 \ldots m \end{array}} . \end{aligned}$$

(B.2)

Denote $Q:=WW^*$. For any $z \in {\mathbb {C}}$, the linearization of $Q- zI_{ln}$ has a simple structure

$$\begin{aligned} \varvec{L}_{Q,z} = \left( \begin{array}{cc} -zI_{ln} &{}\quad W \\ W^* &{}\quad -I_{mn} \end{array} \right) , \end{aligned}$$

(B.3)

and using the representation (B.2) can be viewed as a Kronecker matrix in i.i.d. matrices ${\widehat{W}}_{ij}$. Denote also

$$\begin{aligned} \varvec{L}_{Q,z}(\varvec{c},\varvec{c}^*) := \left( \begin{array}{cc} -zI_{l}\otimes \mathbb {1}_{{\mathcal {S}}}&{}\quad W^{\mathrm {(c)}} \\ (W^{\mathrm {(c)}})^* &{}\quad -I_{m}\otimes \mathbb {1}_{{\mathcal {S}}}\end{array} \right) , \end{aligned}$$

(B.4)

where $W^{\mathrm {(c)}}:=(\frac{1}{\sqrt{l}}c_{ij})_{\begin{array}{c} i=1\ldots l \\ j=1 \ldots m \end{array}}$, and $\{c_{ij}\}$ is a family of freely independent circular elements. By using the properties of the Kronecker random matrices from [3, Theorems 2.4 and 6.1], we get that for any $\varepsilon \in (0,1)$ a.w.o.p.

$$\begin{aligned} \mathrm {Spec}\,(W W^*) \subset \big \{ z \, : \, \mathrm {dist}\big (0,\mathrm {supp}\, \rho ^{z}\big ) \le \varepsilon \big \} , \end{aligned}$$

(B.5)

where $\rho ^{z}$ is the self-consistent density of states satisfying $\mathrm {supp}\, \rho ^{z} = \mathrm {Spec}\,(\varvec{L}^{Q,z}_{0}(\varvec{c},\varvec{c}^*))$ and

$$\begin{aligned} \varvec{L}^{Q,z}_{\omega }(\varvec{c},\varvec{c}^*) = \left( \begin{array}{cc} -\omega I_{l+m}\otimes \mathbb {1}_{{\mathcal {S}}}&{}\quad \varvec{L}_{Q,z}(\varvec{c},\varvec{c}^*) \\ \big (\varvec{L}_{Q,z}(\varvec{c},\varvec{c}^*)\big )^{*} &{} \quad -\omega I_{l+m}\otimes \mathbb {1}_{{\mathcal {S}}}\end{array} \right) . \end{aligned}$$

(B.6)

Inclusion (B.5) can be obtained by repeating the argument from Lemma A.15 leading to (A.55), and taking into account the $(l+m)\times (l+m)$ decomposition in (B.3) and (B.4).

Fix $\delta \in (0,1)$. Then, we claim that there exists a sufficiently small $\varepsilon >0$ such that

$$\begin{aligned} \big \{ z \, : \, \mathrm {dist}\big (0,\mathrm {supp}\, \rho ^{z}\big ) \le \varepsilon \big \} \cap \Big \{z \, : \, \mathrm {Re}z < \bigg (1-\frac{1}{\sqrt{m/l}} \bigg )^{2}(1-\delta ) \Big \} = \emptyset . \end{aligned}$$

(B.7)

Indeed, $W^{\mathrm {(c)}} (W^{\mathrm {(c)}})^*$ follows the free Poisson distribution with rate $l/m > 1$ (equivalent to Marchenko–Pastur distribution with parameter $m/l<1$) with

$$\begin{aligned} \mathrm {Spec}\, (W^{\mathrm {(c)}} (W^{\mathrm {(c)}})^*) = \Big [\Big (1-\sqrt{m/l}\Big )^2,\Big (1+\sqrt{m/l}\Big )^2\Big ] . \end{aligned}$$

(B.8)

There exists $C>0$ such that for any $z\in {\mathbb {C}}$ with $\mathrm {Re}z< (1-\sqrt{m/l})^2(1-\delta )$

$$\begin{aligned} \Big \Vert \Big ( W^{\mathrm {(c)}} (W^{\mathrm {(c)}})^* - z I_l\otimes \mathbb {1}_{{\mathcal {S}}}\Big )^{-1}\Big \Vert _{{\mathbb {C}}^{l\times l}\otimes \, {\mathcal {S}}} \le C , \end{aligned}$$

(B.9)

and thus, by the Schur complement formula, there exists ${\widetilde{C}}>0$ depending only on C, such that

$$\begin{aligned} \big \Vert \big ( \varvec{L}_{Q,z}(\varvec{c},\varvec{c}^*) \big )^{-1}\big \Vert _{{\mathbb {C}}^{(l+m)\times (l+m)}\otimes \, {\mathcal {S}}} \le {\widetilde{C}} . \end{aligned}$$

(B.10)

From (B.6) and (B.10), we get that the resolvent of the Hermitized matrix (B.6) is bounded at $\omega =0$

$$\begin{aligned} \big \Vert \big ( \varvec{L}^{Q,z}_0(\varvec{c},\varvec{c}^*) \big )^{-1}\big \Vert _{{\mathbb {C}}^{2(l+m)\times 2(l+m)}\otimes \, {\mathcal {S}}} = \big \Vert \big ( \varvec{L}_{Q,z}(\varvec{c},\varvec{c}^*) \big )^{-1}\big \Vert _{{\mathbb {C}}^{(l+m)\times (l+m)}\otimes \, {\mathcal {S}}} \le {\widetilde{C}} \end{aligned}$$

(B.11)

uniformly for $\mathrm {Re}z< (1-\sqrt{m/l})^2(1-\delta )$, from which (B.7) can be obtained using the resolvent identity (similarly as in (A.65)–(A.67)).

Finally, (B.5) and (B.7) imply that for any $\delta \in (0,1)$ a.w.o.p. the smallest eigenvalue of the positive definite matrix $WW^*$ is greater than $(1-1/\sqrt{m/l})^2(1-\delta )$, from which (B.1) follows. $\square$