1 Introduction

A (real-valued) square matrix X is asymmetric if it is different from its transpose \(X^T\), and non-normal if \(XX^T\ne X^TX\). Generically, asymmetric matrices are non-normal, and their eigenvalues are much more sensitive to the perturbations of the matrix entries than for their symmetric (hence selfadjoint and normal) counterparts. It is well known that non-normality may raise serious issues when calculating the spectra of such matrices numerically: keeping a fixed precision of calculations might not be sufficient, as some eigenvalues can be ‘ill-conditioned’.

To be more specific, we assume that X can be diagonalized (which for random matrices happens with probability one). Then to each eigenvalue \(\lambda _i\), real or complex (in the latter case being always accompanied by its complex conjugate partner \({\overline{\lambda }}_i\)) correspond two sets of eigenvectors, left \({\mathbf{l}}_i\) and right \({\mathbf{r}}_i\) which can always be chosen to be bi-orthogonal: \({\mathbf{l}}_i^* {\mathbf{r}}_{j}=\delta _{ij}\), where \({\mathbf{l}}_i^*:=\overline{{\mathbf{l}}_i^T}\) stands for Hermitian conjugation. The corresponding eigenproblems are \(X{\mathbf{r}}_i=\lambda _i{\mathbf{r}}_i\) and \(X^T {\mathbf{l}}_i=\lambda _i {\mathbf{l}}_i\). Consider now a matrix \(X'=X+\epsilon P\), where the second term represents an error one makes by storing the matrix entries with a finite precision, with \(\epsilon >0\) controlling the magnitude of the error and P reflecting the matrix structure of the perturbation. In the first order perturbation theory in parameter \(\epsilon \) eigenvalues are shifted by

$$\begin{aligned} |\delta \lambda _i|=\epsilon |{\mathbf{l}}_i^* P {\mathbf{r}}_i|\le \epsilon ||P|| \sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}. \end{aligned}$$
(1)

The latter quantity, \(\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}\), shows that the sensitivity of eigenvalues is essentially controlled by the non-orthogonality of the corresponding left and right eigenvectors. Correspondingly, in the numerical analysis context \(\kappa _i\) is called the eigenvalue condition number of the eigenvalue \(\lambda _i\) [41, 42]. Note also that the Cauchy–Schwarz inequality implies \(\kappa \ge 1\), with \(\kappa =1\) only when X is normal.

It is natural to ask how well-conditioned a ‘typical’ asymmetric matrix is. This question can be most meaningfully answered in the context of Random Matrix Theory (RMT), by defining ‘typical’ as randomly chosen according to a probability measure specified by a particular choice of the ensemble. The simplest yet nontrivial choice is to assume that all entries are mean-zero, independent, identically distributed (i.i.d.) Gaussian numbers. This defines the standard real Ginibre ensemble which we denote \(Gin_1\). Note that the question is equally interesting for matrices with entries that are complex rather than real, defining the complex Ginibre ensemble, which we denote \(Gin_2\). Note that for such an ensemble eigenvalues \(\lambda _i\) are purely complex with probability one.

It is the latter ensemble for which the study of the eigenvalue condition numbers has been initiated two decades ago by Chalker and Mehlig [8, 35]. More precisely, Chalker and Mehlig introduced a matrix of inner products \(O_{ij}=({\mathbf{l}}_i^*{\mathbf{l}}_j)({\mathbf{r}}_j^* {\mathbf{r}}_i)\), which they called ‘eigenvector overlaps’. The diagonal elements of that matrix are simply the squared eigenvalue condition numbers. They further associated with the diagonal elements of the overlap matrix the following single-point correlation function:

$$\begin{aligned} O_1(z)=\left\langle \frac{1}{N}\sum _{k=1}^{N}O_{kk}\delta (z-\lambda _k) \right\rangle _{Gin_2}. \end{aligned}$$
(2)

where the angular brackets stand for the expectation with respect to the probability measure associated with the complex Ginibre ensemble, and \(\delta (z-\lambda _k)\) is the Dirac delta mass at the eigenvalue \(\lambda _k\), so that the empirical density of eigenvalues in the complex plane z reads \(\rho (z)=\frac{1}{N}\sum _{k=1}^{N}\delta (z-\lambda _k).\)

Such \(O_1(z)\) gives the conditional expectations of (squared) \(\kappa \) as \({\mathbb {E}}(\kappa _i^2|z=\lambda _i)=\frac{O_1(z)}{\langle \rho (z)\rangle }\), where \(\langle \rho (z)\rangle \) is the mean spectral density around z [3]. It turned out that in the bulk of the spectrum of the complex Ginibre ensemble the magnitude of a typical diagonal overlap \(O_{ii}\) grows linearly with the size of the matrix N, so one needs to consider a rescaled object \(\tilde{O_1}(z)=\frac{1}{N}O_1(z)\) to obtain a non-trivial limit. In their influential papers [8, 35] Chalker and Mehlig used the ‘formal’ perturbation theory expansion to evaluate asymptotically, for \(N\gg 1\), both the diagonal overlap \(O_1(z)\) and its more general off-diagonal counterpart

$$\begin{aligned} O_2(z_1,z_2)=\left\langle \frac{1}{N}\sum _{k\ne l}^N {O}_{kl}\delta (z_1-\lambda _k)\delta (z_2-\lambda _l)\right\rangle _{Gin_2}\,. \end{aligned}$$

The first mathematically rigorous verification of the Chalker and Mehlig result for the diagonal overlap has been done in [43]. Remarkably, the function \(O_1(z)\) can be efficiently studied within the formalism of free probability [33] which recently allowed to extend the Chalker–Mehlig formulas to a broad class of invariant ensembles beyond the Gaussian case [3, 37]. \(O_1(z)\) is also known for a finite size of the complex Ginibre matrix [8, 35] and products of small Ginibre matrices [7]. It has been recently shown that for complex Ginibre matrices the one- and two-point functions conditioned on an arbitrary number of eigenvalues are related to determinantal point processes [2]. Various features characterizing the rich properties of eigenvectors of nonnormal random matrices have been also studied in [4, 9].

Here it is necessary to mention that the interest in statistical properties of the overlap matrix \(O_{kl}\) and related objects extends much beyond the issues of eigenvalue stability under perturbation and is driven by numerous applications in theoretical and mathematical physics. In particular, non-orthogonality governs transient dynamics in complex systems [30, 32, 40] (see also [16, 34]), analysis of spectral outliers in non-selfadjoint matrices [36], and, last but not least, the description of the Dyson Brownian motion for non-normal matrices [5, 6, 31]. Another steady source of interest in the statistics of eigenvector overlaps is due to its role in chaotic wave scattering. In that context \(O_1(z)\) and \(O_2(z_1,z_2)\) have been studied for a few special models different from Ginibre (both theoretically [19, 26, 27] and very recently experimentally [10, 11]) and in the associated models of random lasing [38, 39]. In the scattering context all eigenvalues are necessarily complex, and the lasing threshold is associated with the eigenvalue with the smallest imaginary part. For that special eigenvalue even the distribution of the overlap \(O_{ii}\) has been studied [39].

In fact, Chalker and Mehlig not only analyzed \(O_1(z)\), but also put forward a conjecture on the tail for the density \({{\mathcal {P}}}\left( {O}_{ii}\right) \) of the distribution of diagonal overlaps \({O}_{ii}\). Namely, based on the exactly solvable case of \(2\times 2\) matrices and numerical simulations for the complex Ginibre case they predicted that for large overlaps the density will exhibit a tail \({{\mathcal {P}}}\left( {O}_{ii}\right) \sim {O}^{-3}_{ii}\), making all the positive integer moments beyond \(O_1(z)\) divergent. This conjecture has been settled only recently with two different methods, by Bourgade and Dubach [5] (where some information about \({O}_{l\ne k}\) was also provided) and by Fyodorov [21]. The latter paper also revealed that for real eigenvalues of real Ginibre matrices \(Gin_1\) the diagonal overlaps \({O}_{ii}\) are distributed with even heavier tail: \({{\mathcal {P}}}\left( {O}_{ii}\right) \sim {O}^{-2}_{ii}\), making even the mean of the overlap divergent.

To address the above distributions it is convenient to introduce the following natural generalization of the equation (2):

$$\begin{aligned} {{\mathcal {P}}}_N(z,t)=\left\langle \sum _{i}^{}\delta (O_{ii}-1-t)\delta (z-\lambda _i)\right\rangle \end{aligned}$$
(3)

interpreted as the joint density function (JDF) of the ‘diagonal’ (or ‘self-overlap’) non-orthogonality factor \(t=O_{ii}-1\) for the right and left eigenvectors corresponding to eigenvalues in the vicinity of a point \(z=x+iy\) in the complex plane. The summation runs over all real eigenvalues for the real ensembles and over all eigenvalues for complex ensembles. As such, it is not a probability density, because it is normalized to the mean total number of (real) eigenvalues. As was shown in [5, 13, 21] the JDF \({{\mathcal {P}}}_N(z,t)\) tends (after an appropriate rescaling of the variables z and t with the size N) to the inverse gamma distribution as \(N\gg 1\):

$$\begin{aligned} \lim _{N\rightarrow \infty } N\,{{\mathcal {P}}}_N(z\sqrt{N},Nt)=\frac{ \langle \rho (z)\rangle }{t} e^{-\frac{{\tilde{O}}_1(z)}{t\langle \rho (z)\rangle }}\left( \frac{{\tilde{O}}_1(z)}{t\langle \rho (z)\rangle }\right) ^{\beta }, \quad |z|<1. \end{aligned}$$
(4)

Here parameter \(\beta =1\) corresponds to the real eigenvalues of real Ginibre matrices (in which case the parameter z should be chosen to be real), \(\beta =2\) to the complex Ginibre case and \(\beta =4\) to the quaternionic Ginibre case. Recall that in (4) the limiting spectral density of real eigenvalues for \(\beta =1\) is \(\langle \rho (z)\rangle =\frac{1}{2\sqrt{2\pi }}\) for the interval \(|z|<1\), whereas the limiting spectral density of complex eigenvalues for \(\beta =2\) is \(\langle \rho (z)\rangle =\pi ^{-1}\) inside the unit circle \(|z|<1\).

For complex Ginibre the limiting expression (4) naturally incorporates the Chalker-Mehlig result. In the formula above \({\tilde{O}}(z)=\pi ^{-1}(1-|z|^2)\), which is the large N limit of the rescaled one-point correlation function. Interestingly, despite the fact that for \(\beta =1\) the mean value \(O_1(z)\) defined via (2) does not exist, the closely related rescaled combination, \({\tilde{O}}_1(z)=\frac{1}{2\sqrt{2\pi }}(1-z^2)\), appears as a parameter in the inverse \(\gamma _1\) distribution and therefore defines the typical value of the diagonal overlap. Further calculations in a few non-Gaussian rotationally invariant matrix ensembles (in particular, associated with ‘truncations’ of unitary matrices) done very recently in [12] suggest that (4) might exhibit a certain degree of universality. Note that the statistics of \(O_{ii}\) for complex eigenvalues of real Ginibre matrices remains an outstanding problem, though it would be natural to expect that also in that case, for a fixed z with a non-vanishing imaginary part, the limit should be the same as for the complex Ginibre case.

Returning to the original question of eigenvalue condition numbers for real-valued matrices, the above results in particular imply that in contrast to well-conditioned eigenvalues of symmetric matrices with \(\kappa =1\) the typical condition numbers in fully asymmetric random matrices grow with matrix size as \(\sqrt{N}\) [21] and show strong fluctuations. One of natural questions is then to ask how those properties evolve for matrices with a controlled degree of asymmetry in their entries. The aim of this work is to answer this question. To this end we consider matrices with i.i.d. real Gaussian entries, such that the entries \(X_{ij}\) and \(X_{ji}\) are correlated. The joint pdf for the elements of this ensemble, known in the literature either as the real partly symmetric Ginibre ensemble, or, alternatively, as the real Elliptic Gaussian ensemble, is given by

$$\begin{aligned} P(X)dX=C_{N}^{-1} \exp \left[ -\frac{1}{2(1-\tau ^2)}Tr (XX^T-\tau X^2)\right] dX. \end{aligned}$$
(5)

Here \(dX=\prod _{i,j=1}^{N}dX_{ij}\) is the flat Lebesgue measure over all matrix elements and the normalization constant reads \(C_N=(2\pi )^{N^2/2}(1+\tau )^{N/2}(1-\tau ^2)^{\frac{N(N-1)}{4}}\). The parameter \(\tau \in [0,1]\) tunes the degree of correlation, \({\mathbb {E}}(X_{ij}X_{ji})=\tau \) for \(i\ne j\), and (5) interpolates between the Real Ginibre Ensemble for \(\tau =0\) and an ensemble of real symmetric matrices (Gaussian Orthogonal Ensemble) for \(\tau =1\). In particularly, it is well known that for large sizes \(N\gg 1\) a nontrivial scaling regime of weak non-Hermiticity arises as long as the product \(N(1-\tau )\) is kept of the order of unity [15, 23,24,25, 28]. It is this regime when non-normality gradually develops, and the condition numbers \(\kappa _i\) start growing away from the value \(\kappa _i=1\). Our considerations allow us to address this regime in a quantitative way.

2 Statement of the Main Results

It turns out that the method of evaluating the JDF in (3) suggested for the Ginibre case in [21] works for the Elliptic ensemble as well, though actual calculations turn out to be significantly more involved. Relegating technical detail to the rest of the paper, in this section we present our main findings.

Our main theorem gives the joint density function of the eigenvalue \(\lambda _i\) and the shifted overlap \(t=O_{ii}-1\) for elliptic matrices of a given size N distributed according to (5). It takes a more compact form when the rescaled variable \(q=(1-\tau )t\) is considered. Therefore, we define \({{\mathcal {P}}}_N^{\tau }(z,q) = (1-\tau )^{-1} {{\mathcal {P}}}_{N}(z,\frac{q}{1-\tau })\).

Theorem 2.1

Let \(X_N\) be an \(N\times N\) random matrix with the probability density function given by (5). Let us define the rescaled and shifted eigenvalue condition number \(q=(1-\tau )(\kappa ^2-1)\). The joint density (3) of a real eigenvalue z and the associated squared condition number expressed via the variable q is given by

$$\begin{aligned}&{{\mathcal {P}}}_N^{\tau }(z,q)=\frac{1}{2(1+\tau )\sqrt{2\pi } \Gamma \left( N-1 \right) } \frac{e^{-\frac{z^2}{2(1+\tau )}\left( 1+\frac{q}{1+q}\right) }}{\sqrt{q(1+q)}}\left( \frac{q}{q+1+\tau }\right) ^{\frac{N}{2}-1} \nonumber \\&\quad \times \left[ \frac{(1+\tau -2z^2)P_{N-2}+2z[R_{N-2}+\tau (N-2)R_{N-3}]}{1+q}\right. \nonumber \\&\quad +\frac{P_{N-2}z^2}{(1+q)^2}+\frac{\tau ^2(1+\tau )^2N(N-2)P_{N-3}}{(1+\tau +q)^2}\nonumber \\&\quad +\left. \frac{(1+\tau )(1-\tau ^2)(N-2)((N-2)P_{N-3}-T_{N-3})}{1+\tau +q} -\frac{2\tau (1+\tau )(N-2)zR_{N-3}}{(1+q)(1+\tau +q)}\right] , \end{aligned}$$
(6)

where the functions: \(P_m:=P_m(z), \, R_m:=R_m(z), \, T_m:=T_m(z)\) are defined in terms of the Hermite polynomials

$$\begin{aligned} \mathrm {He}_m(z)=\frac{(\pm i)^m}{\sqrt{2\pi }} e^{\frac{z^2}{2}}\int _{{\mathbb {R}}}t^m e^{-\frac{t^2}{2}\mp i z t}\mathrm{d}t, \end{aligned}$$
(7)

as

$$\begin{aligned} P_{N}(z)= & {} N!\sum _{k=0}^{N}\frac{\tau ^k}{k!}\left( (k+1)\mathrm {He}_k^2\left( \frac{z}{\sqrt{\tau }}\right) -k\mathrm {He}_{k-1} \left( \frac{z}{\sqrt{\tau }}\right) \mathrm {He}_{k+1}\left( \frac{z}{\sqrt{\tau }}\right) \right) , \end{aligned}$$
(8)
$$\begin{aligned} R_N(z)= & {} \frac{N!}{2}\sum _{k=0}^{N}\frac{\tau ^{k+\frac{1}{2}}}{k!}\left( (k+2)\mathrm {He}_{k+1} \left( \frac{z}{\sqrt{\tau }}\right) \mathrm {He}_{k} \left( \frac{z}{\sqrt{\tau }}\right) -k\mathrm {He}_{k+2}\left( \frac{z}{\sqrt{\tau }}\right) \mathrm {He}_{k-1}\left( \frac{z}{\sqrt{\tau }} \right) \right) , \nonumber \\ \end{aligned}$$
(9)
$$\begin{aligned} T_N(z)= & {} N!\sum _{k=0}^{N}\frac{k \tau ^k}{k!}\left( (k+1)\mathrm {He}_k^2\left( \frac{z}{\sqrt{\tau }}\right) -k\mathrm {He}_{k-1} \left( \frac{z}{\sqrt{\tau }}\right) \mathrm {He}_{k+1}\left( \frac{z}{\sqrt{\tau }}\right) \right) . \end{aligned}$$
(10)

Remark 2.2

Note that for \(\tau =0\) these quantities simplify to \(P_N=e^{z^2}\Gamma \left( N+1,z^2 \right) \), \(R_N=zP_N\) and \(T_N=Nz^2P_{N-1}\), with \(\Gamma \left( N+1,z \right) =\int _z^{\infty }u^Ne^{-x}\mathrm{d}u\), and the known result [21, eq. 2.5] is recovered.

Remark 2.3

The exact mean density of purely real eigenvalues \( \rho ^{(r)}_N(z)\) for the real Elliptic ensemble of even size N is known thanks to Forrester and Nagao [18]. It is given by \(\rho _N^{(r)}(z)=\rho _N^{(1)}(z)+\rho _N^{(2)}(z)\) with

$$\begin{aligned} \rho _{N}^{(1)}(z)&=\frac{1}{\sqrt{2\pi }}e^{-\frac{z^2}{1+\tau }}\sum _{k=0}^{N-2} \frac{\tau ^k}{k!}\mathrm{H}e_k^2\left( \frac{z}{\sqrt{\tau }}\right) , \end{aligned}$$
(11)
$$\begin{aligned} \rho _{N}^{(2)}(z)&=\frac{1}{\sqrt{2\pi }(1+\tau )\Gamma \left( N-1 \right) }e^{-\frac{z^2}{2(1+\tau )}}\tau ^{N-3/2}\mathrm{H}e_{N-1}\left( \frac{z}{\sqrt{\tau }}\right) \nonumber \\&\quad \int _0^z e^{-\frac{u^2}{2(1+\tau )}}\mathrm{H}e_{N-2}\left( \frac{u}{\sqrt{\tau }}\right) \mathrm{d}u. \end{aligned}$$
(12)

For odd N the density can be obtained using the method from [17]. Our expression (6) by its very definition must reproduce the Forrester-Nagao result after integration over the variable t. Performing such an integration analytically is, however, a challenging task which we managed to complete for \(N=2,3,4\). Nevertheless, we checked that performing such an integral numerically for moderate values of N gives results that are indistinguishable from the density of real eigenvalues, see “Appendix A’’.

As the expression (6) is exact, the joint density in the original variable, \({{\mathcal {P}}}_N(z,t)\), can be further analyzed in interesting scaling limits as \(N\rightarrow \infty \). The first of such limits is the so-called bulk scaling corresponding to the eigenvalues inside the limiting support of the spectrum which (after appropriate rescaling \(z\rightarrow \sqrt{N}z\)) for a fixed z and \(0\le \tau <1\) represents an ellipse in the complex plane (hence the name for the ensemble), centered at the origin, with semi-axis \(1-\tau \) along imaginary axis and \(1+\tau \) along the real axis. Since we are dealing only with real eigenvalues, we restrict ourselves to real z such that \(|z|<1+\tau \), where the following asymptotics holds:

Corollary 2.4

(bulk scaling) Define for a fixed \(0\le \tau <1\) and real z satisfying \(|z|<1+\tau \) the limiting scaled JDF as \({{\mathcal {P}}}_{bulk}(z,t)=\lim _{N\rightarrow \infty }N{{\mathcal {P}}}_N (\sqrt{N}z,Nt)\). Then

$$\begin{aligned} {{\mathcal {P}}}_{bulk}(z,t)=\frac{\sqrt{1-\tau ^2}}{2\sqrt{2\pi }} \frac{\left[ 1-\frac{z^2}{(1+\tau )^2}\right] }{t^2}e^{-\frac{1-\tau ^2}{2t} \left[ 1-\frac{z^2}{(1+\tau )^2}\right] }. \end{aligned}$$
(13)

This asymptotics shows that the typical value of the diagonal overlap \(t=O_{ii}-1\) in this regime is always of the order N as \(N\gg 1\), similarly to the behavior for the Ginibre case \(\tau =0\). Moreover, by recalling that the asymptotic density of real eigenvalues in the elliptic case is \(\left\langle \rho (z)\right\rangle =\frac{1}{\sqrt{2\pi (1-\tau ^2)}}\) and introducing \({\tilde{O}}_1(z)=\frac{\sqrt{1-\tau ^2}}{2\sqrt{2\pi }}(1-\frac{z^2}{(1+\tau )^2})\), we see that (13) is exactly of the form (4) for \(\beta =1\).

When approaching the boundary \(|z|=1+\tau \) of the eigenvalue support the typical diagonal overlap \({\tilde{O}}_1(z)\) tends to zero, and in the appropriate scaling vicinity of the boundary it becomes parametrically weaker, as the variable t in such a regime becomes of the order \(\sqrt{N}\):

Corollary 2.5

(edge scaling) Take a fixed \(0\le \tau <1\) and parameterize z and q as \(z=\sqrt{N}(1+\tau )+\delta _{\tau }\sqrt{1-\tau ^2}\) and \(q=\sigma \sqrt{N(1-\tau ^2)}\). Then the limit \({{\mathcal {P}}}_{edge}(\delta _{\tau },\sigma )=\lim _{N\rightarrow \infty } \sqrt{N}{{\mathcal {P}}}_N(z,q)\) exists and is equal to

$$\begin{aligned} {{\mathcal {P}}}_{edge}(\delta _{\tau },\sigma )=\frac{1}{4\pi \sigma ^2(1-\tau ^2)}e^{-\frac{1}{4\sigma ^2}+\frac{\delta _{\tau }}{\sigma }}\left[ e^{-2\delta _{\tau }^2}+\left( \frac{1}{\sigma }-2\delta _{\tau }\right) \int _{2\delta _{\tau }}^{\infty } e^{-\frac{u^2}{2}}\mathrm{d}u \right] . \nonumber \\ \end{aligned}$$
(14)

Note that this form is essentially the same as the one found for the real Ginibre case in [21].

Finally, the ultimate goal of our study is to investigate the weak non-Hermiticity regime occurring for \(N\rightarrow \infty \) when \(\tau \) approaches unity with the rate \(\mathcal {O}(N^{-1})\), so that the parameter \(2N(1-\tau )=a^2\) is fixed. Such parameter therefore controls the deviation from the fully symmetric limit. In this regime of ‘almost symmetric’ matrices already a finite fraction of eigenvalues of the order of N are real, and asymptotically their mean density is given by [15, 25]

$$\begin{aligned} \left\langle \rho (z)\right\rangle =\rho _{sc}(z)\int _0^{1} e^{-\frac{1}{2}As^2} \mathrm{d}s, \quad |z|<2, \end{aligned}$$
(15)

where \(\rho _{sc}(z)=\frac{1}{2\pi }\sqrt{4-z^2}\) is the standard Wigner semicircle density characterizing real symmetric GOE matrices, and \(A=(\pi \rho _{sc}(z)a)^2\).

As anticipated, such a regime turns out to be not only ‘weakly non-Hermitian’, but also ‘weakly non-normal’ as the typical value of the diagonal overlap \(t=O_{ii}-1\) turns out to be of the order of unity in the bulk of the spectrum, namely

Corollary 2.6

Let \(|z|<2\) and \(t\ge 0\) be fixed. Consider the limit \({{\mathcal {P}}}_{weak}(z,t)=\lim _{N\rightarrow \infty } N^{-1/2}{{\mathcal {P}}}_N(z\sqrt{N},t)\) with \(2N(1-\tau )=a^2\) fixed. Then

$$\begin{aligned} {{\mathcal {P}}}_{weak}(z,t)=\frac{A}{2}\rho _{sc}(z)\frac{e^{-\frac{A}{2t}}}{t^2}\int _0^1 e^{-\frac{1}{2}As^2}\left( 1+A+\frac{A}{t}-As^2\right) s^2\mathrm{d}s , \end{aligned}$$
(16)

where \(A=(\pi \rho _{sc}(z)a)^2\) and \(\rho _{sc}(z)=\frac{1}{2\pi }\sqrt{4-z^2}\). See also Fig. 1 for a plot of this distribution.

Remark 2.7

After integration by parts one can rewrite the above as

$$\begin{aligned} {{\mathcal {P}}}_{weak}(z,t)=\frac{A}{2}\rho _{sc}(z)\frac{e^{-\frac{A}{2t}}}{t^2}\left[ \left( \frac{2}{A}-\frac{1}{t}\right) e^{-\frac{A}{2}}+\left( 1+\frac{1}{t}-\frac{2}{A}\right) \int _0^1 e^{-\frac{1}{2}As^2} \mathrm{d}s\right] .\nonumber \\ \end{aligned}$$
(17)

From this form it is easy to check that \(\int _0^{\infty } {{\mathcal {P}}}_{weak}(z,t)\mathrm{d}t \) agrees with the mean density (15), as expected.

We thus conclude that the characteristic tail \({{\mathcal {P}}}_{weak}(z,t)\sim t^{-2}\) is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices, as it survives in the regime of arbitrary weak asymmetry as long as \(a>0\) (Fig. 1).

Fig. 1
figure 1

(Left) 3D plot of \(\mathcal {P}_{weak}(z,t)\). (Right) Section of the plot at \(z=0\) (red line) juxtaposed numerical diagonalization of \(2\cdot 10^4\) matrices of size \(N=500\)

Full size image

3 Derivation of the Main Results

We briefly outline an adaptation of the method of evaluating the JDF in (3) following [21] with necessary modifications.

3.1 Partial Schur Decomposition

Let \(\lambda \) be a real eigenvalue of an \(N\times N\) real matrix \(X_N\). Then it is well known, see, e.g., [14], that the matrix \(X_N\) can be decomposed as

$$\begin{aligned} X_N=O\left( \begin{array}{cc} \lambda &{}\quad {\mathbf{w}}^T \\ 0 &{}\quad X_{N-1} \end{array}\right) O^T=O{\tilde{X}}_NO^T, \end{aligned}$$
(18)

where \({\mathbf{w}}\) is a column vector with \(N-1\) components and \(X_{N-1}\) is a matrix of size \((N-1)\times (N-1)\). The matrix O is the Householder reflection matrix, parameterized as \(O=\mathbf {1}_N - 2{\mathbf{v}}{\mathbf{v}}^T\), where \(\mathbf {1}_N\) is the \(N\times N\) identity matrix and \({\mathbf{v}}\) is an N-dimensional vector of unit norm, with a positive first component. Note that although the left/right eigenvectors of \({\tilde{X}}_N\) corresponding to \(\lambda \) are different from those of \(X_{N}\), the inner products (hence, the eigenvalue condition numbers) are the same. Parameterizing these eigenvectors as \(\tilde{{\mathbf{r}}}_{\lambda }=(1,0,\ldots ,0)^T\) and \(\tilde{{\mathbf{l}}}_{\lambda }=(1,b_1,\ldots ,b_{N-1})=(1,\mathbf {b}^T)\), we immediately obtain for the associated condition number \(\kappa _{\lambda }^2=1+{\mathbf{b}}^T{\mathbf{b}}\). Demanding that \(\tilde{{\mathbf{l}}}_{\lambda }\) is the left eigenvector of \({\tilde{X}}_N\) leads us to the relation \({\mathbf{b}}=(\lambda -X_{N-1}^T)^{-1}{\mathbf{w}}\). As a consequence [21]

$$\begin{aligned} \kappa _{\lambda }^2=1+{\mathbf{w}}^T(\lambda -X_{N-1})^{-1}(\lambda -X_{N-1}^T)^{-1}{\mathbf{w}}. \end{aligned}$$
(19)

The Lebesgue measure on \(X_{N}\) can be decomposed as \(\mathrm{d}X_{N}=|\det (\lambda -X_{N-1})|\mathrm{d}\lambda \mathrm{d}{\mathbf{w}} \mathrm{d}X_{N-1} \mathrm{d}S({\mathbf{v}})\), where \(\mathrm{d}S({\mathbf{v}})\) denotes the uniform measure on the \((N-1)\)-hemisphere [14] (see also supplementary material of [22]).

It turns out to be more technically convenient to concentrate on evaluating a characteristic function \({{\mathcal {L}}}(z,p)=\left<\delta (z-\lambda ) e^{-p{\mathbf{b}}^T{\mathbf{b}}}\right>_{N}\) representing the Laplace transform of the JDF \({{\mathcal {P}}}(z,t)\). Hereafter by \(\left<\ldots \right>_{N}\) we denote the expected value with respect to the probability measure (5) for matrices \(X_N\) of size N.

Lemma 3.1

The characteristic function \({{\mathcal {L}}}(z,p)\) can be represented in the form

$$\begin{aligned} {{\mathcal {L}}}(z,p)=\frac{e^{-\frac{z^2}{2(1+\tau )}}}{2^{\frac{N}{2}}\Gamma \left( \frac{N}{2} \right) \sqrt{1+\tau }}\left<\frac{\det (z-X)(z-X^T)}{\det {}^{1/2}\left[ 2p(1-\tau ^2)+(z-X)(z-X^T)\right] }\right>_{N-1}.\nonumber \\ \end{aligned}$$
(20)

Proof

Substituting the decomposition (18) together with the associated decomposition of the Lebesgue measure into the probability measure of the elliptic ensemble (5) one can easily see that the ensemble average in (20) amounts to performing the following integral:

$$\begin{aligned} {{\mathcal {L}}}(z,p)&=C_{N}^{-1} e^{-\frac{z^2}{2(1+\tau )}}\int \exp \left[ -\frac{1}{2(1-\tau ^2)}Tr (X_{N-1}X_{N-1}^T-\tau X_{N-1}^2)\right] \nonumber \\&\quad \times \exp \left[ -\frac{1}{2(1-\tau ^2)} {\mathbf{w}}^T\left( 1+2p(1-\tau ^2)(z-X_{N-1})^{-1}(z-X^T_{N-1})^{-1}\right) {\mathbf{w}}\right] |\nonumber \\&\quad z-X_{N-1}|\mathrm{d}X_{N-1}\mathrm{d}{\mathbf{w}} \mathrm{d}S({\mathbf{v}}). \end{aligned}$$
(21)

The integral over \({\mathbf{v}}\) yields the volume of the \((N-1)\)-hemisphere \(\int \mathrm{d}S({\mathbf{v}}) =\frac{\pi ^{N/2}}{\Gamma \left( \frac{N}{2} \right) }\) [14]. The integral over \({\mathbf{w}}\) is Gaussian and can be easily performed, giving the factor

$$\begin{aligned}&\frac{ \left[ 2\pi (1-\tau ^2)\right] ^{\frac{N-1}{2}}}{\det {}^{1/2}[1+2p(1-\tau ^2) (z-X_{N-1})^{-1}(z-X_{N-1}^{T})^{-1}]}\nonumber \\&\quad =\frac{\left[ 2\pi (1-\tau ^2)\right] ^{\frac{N-1}{2}} \det {}^{1/2}(z-X_{N-1})(z-X_{N-1}^T)}{\det {}^{1/2}\left[ (2p(1-\tau ^2) +(z-X_{N-1})(z-X_{N-1}^T)\right] }. \end{aligned}$$
(22)

Taking all the multiplicative numerical constants into account and the factor \(|\det (z-X_{N-1})|\) from the Jacobian, we arrive at (20). \(\square \)

3.2 Ratio of Determinants

The problem has been therefore reduced to the calculation of the expectation for the ratio of two random determinants

$$\begin{aligned} D_{N}:=\left<\frac{ \det (z-X)(z-X^T)}{\det {}^{1/2}\left[ (2p(1-\tau ^2)+(z-X)(z-X^T)\right] }\right>_{N}, \end{aligned}$$
(23)

which is evaluated as

Theorem 3.2

$$\begin{aligned}&D_{N}=\frac{2^{-\frac{N}{2}}}{\sqrt{1+\tau }\Gamma \left( \frac{N}{2} \right) } \int _0^{\infty }\frac{\mathrm{d}t}{t} e^{-pt(1-\tau )} e^{-\frac{z^2t}{2(1+\tau )(1+t)}} \left( \frac{t}{1+t}\right) ^{1/2}\left( \frac{t}{1+\tau +t}\right) ^{\frac{N-1}{2}} \nonumber \\&\quad \times \left[ \frac{P_{N-1}(1+\tau -2z^2)+2z(R_{N-1}+\tau (N-1)R_{N-2})}{1+t}+ \frac{P_{N-1}z^2}{(1+t)^2}+ \frac{\tau ^2(1+\tau )^2(N^2-1)P_{N-2}}{(1+\tau +t)^2} \right. \nonumber \\&\quad +\left. \frac{(1+\tau )(1-\tau ^2)(N-1)[(N-1)P_{N-2}-T_{N-2}]}{1+\tau +t} -\frac{2\tau (1+\tau )(N-1)zR_{N-2}}{(1+t)(1+\tau +t)} \right] , \end{aligned}$$
(24)

where \(P_N\), \(R_N\) and \(T_N\) are defined in (8)–(10).

Remark 3.3

Theorem 3.2 immediately implies our main statement, Theorem 2.1: Indeed, by inserting (24) into (20) we see that \({{\mathcal {L}}}(z,p)\) is already represented as a Laplace transform, but of the rescaled variable \(t(1-\tau )\), and (6) follows.

The proof of Theorem 3.2 proceeds via employing the supersymmetry approach to ratios of determinants.

Proof

Let \(\chi , \rho , \theta , \eta \) denote N-component vectors in anticommuting (Grassmann) variables. This allows us to rewrite the determinant in the numerator as a standard Berezin Gaussian integral

$$\begin{aligned} \det (z-X)(z-X^T)=\int \mathrm{d}\chi \mathrm{d}\rho \mathrm{d}\theta \mathrm{d}\eta \exp \left[ -\chi ^T(z-X)\eta -\theta ^T(z-X^T)\rho \right] .\nonumber \\ \end{aligned}$$
(25)

The inverse square root of the determinant of a symmetric positive definite matrix A can be represented as a standard Gaussian integral. Namely, introducing N-component real vectors \({\mathbf{s}}_1,{\mathbf{s}}_2\) we can write

$$\begin{aligned}&\det {}^{-\frac{1}{2}}\left[ 2p(1-\tau ^2)+(z-X)(z-X^T)\right] \nonumber \\&\quad =\frac{1}{(2\pi )^{N}}\int \mathrm{d}{\mathbf{s}}_1 \mathrm{d}{\mathbf{s}}_2 \exp \left[ -\frac{1}{2}({\mathbf{s}}_1^T {\mathbf{s}}_2^T)\left( \begin{array}{cc} u &{}\quad i(z-X) \\ i(z-X^T) &{}\quad u \end{array}\right) \left( \begin{array}{c} {\mathbf{s}}_1 \\ {\mathbf{s}}_2 \end{array}\right) \right] , \end{aligned}$$
(26)

where we denoted \(u^2=2p(1-\tau ^2)\). This provides a representation of the right-hand side in (24) in the form

$$\begin{aligned} D_{N}&=\frac{1}{(2\pi )^N}\int \mathrm{d}\chi \mathrm{d}\rho \mathrm{d}\theta \mathrm{d}\eta \mathrm{d}{} \mathbf{s} _1 \mathrm{d}{} \mathbf{s} _2\nonumber \\&\quad \exp \left[ -z(\chi ^T\eta +\theta ^T\rho )-\frac{1}{2}(u{\mathbf{s}}_1^T{\mathbf{s}}_1+u {\mathbf{s}}_2^T{\mathbf{s}}_2+2i z {\mathbf{s}}_1^T{\mathbf{s}}_2)\right] \nonumber \\&\quad \times \left\langle e^{Tr X(\theta \rho ^T-\eta \chi ^T+i{\mathbf{s}}_2{\mathbf{s}}_1^T)}\right\rangle _N. \end{aligned}$$
(27)

The identity \(\left\langle e^{-Tr XA}\right\rangle _N=e^{\frac{1}{2}Tr (AA^T+\tau A^2)}\) allows us to perform the ensemble average. This in turn produces terms that are quartic in Grassmann variables, which we further bilinearize by employing a few auxiliary Gaussian integrals, the step known as the Hubbard–Stratonovich transformation:

$$\begin{aligned} e^{\theta ^T\eta \rho ^T\chi }= & {} \frac{1}{\pi }\int _{{\mathbb {C}}} \mathrm{d}^2a e^{-|a|^2+a\theta ^T\eta +{\bar{a}}\rho ^T\chi },\nonumber \\&\quad e^{\tau \chi ^T\theta \rho ^T\eta } = \frac{1}{\pi } \int _{{\mathbb {C}}} \mathrm{d}^2 b e^{-|b|^2+\sqrt{\tau }b\chi ^T\theta +\sqrt{\tau }{\bar{b}}\rho ^T\eta }, \end{aligned}$$
(28)
$$\begin{aligned} e^{-\frac{\tau }{2}(\rho ^T\theta )^2}= & {} \frac{1}{\sqrt{2\pi }} \int _{{\mathbb {R}}} e^{-\frac{c^2}{2}+ic\sqrt{\tau }\rho ^T\theta }\mathrm{d}c,\nonumber \\&\qquad e^{-\frac{\tau }{2}(\chi ^T\eta )^2} = \frac{1}{\sqrt{2\pi }} \int _{{\mathbb {R}}} e^{-\frac{f^2}{2}-if\sqrt{\tau }\chi ^T\eta }\mathrm{d}f, \end{aligned}$$
(29)

where we use the notation \(\mathrm{d}^2z=\mathrm{d}x\, \mathrm{d}y\) for \(z=x+iy\).

Applying these transformations converts all integrations over anticommuting variables into a Gaussian Berezin integral which we can write as \(\int \mathrm{d}\chi \mathrm{d}\rho \mathrm{d}\theta \mathrm{d}\eta e^{-\frac{1}{2}\xi ^T M \xi }\), where \(\xi ^T=(\chi ^T\eta ^T\theta ^T \rho ^T)\) and the antisymmetric matrix M is given by

$$\begin{aligned} M=\left( \begin{array}{cccc} 0 &{}\quad g-iA &{}\quad -b\sqrt{\tau } &{}\quad {\bar{a}} \\ -g+iA^T &{}\quad 0 &{}\quad a &{}\quad {\bar{b}}\sqrt{\tau } \\ b\sqrt{\tau } &{}\quad -a &{}\quad 0 &{}\quad h-i A^T \\ -{\bar{a}} &{}\quad -{\bar{b}}\sqrt{\tau } &{}\quad -h+iA &{}\quad 0 \end{array}\right) . \end{aligned}$$
(30)

Here we denoted \(g=z+if\sqrt{\tau }\), \(h=z+ic\sqrt{\tau }\) for brevity, and introduced the rank-two matrix \(A={\mathbf{s}}_1{\mathbf{s}}_2^T+\tau {\mathbf{s}}_2 {\mathbf{s}}_1^T\).

The Berezin Gaussian integration yields a Pfaffian of the matrix M. The Pfaffian is evaluated using the fact that through the Schur decomposition A is equivalent to \({\tilde{A}}\oplus 0_{N-2}\), where \(0_{N-2}\) denotes the matrix of size \(N-2\) filled with zeros, and

$$\begin{aligned} {\tilde{A}} = \begin{pmatrix} \lambda _1 &{} \quad t \\ 0 &{} \quad \lambda _2 \end{pmatrix}. \end{aligned}$$
(31)

As a consequence, \(Pf (M) = Pf (F)^{N-2} Pf (G)\), where the matrices of size 4 and 8 read explicitly

$$\begin{aligned}&F = \left( \begin{array}{cccc} 0 &{}\quad g &{}\quad -b\sqrt{\tau } &{}\quad {\bar{a}} \\ -g &{}\quad 0 &{}\quad a &{}\quad {\bar{b}}\sqrt{\tau } \\ b\sqrt{\tau } &{}\quad -a &{}\quad 0 &{}\quad h \\ -{\bar{a}} &{}\quad -{\bar{b}}\sqrt{\tau } &{}\quad -h &{}\quad 0 \end{array}\right) ,\nonumber \\&\qquad G=F\otimes \mathbf {1}_2 + \left( \begin{array}{cccc} 0 &{}\quad - i{\tilde{A}} &{}\quad 0 &{}\quad 0 \\ i {\tilde{A}}^T &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad i {\tilde{A}}^T \\ 0 &{}\quad 0 &{}\quad -i {\tilde{A}} &{}\quad 0 \end{array}\right) . \end{aligned}$$
(32)

The Pfaffian reads

$$\begin{aligned} \mathrm {Pf}(M)&=(|a|^2+\tau |b|^2+gh)^{N-2}\left[ (|a|^2+\tau |b|^2+gh)^{2}-(|a|^2+\tau |b|^2+gh)i(g+h)Tr A \right. \nonumber \\&\left. -|a|^2 Tr AA^T-\tau |b|^2Tr A^2-(g^2+h^2)\det A- gh(Tr A)^2-i (g+h) \det A \, Tr A+\det {}^2A\right] . \end{aligned}$$
(33)

We notice that

$$\begin{aligned} \begin{array}{cclccl} Tr A &{}=&{} (1+\tau ) {\mathbf{s}}_1^T{\mathbf{s}}_2,&{}\quad Tr A^2 &{}=&{} (1+\tau ^2)({\mathbf{s}}_1^T{\mathbf{s}}_2)^2+2\tau ({\mathbf{s}}_1^T{\mathbf{s}}_1)({\mathbf{s}}_2^T{\mathbf{s}}_2), \\ Tr AA^T &{}=&{} (1+\tau ^2)({\mathbf{s}}_1^T{\mathbf{s}}_1)({\mathbf{s}}_2^T{\mathbf{s}}_2) + 2\tau ({\mathbf{s}}_1^T{\mathbf{s}}_2)^2, &{} \quad \det A &{}=&{} \tau ({\mathbf{s}}_1^T{\mathbf{s}}_2)^2-\tau ({\mathbf{s}}_1^T{\mathbf{s}}_1)({\mathbf{s}}_2^T{\mathbf{s}}_2). \end{array} \end{aligned}$$

Therefore, the resulting expression depends on the vectors \({\mathbf{s}}_1\) and \({\mathbf{s}}_2\) only via their scalar products, so it is convenient to parameterize integrals by the entries of the associated Gram matrix [20, Section 2] [29, Theorem 1a in Appendix D]

$$\begin{aligned} {\hat{Q}}=\begin{pmatrix} Q_1 &{}\quad Q \\ Q &{}\quad Q_2 \end{pmatrix}, \qquad {\hat{Q}}_{ij}=({\mathbf{s}}^T_i{\mathbf{s}}_j), \quad i,j=1,2. \end{aligned}$$
(34)

The Jacobian of this change of variables is \((\det {\hat{Q}})^{\frac{N-3}{2}}\), while the integration over redundant angular variables yields the factor \(C_{N,2}^{(o)} =\frac{2^{N-2}\pi ^{N-1}}{\Gamma \left( N-1 \right) }\) [29]. The range of integration is restricted by the non-negativity conditions \(Q_1\ge 0,\,Q_2\ge 0, \det {\hat{Q}}=Q_1Q_2-Q^2\ge 0\). Following [21] it is convenient to change variables into \(r=(\det {\hat{Q}})^{1/2}\) and parameterize the integration region by \(Q_2=\frac{r^2+Q^2}{Q_1}\). The change of measure reads \(\mathrm{d}Q_1 \mathrm{d}Q_2 \mathrm{d}Q=2\frac{\mathrm{d}Q_1}{Q_1}r \mathrm{d}r \mathrm{d}Q\). After rescaling \(Q_1\rightarrow uQ_1\), we have

$$\begin{aligned} D_{N}&=\frac{1}{4\pi ^4\Gamma \left( N-1 \right) }\int _{{\mathbb {C}}} \mathrm{d}^2a\int _{{\mathbb {C}}}\mathrm{d}^2b\, e^{-|a|^2-|b|^2} \int _{{\mathbb {R}}^2}\mathrm{d}c\,\mathrm{d}f\, e^{-\frac{c^2}{2}-\frac{f^2}{2}} \nonumber \\&\quad \times \,\int _{-\infty }^{\infty }\mathrm{d}Q\int _{0}^{\infty }\frac{\mathrm{d}Q_1}{Q_1} \int _0^{\infty }r^{N-2}\mathrm{d}r \, e^{-\frac{1}{2}\left( u^2Q_1+\frac{r^2+Q^2}{Q_1}+2izQ+r^2+Q^2(1+\tau )\right) } {\mathrm{Pf}}(M). \end{aligned}$$
(35)

After the change of variables (34), \(\mathrm {Pf} (M)\) can be expressed as

$$\begin{aligned}&(|a|^2+\tau |b|^2+gh)^{N-2}\left[ (|a|^2+\tau |b|^2+gh)^2\right. \nonumber \\&\quad \left. -(|a|^2+\tau |b|^2+gh) (iQ(1+\tau )(g+h)+Q^2(1+\tau )^2)- \right. \nonumber \\&\quad \left. -|a|^2(1+\tau ^2)r^2-2\tau ^2r^2|b|^2+\tau r^2(g^2+h^2)-i\tau (g+h)Q(1+\tau )r^2+\tau ^2 r^4\right] . \end{aligned}$$
(36)

The integrals over a, b, c, f are performed in the following way. Let us denote

$$\begin{aligned} P_N=\frac{1}{2\pi ^3}\int \mathrm{d}^2a \mathrm{d}^2b \mathrm{d}c \mathrm{d}f e^{-|a|^2-|b|^2-\frac{c^2}{2}-\frac{f^2}{2}} (|a|^2+\tau |b|^2+gh)^{N}. \end{aligned}$$
(37)

Expanding the expression in the bracket and using the binomial theorem twice, we obtain

$$\begin{aligned} P_{N}=N!\sum _{k=0}^{N}\tau ^k\sum _{m=0}^{k}\frac{1}{m!}\mathrm {He}_{m}^2 \left( \frac{z}{\sqrt{\tau }}\right) , \end{aligned}$$
(38)

where \(\mathrm {He}_{m}(x)=(2\pi )^{-1/2}\int _{-\infty }^{\infty } e^{-\frac{y^2}{2}}(x+iy)^m \mathrm{d}y\) are the monic Hermite polynomials. The internal sum can be performed via the Christoffel–Darboux formula, finally yielding

$$\begin{aligned} P_N=N!\sum _{k=0}^{N}\frac{\tau ^k}{k!}\left( (k+1)\mathrm {He}_k^2\left( \frac{z}{\sqrt{\tau }}\right) -k \mathrm {He}_{k-1}\left( \frac{z}{\sqrt{\tau }}\right) \mathrm {He}_{k+1}\left( \frac{z}{\sqrt{\tau }}\right) \right) .\nonumber \\ \end{aligned}$$
(39)

Note that \(P_{N}\) can be interpreted as the expectation of the squared characteristic polynomial \(\left<\det (z-X)(z-X^T)\right>_{N}\) and in this capacity has been already studied for the Real Elliptic ensemble [1]. All other integrals over a, b, c, f in (35) are performed in a similar way. After exploiting the three term recurrence for Hermite polynomials \(\mathrm {He}_{N+1}(x)=x\mathrm {He}_{N}(x)-N\mathrm {He}_{N-1}(x)\), the integrals are evaluated to

$$\begin{aligned}&P_{N}-P_{N-1}Q^2(1+\tau )^2+r^4\tau ^2P_{N-2}\nonumber \\&-r^2[(N-1)(1+\tau ^2)+4\tau ^2] P_{N-2}-2iQ(1+\tau )R_{N-1} \nonumber \\&\quad + 2r^2\tau (z-iQ(1+\tau ))R_{N-2}+(1-\tau ^2)r^2T_{N-2}, \end{aligned}$$
(40)

where \(R_N\) and \(T_N\) are defined by (9) and (10). Note also that \(R_{N}(z)=\frac{1}{2(N+1)}\frac{\mathrm{d}P_{N+1}(z)}{\mathrm{d}z}\). It is convenient to exploit the structure of (40) and exponent in (35) and rescale further \(Q\rightarrow \frac{Q}{1+\tau }\) and, similarly, \(Q_1\rightarrow \frac{Q_1}{1+\tau }\). Recalling that \(u^2=2p(1-\tau ^2)\), one then arrives at

$$\begin{aligned} D_{N}&=\frac{1}{2\pi (1+\tau )\Gamma \left( N-1 \right) }\int _0^{\infty }\frac{\mathrm{d}Q_1}{Q_1}e^{-pQ_1(1-\tau )}\int _{{\mathbb {R}}}\mathrm{d}Q e^{-\frac{1}{2(1+\tau )}\left( Q^2\frac{1+Q_1}{Q_1}+2izQ\right) }\nonumber \\&\quad \int _0^{\infty }r^{N-2} \mathrm{d}r e^{-\frac{r^2}{2}\frac{Q_1+1+\tau }{Q_1}} \left[ P_N-P_{N-1}Q^2+r^4\tau ^2 P_{N-2}-r^2[(N-1)(1+\tau ^2)+4\tau ^2]P_{N-2} \right. \nonumber \\&\quad \left. -2iQR_{N-1}+2r^2\tau (z-iQ)R_{N-2}+ (1-\tau ^2)r^2T_{N-2}\right] . \end{aligned}$$
(41)

The integrations over Q and r are elementary. The integral over Q is Gaussian, while the one over r is of the type \(\int _0^{\infty }r^{N-2} e^{-ar^2/2}\mathrm{d}r= \frac{1}{2}\left( \frac{2}{a}\right) ^{\frac{N-1}{2}}\Gamma \left( \frac{N-1}{2} \right) \). The remaining integral over \(Q_1\) formally looks logarithmically divergent. To see the cancellation of the divergent part, one should exploit a non-trivial identity

$$\begin{aligned}&P_N-P_{N-1}(1+\tau -z^2)-(N-1)[2\tau ^2+N-1]P_{N-2}\nonumber \\&\quad -2zR_{N-1}+(1-\tau ^2)(N-1)T_{N-2}=0, \end{aligned}$$
(42)

which is verified in “Appendix B”. After elementary but tedious algebraic manipulations with the help of Mathematica in rearranging terms, we finally obtain

$$\begin{aligned} D_N&=\frac{2^{-\frac{N}{2}}}{\sqrt{1+\tau }\Gamma \left( \frac{N}{2} \right) } \int _0^{\infty } \frac{\mathrm{d}Q_1}{Q_1} e^{-pQ_1(1-\tau )}\nonumber \\&e^{-\frac{z^2Q_1}{2(1+\tau )(1+Q_1)}} \left( \frac{Q_1}{1+Q_1}\right) ^{\frac{1}{2}} \left( \frac{Q_1}{1+\tau +Q_1} \right) ^{\frac{N-1}{2}} \nonumber \\&\quad \times \left[ \frac{P_{N-1}(1+\tau -2z^2)+2z(R_{N-1}+\tau (N-1)R_{N-2})}{1+Q_1}\right. \nonumber \\&\quad +\frac{P_{N-1}z^2}{(1+Q_1)^2}+\frac{\tau ^2(1+\tau )^2(N^2-1)P_{N-2}}{(1+\tau +Q_1)^2} \nonumber \\&\quad \left. + \frac{(1+\tau )(1-\tau ^2)(N-1)[(N-1)P_{N-2}-T_{N-2}]}{1+\tau +Q_1} -\frac{2\tau (1+\tau )(N-1)zR_{N-2}}{(1+Q_1)(1+\tau +Q_1)} \right] . \end{aligned}$$
(43)

\(\square \)

3.3 Asymptotic Analysis

As \(P_N\), \(R_N\) and \(T_N\) are the building blocks of the determinant, we consider here their large-N asymptotics. First, we find convenient integral representations, which should allow the use of the Laplace method. For this we start from (39) and, using the integral representation for Hermite polynomials in (7) with both signs, we obtain

$$\begin{aligned} P_N=\frac{N!}{2\pi \tau }e^{\frac{z^2}{\tau }}\sum _{k=0}^{N}\frac{1}{k!} \int _{{\mathbb {R}}^2} \mathrm{d}t \mathrm{d}s e^{-\frac{t^2+s^2}{2\tau }-\frac{iz}{\tau }(t-s)} [(k+1)t^ks^k+k t^{k+1} s^{k-1}].\nonumber \\ \end{aligned}$$
(44)

The sum is evaluated using \(\sum _{k=0}^{N}\frac{x^k}{k!}=e^{x}\frac{\Gamma \left( N+1,x \right) }{\Gamma \left( N+1 \right) }\), where \(\Gamma \left( N+1,x \right) =\int _x^{\infty }u^{N}e^{-u}\mathrm{d}u\). This yields

Lemma 3.4

$$\begin{aligned} P_{N}(z)&=\frac{N!}{2\pi \tau } e^{\frac{z^2}{\tau }}\int _{{\mathbb {R}}^2} \mathrm{d}t \mathrm{d}s e^{-\frac{t^2+s^2}{2\tau }-\frac{iz}{\tau }(t-s)+ts}\nonumber \\&\quad \left( \frac{\Gamma \left( N+1,ts \right) }{N!}+t(s+t)\frac{\Gamma \left( N,ts \right) }{(N-1)!}\right) . \end{aligned}$$
(45)

An analogous procedure applied to \(T_N\) gives

Lemma 3.5

$$\begin{aligned} T_{N}= & {} \frac{N!}{2\pi \tau }e^{\frac{z^2}{\tau }}\int _{{\mathbb {R}}^2}\mathrm{d}t \mathrm{d}s e^{-\frac{t^2+s^2}{2\tau }-\frac{iz}{\tau }(t-s)+ts}\nonumber \\&\left( (t^2+2ts) \frac{\Gamma \left( N,ts \right) }{(N-1)!}+t^2s(t+s)\frac{\Gamma \left( N-1,ts \right) }{(N-2)!}\right) . \end{aligned}$$
(46)

3.3.1 Bulk Scaling

Let us give the proof of Corollary 2.4.

Proof

After rescaling \(z\rightarrow z\sqrt{N}\), \(t\rightarrow t\sqrt{N}\) and \(s\rightarrow s\sqrt{N}\), and then changing the integration variables \((t,s)\rightarrow (p,q)\) as \((t+s)/\sqrt{2}=p\) and \((t-s)/\sqrt{2}=q\) the equation (45) takes the following form:

$$\begin{aligned} P_N(z\sqrt{N})= & {} \frac{ N! N}{2\pi \tau }e^{N\frac{z^2}{\tau }}\int _{{\mathbb {R}}}\mathrm{d}p\,e^{-N\frac{p^2}{2} \left( \frac{1}{\tau }-1\right) } \int _{{\mathbb {R}}}\mathrm{d}q e^{-N\left( \frac{q^2}{2}\left( \frac{1}{\tau }+1\right) +\frac{iz\sqrt{2}}{\tau }q\right) } \nonumber \\&\times \left( \theta _{N}\left( \frac{p^2-q^2}{2}\right) +Np^2\theta _{N-1} \left( \frac{p^2-q^2}{2}\frac{N}{N-1}\right) \right) , \end{aligned}$$
(47)

where we denoted \(\theta _{N}(x)=\frac{\Gamma \left( N+1,Nx \right) }{\Gamma \left( N+1 \right) }\). Note that for any N this function is bounded: \(\theta _{N}(x)\le 1\), and in the limit \(N\rightarrow \infty \) for a fixed real x we have \(\theta _{N}(x)\rightarrow \theta _{\infty }(x)\), where \( \theta _{\infty }(x)=1\) for \(x<1\) and 0 otherwise.

For \(N\gg 1\) the integral over p can be most straightforwardly evaluated by the Laplace method, yielding that the leading contribution to \(P_N(z\sqrt{N})\) can be written as

$$\begin{aligned} P_N(z\sqrt{N})\sim & {} \frac{ N! \sqrt{N}}{\sqrt{2\pi \tau (1-\tau )}}\,e^{N\frac{z^2}{\tau }} \int _{{\mathbb {R}}}\mathrm{d}q e^{-N\left( \frac{q^2}{2}\left( \frac{1}{\tau }+1\right) +\frac{iz\sqrt{2}}{\tau }q\right) }\nonumber \\&\left( \theta _{N}\left( \frac{-q^2}{2}\right) +\frac{\tau }{1-\tau }\theta _{N-1} \left( \frac{-q^2}{2}\right) \right) . \end{aligned}$$
(48)

For large N the q-integral above can be performed (for a fixed, N-independent real value of z) by the standard saddle point method, with the saddle point position given by \(q=q_*:=-\frac{iz\sqrt{2}}{1+\tau }\) yielding the required asymptotic formula:

$$\begin{aligned} P_{N}(z\sqrt{N})\sim \frac{N!}{\sqrt{1-\tau ^2}(1-\tau )} e^{\frac{Nz^2}{1+\tau }} \theta _{\infty }\left( z^2/(1+\tau )^2\right) . \end{aligned}$$
(49)

The same type of reasoning applied to (46) gives

$$\begin{aligned} T_{N}(z\sqrt{N}) \sim \frac{N!}{(1-\tau ^2)^{3/2}}\frac{Nz^2}{1+\tau } e^{\frac{Nz^2}{1+\tau }}\theta _{\infty }\left( z^2/(1+\tau )^2\right) . \end{aligned}$$
(50)

Finally, the asymptotics

$$\begin{aligned} R_{N}(z\sqrt{N})\sim \frac{N!z\sqrt{N}}{(1-\tau ^2)^{3/2}} e^{\frac{Nz^2}{1+\tau }}\theta _{\infty }\left( z^2/(1+\tau )^2\right) \end{aligned}$$
(51)

is obtained from (49) using the fact that \(R_{N}(z)=\frac{1}{2(N+1)}\frac{\mathrm{d}P_{N+1}(z)}{\mathrm{d}z}\).

Upon inserting this asymptotics into (6) and rescaling \(q\rightarrow Nq\) it is clear that only the second to last term in the square bracket provides the leading order contribution, which happens to be

$$\begin{aligned}&\frac{(1+\tau )(1-\tau ^2)(N-1)[(N-1)P_{N-2}-T_{N-2}]}{1+\tau +Nq} \nonumber \\&\quad \sim \frac{(N-1)![(1+\tau )^2-z^2]}{q\sqrt{1-\tau ^2}} e^{\frac{Nz^2}{1+\tau }}. \end{aligned}$$
(52)

As a consequence, the joint pdf reads

$$\begin{aligned} N\mathcal {P}_N(z\sqrt{N},Nq)=\frac{\sqrt{\frac{1+\tau }{1-\tau }}(1-\frac{z^2}{(1+\tau )^2})}{2\sqrt{2\pi }q^2}e^{-\frac{1+\tau }{2q}\left( 1-\frac{z^2}{(1+\tau )^2}\right) }. \end{aligned}$$
(53)

Changing variables to \(t=\frac{q}{1-\tau }\), one immediately recovers Corollary 2.4. \(\square \)

3.3.2 Edge Scaling

When z is tuned to values parametrically close to \(z=\pm (1+\tau )\) where the step-function argument in equations (49)–(52) is close to unity with a distance \(\mathcal {O}(N^{-1/2})\), the corresponding asymptotics need to be evaluated with higher accuracy. Such a regime is known as the edge scaling, which features in Corollary 2.5 which we now prove.

Proof

In the proof we choose the vicinity of \(z=1+\tau \). Correspondingly, in (47) we now scale \(z=1+\tau +\frac{w}{\sqrt{N}}\), where w is of order 1. The transition from (47) to (48) remains the same as before. Now we use the integral representation of the incomplete gamma function \(\Gamma \left( N,x \right) =x^N\int _1^{\infty } u^{N-1} e^{-ux}\mathrm{d}u\), which helps to rewrite the integral (48) in the form

$$\begin{aligned} P_N(z)&\sim \frac{N! N^{N+1}}{4\pi \tau \Gamma \left( N \right) } e^{N\frac{z^2}{\tau }} \int _{{\mathbb {R}}} \mathrm{d}q\,\nonumber \\&\quad e^{-\frac{N(1+\tau )}{2\tau } (q^2+i2\sqrt{2}q) -\frac{iqw\sqrt{2N}}{\tau }} \int _1^{\infty }\mathrm{d}u\, e^{-\frac{Nu}{2}(-q^2)+N\ln \left[ \frac{u}{2}(-q^2)\right] }\times \nonumber \\&\quad \left( \frac{-q^2}{2}+\frac{\tau }{1-\tau }\frac{1}{u}\right) . \end{aligned}$$
(54)

An inspection shows that whereas the \(q-\)integration is dominated by the contribution from the saddle point \(q=-\sqrt{2}i\), the last u-integral is dominated by the vicinity of \(u=1\) of the width \(O(N^{-1/2})\). Parameterizing in such a vicinity \(u=1+\frac{v}{\sqrt{N}}\) one then arrives at the leading-order asymptotics

$$\begin{aligned} P_N\sim \frac{N! N^{N-1/2}}{(1-\tau )^2\Gamma \left( N \right) } e^{\frac{w^2}{1+\tau }+2w\sqrt{N}+N\tau } \int _0^{\infty } e^{-\frac{1+\tau }{2(1-\tau )}\left( v+\frac{2w}{1+\tau }\right) ^2}\mathrm{d}v. \end{aligned}$$
(55)

After the change of variables \(u=\sqrt{\frac{1+\tau }{1-\tau }}(v+\frac{2w}{1+\tau })\) and the use of Stirling’s approximation \(\Gamma \left( N+1 \right) \sim \sqrt{2\pi }N^{N+\frac{1}{2}} e^{-N}\), we obtain

$$\begin{aligned} P_{N}\sim \frac{N! e^{\frac{z^2}{1+\tau }}}{(1-\tau )\sqrt{2\pi (1-\tau ^2)}} \int \limits _{\frac{2w}{\sqrt{1-\tau ^2}}}^{\infty } e^{-\frac{u^2}{2}}\mathrm{d}u. \end{aligned}$$
(56)

The last integral is related to the complementary error function \(\mathrm {erfc}(x)=\frac{2}{\sqrt{\pi }}\int _x^{\infty } e^{-t^2}\mathrm{d}t\). Using \(R_N(z)=\frac{1}{2(N+1)}\frac{\mathrm{d}P_{N+1}(z)}{\mathrm{d}z}\) we obtain that in such a regime asymptotically \(R_{N}\sim \sqrt{N}P_{N}\). From the asymptotics of (52) one expects that the leading order contributions from \((N+1)P_N\) and \(T_N\) cancel. Therefore, one needs to work with the appropriate integral representation. Combining (45) and (46) and following the analogous reasoning as above we obtain

$$\begin{aligned} (N+1)P_N-T_N\sim \frac{N!\sqrt{N}}{\sqrt{2\pi }(1-\tau ^2)} e^{\frac{z^2}{1+\tau }}\left[ e^{-\frac{2w^2}{1-\tau ^2}}-\frac{2w}{\sqrt{1-\tau ^2}} \int _{\frac{2w}{\sqrt{1-\tau ^2}}}^{\infty } e^{-\frac{u^2}{2}}\mathrm{d}u\right] ,\nonumber \\ \end{aligned}$$
(57)

which is of the same order as \(\sqrt{N}P_N\), as expected. To get the correct asymptotics at the edge, we rescale \(q\rightarrow q\sqrt{N}\) in (6). It is now clear that the first term in the square bracket in (6) is subleading and the contribution of other terms is of the same order. The asymptotics of \(\left( \frac{q\sqrt{N}}{1+\tau +q\sqrt{N}}\right) ^{\frac{N}{2}-1}\) is calculated as

$$\begin{aligned} e^{-\frac{N}{2}\ln \left( 1+\frac{1+\tau }{q\sqrt{N}}\right) }\sim e^{-\frac{(1+\tau )\sqrt{N}}{2q}+\frac{(1+\tau )^2}{4q^2}}, \end{aligned}$$
(58)

and we obtain

$$\begin{aligned} \mathcal {P}\sim \frac{1}{4\pi q^2 \sqrt{N}}e^{\frac{w}{q}-\frac{1-\tau ^2}{4q^2}}\left[ e^{-\frac{2w^2}{1-\tau ^2}}+\left( \frac{\sqrt{1-\tau ^2}}{q}-\frac{2w}{\sqrt{1-\tau ^2}}\right) \int _{\frac{2w}{1-\tau ^2}} e^{-\frac{u^2}{2}} \mathrm{d}u\right] .\nonumber \\ \end{aligned}$$
(59)

After denoting \(w=\delta _{\tau }\sqrt{1-\tau ^2}\) and \(q=\sigma \sqrt{1-\tau ^2}\), the statement of Corollary 2.5 follows.

\(\square \)

3.3.3 Weak Non-Hermiticity

Our final goal is to provide the proof of Corollary2.6.

Proof

We start again from (47) keeping z fixed and \(N-\)independent like before in the bulk case, but for the weak non-Hermiticity regime we replace \(\tau =1-\frac{a^2}{2N}\). It is then immediately obvious that the p-integral is no longer dominated by the small vicinity \(p\sim N^{-1/2}\), but rather by the integration range of order unity. Then a quick inspection shows that for extracting the leading asymptotics in the large N limit one can effectively replace (47) by

$$\begin{aligned} P_N(z\sqrt{N})\sim \frac{ N! N^2}{2\pi \tau }e^{N\frac{z^2}{\tau }}\int _{{\mathbb {R}}}\mathrm{d}p\,p^2 e^{-\frac{p^2a^2}{4}} \int _{{\mathbb {R}}}\mathrm{d}q e^{-N\left( q^2+iz\sqrt{2}q\right) } \theta _{\infty }\left( \frac{p^2-q^2}{2}\right) . \end{aligned}$$
(60)

Performing the integral over q by the saddle point method, we see that the range of integration over p is given by \(|p|<\frac{\sqrt{4-z^2}}{\sqrt{2}}\). After a few simple changes of variables and straightforward manipulations one arrives at

$$\begin{aligned} P_N(z\sqrt{N})\sim \frac{N! N^{3/2}(4-z^2)^{3/2}}{2\sqrt{2\pi }} e^{\frac{Nz^2}{2}} \int _0^{1} e^{-\frac{s^2a^2}{2}\left( 1-\frac{z^2}{4}\right) } s^2 \mathrm{d}s. \end{aligned}$$
(61)

The asymptotic behavior of \(R_N\) simply follows from the relation \(R_{N}(z)=\frac{1}{2(N+1)}\frac{\mathrm{d} P_{N+1}(z)}{\mathrm{d}z}\) and is related to the asymptotics of \(P_N\) as \(R_N=\frac{z\sqrt{N}}{2}P_N\). Asymptotics of \(T_N\) analogously follows from its integral representation and reads

$$\begin{aligned} T_{N}(z\sqrt{N})\sim \frac{N!N^{5/2}(4-z^2)^{3/2}}{2\sqrt{2\pi }} e^{\frac{Nz^2}{2}}\int _0^1 e^{-\frac{s^2a^2}{2}\left( 1-\frac{z^2}{4}\right) } \frac{4s^4-z^2s^4+z^2s^2}{4}\mathrm{d}s. \end{aligned}$$
(62)

Note that in (6) we used the rescaled quantity \(q=(1-\tau )t\). Therefore, for the correct asymptotics, we need to rescale \(q\rightarrow \frac{2N}{a^2}t\). This shows that all terms in the square bracket are of the same order. Direct use of the asymptotic forms (61) and (62) leads to (16). \(\square \)