CLT for Non-Hermitian Random Band Matrices with Variance Profiles

Abstract

We study the linear eigenvalue statistics of a non-Hermitian random band matrix with a continuous variance profile $w_{\nu }(x)$ and increasing bandwidth $b_{n}$. We show that the fluctuations of the linear eigenvalue statistics converges to $N(0,\sigma _{f}^{2}(\nu ))$, where $\nu =\lim _{n\rightarrow \infty }(2b_{n}/n)\in [0,1]$ and f is an analytic test function. We obtain explicit formulae of $\sigma _{f}^{2}(\nu )$ in two different cases, namely when $\nu \in (0,1]$ and when $\nu = 0$. In addition, we show that $\sigma _{f}^{2}(\nu )\rightarrow \sigma _{f}^{2}(0)$ as $\nu \downarrow 0$. In particular by setting $\nu =1$, we obtain the result for full non-Hermitian matrices with a constant variance profile, which was previously found by Rider and Silverstein (Ann Probab 34:2118–2143, 2006).

Appendices

Appendix A

In this section, we discuss about the norm of non-Hermitian random matrices. A sharp almost sure bound on the spectral radius of non-Hermitian random matrices can be found in [20, 21].

Theorem A.1

[21] Let $M=(m_{ij})_{n\times n}$ be a sequence of $n\times n$ random matrices with $m_{ij};1\le i,j\le n$ real valued i.i.d. for each n. Assume that for each n,

• (i) $\mathbb {E}[m_{11}]=0$

• (ii) $\mathbb {E}[m_{11}^{2}]=\sigma ^{2}$

• (iii) $\mathbb {E}[|m_{11}|^{p}]\le p^{c p}$ for all $p\ge 2$ and for some $c>0$.

Let

$$\begin{aligned} \rho _{n}=\max _{1\le i\le n}\{|\lambda _{i}(M/\sqrt{n})|\}. \end{aligned}$$

Then $\limsup _{n\rightarrow \infty }\rho _{n}\le \sigma$ almost surely.

We see that in our context if a full random matrix satisfies the condition 3.1, then it also satisfies the above condition and as a result, $\rho _{n}\le 1$ almost surely as $n\rightarrow \infty$. However, the above theorem does not take a variance profile into account. The following theorem from [7] estimates the norm of a symmetric random matrix with a variance profile.

Theorem A.2

[7, Corollary 3.5] Let X be a $n\times n$ real symmetric matrix with $X_{ij}=\xi _{ij}w_{ij}$, where $\{\xi _{ij}:i\ge j\}$ are independent centered random variables and $\{w_{ij}:i\ge j\}$ are give scalars. If $\mathbb {E}[|\xi _{ij}|^{2p}]^{1/2p}\le C p^{\beta /2}$ for some $C,\beta >0$ and all p, i, j, then

$$\begin{aligned} \mathbb {E}\Vert X\Vert \le C'\max _{i}\sqrt{\sum _{j=1}^{n}w_{ij}^{2}}+C'\max _{i,j}|w_{ij}|\log ^{(\beta \wedge 1)/2}n, \end{aligned}$$

where C depends on $C,\beta$ only.

Using the above theorem along with Poincaré inequality, we have the following lemma.

Lemma A.3

Let M be an $n\times n$ random matrix as in the Theorem 3.2. Then there exists $\rho \ge 1$ such that

$$\begin{aligned} \mathbb {P}\left( \Vert M\Vert>\rho /4+t\right) \le K\exp \left( -\sqrt{\frac{\alpha c_{n}}{2\omega }}t\right) ,\;\;\forall \;t>0, \end{aligned}$$

(A.1)

where $K>0$ is a universal constant. In particular,

$$\begin{aligned}&\mathbb {P}(\Vert M\Vert >\rho \;\text {infinitely often})=0, \end{aligned}$$

(A.2)

$$\begin{aligned} \text {and}\;&\;\mathbb {E}[\Vert M\Vert ^{l}]\le \rho ^{l}+K\Gamma (l+1)\left( \frac{\alpha c_{n}}{2\omega }\right) ^{-l/2}\exp \left( \sqrt{\frac{\alpha c_{n}}{2\omega }}\frac{\rho }{4}\right) . \end{aligned}$$

(A.3)

Proof

First of all, we may write $M=M_{R}+iM_{I}$, where both $M_{R}$ and $M_{I}$ are real valued matrices. Then we can estimate $\Vert M\Vert \le \Vert M_{R}\Vert +\Vert M_{I}\Vert$. Therefore without loss of generality, let us consider M be a real valued matrix. Consider

$$\begin{aligned} \tilde{M}:=\left[ \begin{array}{cc} O &{} M\\ M^{*} &{} O \end{array} \right] , \end{aligned}$$

and apply Theorem A.2 on ${\tilde{M}}$ to obtain the same bound for $\mathbb {E}\Vert \tilde{M}\Vert (=\mathbb {E}\Vert M\Vert )$. In our case, $w_{ij}^{2}=\frac{1}{c_{n}}w_{\nu }((i-j)/n)$ or $w_{ij}^{2}=\frac{1}{c_{n}}w_{0}((i-j)/n)$ as described in Definition 2.1. Since $c_{n}\ge \log ^{3}n$ and w is a piece-wise continuous function, $\lim _{n\rightarrow \infty }\sum _{j=1}^{n}w_{ij}^{2}=\int w(x)\;dx=1$ and $\max _{i,j}|w_{ij}|\log ^{(\beta \wedge 1)/2}n\rightarrow 0$. As a result, there exists $\rho \ge 1$ such that

$$\begin{aligned} \limsup \mathbb {E}\Vert M\Vert \le \rho /4. \end{aligned}$$

Here we note that we need $c_{n}$ to grow at least as $\log n$. In fact, this is a sharp condition. Otherwise, the matrix norm may be unbounded [11, 27].

Secondly, $\Vert M\Vert \le \sqrt{\sum _{i,j}|m_{ij}|^{2}}$ implies that $h(M):= \Vert M\Vert$ is a Lipschitz$_{1}$ function. Therefore applying the properties of Poincaré inequality as described in Definition B.1, we have

$$\begin{aligned}&\mathbb {P}\left( |\Vert M\Vert -\mathbb {E}\Vert M\Vert |>t\right) \le K\exp \left( -\sqrt{\frac{\alpha c_{n}}{2\omega }}t\right) \\ \text {i.e.} \;&\;\mathbb {P}\left( \Vert M\Vert >\rho /4+t\right) \le K\exp \left( -\sqrt{\frac{\alpha c_{n}}{2\omega }}t\right) , \end{aligned}$$

where $\omega =\sup _{x}w(x)$.

Equation (A.2) can be seen from the equation (A.1) along with the application of Borel–Cantelli lemma with $t=3\rho /4$. Equation (A.3) can be justified as follows,

$$\begin{aligned} \mathbb {E}[\Vert M\Vert ^{l}]&=\int _{0}^{\rho /4}lu^{l-1}\mathbb {P}(\Vert M\Vert>u)\;du+\int _{\rho /4}^{\infty }lu^{l-1}\mathbb {P}(\Vert M\Vert >u)\;du\\&\le \rho ^{l}+Kl\exp \left( \sqrt{\frac{\alpha c_{n}}{2\omega }}\frac{\rho }{4}\right) \int _{\rho /4}^{\infty }u^{l-1}\exp \left( -\sqrt{\frac{\alpha c_{n}}{2\omega }}u\right) \;du\\&\le \rho ^{l}+K\Gamma (l+1)\left( \frac{\alpha c_{n}}{2\omega }\right) ^{-l/2}\exp \left( \sqrt{\frac{\alpha c_{n}}{2\omega }}\frac{\rho }{4}\right) . \end{aligned}$$

We would like to remark that the lemma is also true for the column removed matrices $M^{(k)}$, and the same proof will go through. $\square$

We finally would like to remark that although the Theorem A.2 gives a constant bound on the norm of the matrix with a variance profile, the constant is not that sharp unlike Theorem A.1. However, we expect that for matrices with continuous variance profile, the correct norm bound should be $\limsup _{n\rightarrow \infty }\sum _{i=1}^{n}w_{ij}^{2}$. This was remarked in [29, Remark 4.11]. In our case, this limit is equal to 1. We have mentioned in Remark 4.2 that eventually it suffices to take $z,\eta \in \partial \mathbb {D}_{1}$ only.

Appendix B

Here we list down some key results. Interested readers may find the proofs in the included references.

Definition B.1

(Poincaré inequality). A complex random variable $\xi$ is said to satisfy Poincaré inequality with constant $\alpha \in (0,\infty )$ if for any differentiable function $h:\mathbb {C}\rightarrow \mathbb {C}$, we have $\text {Var}(h(\xi ))\le \frac{1}{\alpha }\mathbb {E}[|h'(\xi )|^{2}]$. Here $\mathbb {C}$ is identified with $\mathbb {R}^{2}$.

Here are some properties of Poincaré inequality

• (1) If $\xi$ satisfies Poincaré inequality with constant $\alpha$, then $c\xi$ also satisfies Poincaré inequality with constant $\alpha / c^{2}$ for any $c\in \mathbb {R}\backslash \{0\}$.

• (2) If two independent random variables $\xi _{1},\xi _{2}$ satisfy the Poincaré inequality with the same constant $\alpha$, then for any differentiable function $h:\mathbb {C}^{2}\rightarrow \mathbb {C}$, $\text {Var}(h(\xi ))\le \frac{1}{\alpha }\mathbb {E}[\Vert \nabla h\Vert _{2}^{2}]$ i.e. $\xi :=(\xi _{1},\xi _{2})$ also satisfies Poincaré inequality with the same constant $\alpha$.

• (3) [4, Lemma 4.4.3] If $\xi \in \mathbb {C}^{N}$ satisfies Poincaré inequality with constant $\alpha$, then for any differentiable function $h:\mathbb {C}^{N}\rightarrow \mathbb {C}$

  $$\begin{aligned} \mathbb {P}(|h(\xi )-\mathbb {E}[h(\xi )]|>t)\le K\exp \left\{ -\frac{\sqrt{\alpha }t}{\sqrt{2}\Vert \Vert \nabla h\Vert _{2}\Vert _{\infty }} \right\} , \end{aligned}$$

  where $K=-2\sum _{i=1}^{\infty }2^{i}\log (1-2^{-2i-1})$. Here $\mathbb {C}^{N}$ is identified with $\mathbb {R}^{2N}$. In the above, $\Vert \nabla h(x)\Vert _{2}$ denotes the $\ell ^{2}$ norm of the N-dimensional vector $\nabla h(x)$ at $x\in \mathbb {C}^{N}$; and $\Vert \Vert \nabla h\Vert _{2}\Vert _{\infty }=\sup _{x\in \mathbb {C}^{N}}\Vert \nabla h(x)\Vert _{2}$. In particular if $h:\mathbb {C}^{N}\rightarrow \mathbb {C}$ is a Lipschitz function with lipschitz constant $\Vert h\Vert _{Lip}$, then $\Vert \Vert \nabla h\Vert _{2}\Vert _{\infty }\le \Vert h\Vert _{Lip}$.

For example, Gaussian random variables and compactly supported continuous random variables satisfy Poincaré inequality.

Lemma B.2

[10, Theorem 35.12] Let $\{\xi _{n,k}\}_{1\le k\le n}$ be a martingale difference array with respect to a filtration $\{\mathcal {F}_{k,n}\}_{1\le k\le n}$. Suppose for any $\delta >0$,

• (i) $\lim _{n\rightarrow \infty }\sum _{k=1}^{n}\mathbb {E}[\xi _{n,k}^{2}\mathbf{1} _{|\xi _{n,k}|>\delta }]=0$,

• (ii) $\sum _{k=1}^{n}\mathbb {E}[\xi _{n,k}^{2}|\mathcal {F}_{n,k-1}]{\mathop {\rightarrow }\limits ^{p}}\sigma ^{2}$ as $n\rightarrow \infty$.

Then $\sum _{k=1}^{n}\xi _{n,k}{\mathop {\rightarrow }\limits ^{d}}\mathcal {N}_{1}(0,\sigma ^{2})$.

Lemma B.3

([41], Sherman-Morrison formula). Let A and $A+ve_{k}^{t}$ be two invertible matrices, where $v\in \mathbb {C}^{n}$. Then

$$\begin{aligned} (A+ve_{k}^{t})^{-1}v=\frac{A^{-1}v}{1+e_{k}^{t}A^{-1}v}. \end{aligned}$$