---
abstract: |
    For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves.

    Our result also generalizes earlier results of Götze–Tikhomirov [@GTprod] and O’Rourke–Soshnikov [@OS] concerning the product of independent iid random matrices.
address:
- 'Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309 '
- 'Department of Mathematics, UCLA '
- 'Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616-8633 '
- 'Department of Mathematics, Yale University, New Haven, CT 06520, USA'
author:
- 'Sean O’Rourke'
- David Renfrew
- Alexander Soshnikov
- Van Vu
title: Products of independent elliptic random matrices
---

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Introduction
============

We begin by recalling that the *eigenvalues* of a $N \times N$ matrix ${\mathbf}{M}$ are the roots in $\mathbb{C}$ of the characteristic polynomial $\det( {\mathbf}{M} - z {\mathbf}{I})$, where ${\mathbf}{I}$ is the identity matrix. We let $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ denote the eigenvalues of ${\mathbf}{M}$. In this case, the *empirical spectral measure* $\mu_{{\mathbf}{M}}$ is given by $$\mu_{\mathbf}{M} := \frac{1}{N} \sum_{i=1}^N \delta_{\lambda_i({\mathbf}{M})}.$$ The corresponding *empirical spectral distribution* (ESD) is given by $$F^{{\mathbf}{M}}(x,y) := \frac{1}{N} \# \left\{1 \leq i \leq N: {\operatorname{Re}}(\lambda_i({\mathbf}{M})) \leq x, {\operatorname{Im}}(\lambda_i({\mathbf}{M})) \leq y \right\}.$$ Here $\# E$ denotes the cardinality of the set $E$.

If the matrix ${\mathbf}{M}$ is Hermitian, then the eigenvalues $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ are real. In this case the ESD is given by $$F^{{\mathbf}{M}}(x) := \frac{1}{N} \# \left\{ 1 \leq i \leq N : \lambda_i({\mathbf}{M}) \leq x \right\}.$$

One of the simplest random matrix ensembles is the class of random matrices with independent and identically distributed (iid) entries.

Let $\xi$ be a complex random variable. We say ${\mathbf}{Y}_N$ is an $N \times N$ *iid random matrix* with atom variable $\xi$ if the entries of ${\mathbf}{Y}_N$ are iid copies of $\xi$.

When $\xi$ is a standard complex Gaussian random variable, ${\mathbf}{Y}_N$ can be viewed as a random matrix drawn from the probability distribution $${\mathbb{P}}(d {\mathbf}{M}) = \frac{1}{\pi^{N^2}} e^{- \operatorname{tr}({\mathbf}{M} {\mathbf}{M}^\ast)} d {\mathbf}{M}$$ on the set of complex $N \times N$ matrices. Here $d {\mathbf}{M}$ denotes the Lebesgue measure on the $2N^2$ real entries $$\{ {\operatorname{Re}}(m_{ij}) : 1 \leq i, j \leq N\} \cup \{ {\operatorname{Im}}(m_{ij}) : 1 \leq i, j \leq N\}$$ of ${\mathbf}{M}=(m_{ij})_{i,j=1}^N$. The measure ${\mathbb{P}}(d {\mathbf}{M})$ is known as the [*complex Ginibre ensemble*]{}. The [*real Ginibre ensemble*]{} is defined analogously. Following Ginibre [@Gi], one may compute the joint density of the eigenvalues of a random matrix ${\mathbf}{Y}_N$ drawn from the complex Ginibre ensemble. Indeed, $(\lambda_1({\mathbf}{Y}_N), \ldots, \lambda_N({\mathbf}{Y}_N))$ has density $$\label{eq:ginibre}
    p_N(z_1, \ldots, z_N) := \frac{1}{\pi^N \prod_{i=1}^N k! } \exp \left( - \sum_{k=1}^N |z_k|^2 \right) \prod_{1 \leq i < j \leq N} |z_i - z_j |^2.$$

Mehta [@M; @M:B] used the joint density function to compute the limiting spectral measure of the complex Ginibre ensemble. In particular, he showed that if ${\mathbf}{Y}_N$ is drawn from the complex Ginibre ensemble, then the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges to the [*circular law*]{} $F_{\mathrm{circ}}$ as $N \to \infty$, where $$F_{\mathrm{circ}}(x,y) := \mu_{\mathrm{circ}} \left( \left\{ z \in \mathbb{C} : {\operatorname{Re}}(z) \leq x, {\operatorname{Im}}(z) \leq y \right\} \right)$$ and $\mu_{\mathrm{circ}}$ is the uniform probability measure on the unit disk in the complex plane. Edelman [@Ed-cir] verified the same limiting distribution for the real Ginibre ensemble.

For the general (non-Gaussian) case, there is no formula for the joint distribution of the eigenvalues and the problem appears much more difficult. The universality phenomenon in random matrix theory asserts that the spectral behavior of an iid random matrix does not depend on the distribution of the atom variable $\xi$ in the limit $N \rightarrow \infty$. In other words, one expects that the circular law describes the limiting ESD of a large class of random matrices (not just Gaussian matrices).

An important result was obtained by Girko [@G1; @G2] who related the empirical spectral measure of a non-Hermitian matrix to that of a family of Hermitian matrices. Using this Hermitization technique, Bai [@Bcirc; @BSbook] gave the first rigorous proof of the circular law for general (non-Gaussian) distributions. He proved the result under a number of moment and smoothness assumptions on the atom variable $\xi$, and a series of recent improvements were obtained by Götze and Tikhomirov [@GTcirc], Pan and Zhou [@PZ] and Tao and Vu [@TVcirc; @TVesd]. In particular, Tao and Vu [@TVbull; @TVesd] established the law with the minimum assumption that $\xi$ has finite variance.

\[thm:tvcirc\] Let $\xi$ be a complex random variable with mean zero and unit variance. For each $N \geq 1$, let ${\mathbf}{Y}_N$ be a $N \times N$ iid random matrix with atom variable $\xi$. Then the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the circular law $F_{\mathrm{circ}}$ as $N \rightarrow \infty$.

More recently, Götze and Tikhomirov [@GTprod] consider the ESD of the product of $m$ independent iid random matrices. They show that, as the sizes of the matrices tend to infinity, the limiting distribution is given by $F_m$, where $F_m$ is supported on the unit circle in the complex plane and has density $f_m$ given by $$\label{plotnost}
    f_m(z) := \left\{
                \begin{array}{ll}
                    \frac{1}{m \pi} |z|^{\frac{2}{m}-2}, & \text{ for } |z|\leq 1 \\
                    0, & \text{ for } |z| > 1
                \end{array} \right.$$ in the complex plane. It can be verified directly, that if $\psi$ is a random variable distributed uniformly on the unit disk in the complex plane, then $\psi^m$ has distribution $F_m$.

\[thm:GTprod\] Let $m \geq 1$ be an interger, and assume $\xi_1, \ldots, \xi_m$ are complex random variables with mean zero and unit variance. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ iid random matrix with atom variable $\xi_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. Define the product $${\mathbf}{P}_N := N^{-m/2} {\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}.$$ Then ${\mathbb{E}}F^{{\mathbf}{P}_N}$ converges to $F_m$ as $N \to \infty$.

The convergence of $F^{{\mathbf}{P}_N}$ to $F_m$ in Theorem \[thm:GTprod\] was strengthened to almost sure convergence in [@B; @OS]. The Gaussian case was originally considered by Burda, Janik, and Waclaw [@BJW]; see also [@Bsurv]. We refer the reader to [@AB; @ABK; @AIK; @AIK2; @AKW; @AS; @BJLNS; @F; @F2] and references therein for many other interesting results concerning products of Gaussian random matrices.

New results
===========

In this paper, we generalize Theorem \[thm:GTprod\] by considering products of independent real elliptic random matrices. Elliptic random matrices were originally introduced by Girko [@Gorig; @Gten] in the 1980s.

\[def:elliptic\] Let $(\xi_1, \xi_2)$ be a random vector in $\mathbb{R}^2$, and let $\zeta$ be a real random variable. We say ${\mathbf}{Y}_N = (y_{ij})_{i,j=1}^N$ is a $N \times N$ *real elliptic random matrix* with atom variables $(\xi_1,\xi_2), \zeta$ if the following conditions hold.

-   (independence) $\{ y_{ii} : 1 \leq i \leq N\} \cup \{ (y_{ij}, y_{ji}) : 1 \leq i < j \leq N\}$ is a collection of independent random elements.

-   (off-diagonal entries) $\{ (y_{ij}, y_{ji}) : 1 \leq i < j \leq N\}$ is a collection of iid copies of $(\xi_1,\xi_2)$.

-   (diagonal entries) $\{y_{ii} : 1 \leq i \leq N\}$ is a collection of iid copies of $\zeta$.

Real elliptic random matrices generalize iid random matrices. Indeed, if $\xi_1, \xi_2, \zeta$ are iid, then ${\mathbf}{Y}_N$ is just an iid random matrix. On the other hand, if $\xi_1 = \xi_2$ almost surely, then ${\mathbf}{Y}_N$ is a real symmetric matrix. In this case, the eigenvalues of ${\mathbf}{Y}_N$ are real and ${\mathbf}{Y}_N$ is known as a *real symmetric Wigner matrix* [@W].

Suppose $\xi_1, \xi_2$ have mean zero and unit variance. Set $\rho := {\mathbb{E}}[\xi_1 \xi_2]$. When $|\rho| < 1$ and $\zeta$ has mean zero and finite variance, it was shown in [@NgO] that the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the elliptic law $F_\rho$ as $N \to \infty$, where $$F_\rho(x,y) = \mu_{\rho}( \{z \in \mathbb{C} : {\operatorname{Re}}(z) \leq x, {\operatorname{Im}}(z) \leq y\})$$ and $\mu_\rho$ is the uniform probability measure on the ellipsoid $$\mathcal{E}_{\rho} = \left\{ z \in {\mathbb{C}}: \frac{{\operatorname{Re}}(z)^2}{(1+\rho)^2} +\frac{{\operatorname{Im}}(z)^2}{(1-\rho)^2} < 1 \right\} .$$ This is a natural generalization of the circular law (Theorem \[thm:tvcirc\]). Figure \[fig:elliptic\] displays a numerical simulation of the eigenvalues of a real elliptic random matrix.

In this note, we consider the product ${\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}$ of $m$ independent real elliptic random matrices. In particular, we assume each real elliptic random matrix ${\mathbf}{Y}_{N,k}$ has atom variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ which satisfy the following conditions.

\[ass:atom\] There exists $\tau > 0$ such that the following conditions hold.

1.  $\xi_{k,1}, \xi_{k,2}$ both have mean zero and unit variance.

2.  ${\mathbb{E}}| \xi_{k,1}|^{2 + \tau} + {\mathbb{E}}|\xi_{k,2}|^{2 + \tau} < \infty$. \[item:tau\]

3.  $\rho_k := {\mathbb{E}}[\xi_{k,1} \xi_{k,2}]$ satisfies $|\rho_k| < 1$. \[item:rhoass\]

4.  $\zeta_k$ has mean zero and finite variance.

In our main result below, we show that the limiting distribution $F_m$ (with density given by ) from Theorem \[thm:GTprod\] for the product of independent iid random matrices is also the limiting distribution for the product of independent elliptic random matrices. In other words, the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. This type of universality was first considered by Burda, Janik, and Waclaw in [@BJW] for matrices with Gaussian entries; see also [@Bsurv]. Figure \[fig:plots\] displays several numerical simulations which illustrate this phenomenon.

\[thm:simple\] Let $m > 1$ be an integer. For each $1 \leq k \leq m$, let $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ be real random elements that satisfy Assumption \[ass:atom\]. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real elliptic random matrix with atom variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. Then the ESD of the product $${\mathbf}{P}_N := N^{-m/2} {\mathbf}{Y}_{N,1} \cdots {\mathbf}{Y}_{N,m}$$ converges almost surely to $F_m$ (with density given by ) as $N \to \infty$.

More generally, we establish a version of Theorem \[thm:simple\] where each elliptic random matrix ${\mathbf}{Y}_{N,k}$ is perturbed by a deterministic, low rank matrix ${\mathbf}{A}_{N,k}$ with small Hilbert-Schmidt norm. In fact, Theorem \[thm:simple\] will follow from Theorem \[thm:main\] below. We recall that, for any $m \times n$ matrix ${\mathbf}{M}$, the Hilbert-Schmidt norm $\|{\mathbf}{M}\|_2$ is given by the formula $$\label{eq:def:hs}
    \|{\mathbf}{M}\|_2 := \sqrt{ \operatorname{tr}({\mathbf}{M} {\mathbf}{M}^\ast) } = \sqrt{ \operatorname{tr}({\mathbf}{M}^\ast {\mathbf}{M})}.$$

\[thm:main\] Let $m > 1$ be an integer. For each $1 \leq k \leq m$, let $(\xi_{k,1}, \xi_{k,2}), \zeta_k$ be real random elements that satisfy Assumption \[ass:atom\]. For each $N \geq 1$ and $1 \leq k \leq m$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real elliptic random matrix with atom variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. For each $1 \leq k \leq m$, let ${\mathbf}{A}_{N,k}$ be a $N \times N$ deterministic matrix, and assume $$\label{eq:Aassump}
    \max_{1 \leq k \leq m} \operatorname{rank}({\mathbf}{A}_{N,k}) = O(N^{1-{\varepsilon}}) \quad \text{and} \quad \sup_{N \geq 1} \max_{1 \leq k \leq m} \frac{1}{N^2} \| {\mathbf}{A}_{N,k} \|_2 < \infty,$$ for some ${\varepsilon}> 0$. Then the ESD of the product $$\label{def:PN}
    {\mathbf}{P}_N := N^{-m/2} \prod_{k=1}^m ( {\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k})$$ converges almost surely to $F_m$ (with density given by ) as $N \to \infty$.

We conjecture that items and from Assumption \[ass:atom\] are not required for Theorem \[thm:main\] to hold. Indeed, in view of Theorem \[thm:tvcirc\] and [@NgO], it is natural to conjecture that $\xi_1, \xi_2$ need only have two finite moments. Also, our proof of Theorem \[thm:main\] can almost be completed under the assumption that $-1 < \rho_k \leq 1$. We only require that $\rho_k \neq 1$ in Section \[sec:singularvalue\] in order to control the least singular value of matrices of the form ${\mathbf}{Y}_{N,k} + {\mathbf}{F}_{N,k}$, where ${\mathbf}{F}_{N,k}$ is a deterministic matrix whose entries are bounded by $N^{\alpha}$, for some $\alpha > 0$. See Remark \[rem:lsv\] and Theorem \[thm:prodWigner\] below for further details.

Among other things, the perturbation by ${\mathbf}{A}_{N,k}$ in Theorem \[thm:main\] allows one to consider elliptic random matrices with nonzero mean. Indeed, let $\mu_k$ be a real number, and assume each entry of ${\mathbf}{A}_{N,k}$ takes the value $\mu_k$. Then ${\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k}$ is an elliptic random matrix whose atom variables have mean $\mu_k$.

In [@GNT], a result similar to Theorem \[thm:simple\] is proved under a different set of assumptions.

As noted above, when $\rho_k = 1$, the matrix ${\mathbf}{Y}_{N,k}$ is known as a real symmetric Wigner matrix. Theorem \[thm:main\] requires that $|\rho_k| < 1$, but in the special case when $m=2$, we are able to extend our proof to show that the same result holds for the product of two independent real symmetric Wigner matrices.

\[thm:prodWigner\] Let $\xi_{1,1}, \xi_{2,1}$ be real random variables with mean zero and unit variance, and which satisfy $${\mathbb{E}}|\xi_{1,1}|^{2+\tau} + {\mathbb{E}}|\xi_{2,1}|^{2+\tau} < \infty$$ for some $\tau > 0$. For each $N \geq 1$ and $k=1,2$, let ${\mathbf}{Y}_{N,k}$ be an $N \times N$ real symmetric matrix whose diagonal entries and upper diagonal entries are iid copies of $\xi_{k,1}$, and assume ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ are independent. Then the ESD of the product $${\mathbf}{P}_N := N^{-1} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2}$$ converges almost surely to $F_2$ (with density given by when $m=2$) as $N \to \infty$.

Overview and outline
--------------------

We begin by outlining the proof of Theorem \[thm:main\]. Instead of directly considering $ {\mathbf}{P}_N := N^{-m/2} \prod_{k=1}^m ( {\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k})$, we introduce a linearized random matrix, ${\mathbf}{Z}_N:= \frac{1}{\sqrt{N}} ({\mathbf}{Y}_N + {\mathbf}{A}_N)$, where ${\mathbf}{Y}_N$ and ${\mathbf}{A}_N$ are $mN \times mN$ block matrices of the form $$\label{def:YN}
    {\mathbf}{Y}_N := \begin{bmatrix} 
                0 &  {\mathbf}{Y}_{N,1} &     &            & 0       \\
                                0 & 0    & {\mathbf}{Y}_{N,2} &            & 0        \\      
                                &      & \ddots & \ddots     &         \\
                                0 &      &     &          0 & {\mathbf}{Y}_{N,m-1} \\
                            {\mathbf}{Y}_{N,m} &      &     &            &  0      
    \end{bmatrix}$$ and $$\label{def:AN}
    {\mathbf}{A}_N := \begin{bmatrix} 
                0 &  {\mathbf}{A}_{N,1} &     &            & 0       \\
                                0 & 0    & {\mathbf}{A}_{N,2} &            & 0        \\      
                                &      & \ddots & \ddots     &         \\
                                0 &      &     &          0 & {\mathbf}{A}_{N,m-1} \\
                            {\mathbf}{A}_{N,m} &      &     &            &  0      
            \end{bmatrix}.$$

The following theorem gives the limiting distribution of ${\mathbf}Z_N$, from which we will deduce our main theorem as a corollary.

\[thm:circular\] Under the assumptions of Theorem \[thm:main\], the ESD of ${\mathbf}{Z}_N := \frac{1}{\sqrt{N}} ({\mathbf}{Y}_N + {\mathbf}{A}_N)$ converges almost surely to the circular law $F_{\mathrm{circ}}$ as $N \to \infty$.

In Section \[sec:linear\], we show that Theorem \[thm:main\] is a short corollary of Theorem \[thm:circular\]. This same linearization trick was used in [@OS] to study products of non-Hermitian matrices with iid entries. Similar techniques were also used in [@A; @HT] to study general self-adjoint polynomials of self-adjoint random matrices.

Sections \[sec:resolvent\], \[sec:singularvalue\], and \[sec:complete\] are dedicated to proving Theorem \[thm:circular\]. Following the ideas of Girko [@G1; @G2], we compute the limiting spectral measure of a non-Hermitian random matrix ${\mathbf}{M}$, by employing the method of *Hermitizaition*. Given an $N \times N$ matrix ${\mathbf}{M}$, we recall that the empirical spectral measure of ${\mathbf}{M}$ is given by $$\mu_{\mathbf}{M} := \frac{1}{N} \sum_{i=1}^N \delta_{\lambda_i({\mathbf}{M})},$$ where $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ are the eigenvalues of ${\mathbf}{M}$. We let $\nu_{{\mathbf}{M}}$ denote the symmetric empirical measure built from the singular values of ${\mathbf}{M}$. That is, $$\nu_{{\mathbf}{M}} := \frac{1}{2N} \sum_{i =1}^N \left( \delta_{\sigma_i({\mathbf}{M})} + \delta_{-\sigma_i({\mathbf}{M})} \right),$$ where $\sigma_1({\mathbf}{M}) \geq \cdots \geq \sigma_N({\mathbf}{M}) \geq 0$ are the singular values of ${\mathbf}{M}$. In particular, $$\sigma_1({\mathbf}{M}) := \sup_{\|x\| = 1} \| {\mathbf}{M} x \|$$ is the largest singular value of ${\mathbf}{M}$ and $$\sigma_N({\mathbf}{M}) := \inf_{\|x\| = 1} \| {\mathbf}{M} x \|$$ is the smallest singular value, both of which will play a key role in our analysis below.

The key observation of Girko [@G1; @G2] relates the empirical spectral measure of a non-Hermitian matrix to that of a Hermitian matrix. To illustrate the connection, consider the *Cauchy–Stieltjes transform* $s_N$ of the measure $\mu_{{\mathbf}{M}}$, where ${\mathbf}{M}$ is an $N \times N$ matrix, given by $$s_N(z) := \frac{1}{N} \sum_{i=1}^N \frac{1}{ \lambda_i({\mathbf}{M}) - z} = \int_{\mathbb{C}} \frac{1}{x -z } \mu_{{\mathbf}{M}} (dx),$$ for $z \in \mathbb{C}$. Since $s_N$ is analytic everywhere except at the poles (which are exactly the eigenvalues of ${\mathbf}{M}$), the real part of $s_N$ determines the eigenvalues. Let $\sqrt{-1}$ denote the imaginary unit, and set $z = s + \sqrt{-1} t$. Then we can write the real part of $s_N(z)$ as $$\begin{aligned}
    {\operatorname{Re}}( s_N(z)) &= \frac{1}{N} \sum_{i=1}^N \frac{ {\operatorname{Re}}(\lambda_i({\mathbf}{M})) - s }{ \left| \lambda_i({\mathbf}{M}) - z \right|^2} \\
        &= \frac{-1}{2N} \sum_{i=1}^N \frac{\partial}{\partial s} \log \left| \lambda_i({\mathbf}{M}) - z \right|^2 \\
        &= \frac{-1}{2N} \frac{\partial}{\partial s}  \log \det \left( {\mathbf}{M} - z {\mathbf}{I} \right) \left( {\mathbf}{M} - z{\mathbf}{I} \right)^{\ast} \\
        &= -\frac{\partial}{\partial s} \int_{\mathbb{C}} \log x^2  \nu_{{\mathbf}{M} - z {\mathbf}{I}}(dx), \end{aligned}$$ where ${\mathbf}{I}$ denotes the identity matrix. In other words, the task of studying $\mu_{{\mathbf}{M}}$ reduces to studying the measures $\{\nu_{{\mathbf}{M} - z {\mathbf}{I}}\}_{z \in \mathbb{C}}$. The difficulty now is that the $\log$ function has two poles, one at infinity and one at zero. The largest singular value can easily be bounded by a polynomial in $N$. The main difficulty is controlling the least singular value.

In order to study $\nu_{{\mathbf}M}$ it is useful to note that it is also the empirical spectral measure for the Hermitization of ${\mathbf}M$. The *Hermitization* of ${\mathbf}M$ is defined to be $${\mathbf}H:=\begin{bmatrix} 
                0 &  {\mathbf}M\\
                            {\mathbf}M^*&  0
            \end{bmatrix}.$$ For an $N \times N$ matrix, the Stieltjes transform of $\nu_{{\mathbf}{M} - z {\mathbf}{I}}$ is also the trace of the Hermitized resolvent. That is, for $\eta \in \mathbb{C}^{+} := \{ w \in \mathbb{C} : {\operatorname{Im}}(w) > 0\}$, we have $$\int \frac{ 1  }{x-\eta}\nu_{{\mathbf}{M} - z {\mathbf}{I}}(dx) = \frac{1}{2N} \operatorname{tr}({\mathbf}R( {\mathbf}q)),$$ where $${\mathbf}{R}({\mathbf}q) := ( {\mathbf}{H} -   {\mathbf}{q} \otimes {\mathbf}{I}_N)^{-1}, \quad {\mathbf}{q} := \begin{bmatrix}  \eta  & z  \\ \bar{z}  & \eta  \end{bmatrix}.$$ Here ${\mathbf}{q} \otimes {\mathbf}{I}_N$ denotes the Kronecker product of the matrix ${\mathbf}{q}$ and the identity matrix ${\mathbf}{I}_N$.

Typically, in order to estimate the measures $\nu_{{\mathbf}{M} -z {\mathbf}{I}}$, one shows that the Stieltjes transform approximately satisfies a fixed point equation. Then one can show that this Stieltjes transform is close to the Stieltjes transform that exactly solves the fixed point equation. Because of the dependencies between entries in the matrix ${\mathbf}Z_N$, directly computing the trace of the resolvent of the Hermitization of ${\mathbf}Z_N$ is troublesome. To circumvent this issue, in Section \[sec:resolvent\], instead of taking the trace of the resolvent, we instead take the partial trace and consider a $2m \times 2m$ matrix-valued Stieltjes transform. Then we show this partial trace approximately satisfies a matrix-valued fixed point equation.

In Section \[sec:singularvalue\], we deduce a bound for the least singular value of the matrix $\frac{1}{\sqrt{N}}({\mathbf}{Y}_N + {\mathbf}{A}_N) - z {\mathbf}{I}$ from the known bounds on the least singular values of the individual matrices ${\mathbf}Y_{N,k} +{\mathbf}A_{N,k}$. We finally complete the proof of Theorem \[thm:circular\] in Section \[sec:complete\].

The proof of Theorem \[thm:prodWigner\] is very similar to the proof of Theorem \[thm:main\]. In fact, there are only a few places in the proof of Theorem \[thm:main\] where the condition $|\rho_k| < 1$ is required. We prove Theorem \[thm:prodWigner\] in Section \[sec:Wigner\].

A remark from free probability
------------------------------

Notation
--------

We use asymptotic notation (such as $O,o,\Omega$) under the assumption that $N \rightarrow \infty$. We use $X \ll Y, Y \gg X, Y=\Omega(X)$, or $X = O(Y)$ to denote the bound $X \leq CY$ for all sufficiently large $N$ and for some constant $C$. Notations such as $X \ll_k Y$ and $X=O_k(Y)$ mean that the hidden constant $C$ depends on another constant $k$. We always allow the implicit constants in our asymptotic notation to depend on the integer $m$ from Theorem \[thm:main\]; we will not denote this dependence with a subscript. $X=o(Y)$ or $Y=\omega(X)$ means that $X/Y \rightarrow 0$ as $N \rightarrow \infty$.

$\|{\mathbf}{M}\|$ is the spectral norm of the matrix ${\mathbf}{M}$. $\|{\mathbf}{M}\|_2$ denotes the Hilbert-Schmidt norm of ${\mathbf}{M}$ (defined in ). We let ${\mathbf}{I}_N$ denote the $N \times N$ identity matrix. Often we will just write ${\mathbf}{I}$ for the identity matrix when the size can be deduced from the context.

We write a.s., a.a., and a.e. for almost surely, Lebesgue almost all, and Lebesgue almost everywhere respectively. We use $\sqrt{-1}$ to denote the imaginary unit and reserve $i$ as an index. We let ${\ensuremath{\mathbf{1}_{{E}}}}$ denote the indicator function of the event $E$.

We let $C$ and $K$ denote constants that are non-random and may take on different values from one appearance to the next. The notation $K_p$ means that the constant $K$ depends on another parameter $p$. We always allow the constants $C$ and $K$ to depend on the integer $m$ from Theorem \[thm:main\]; we will not denote this dependence with a subscript.

In view of Theorem \[thm:main\] and Assumption \[ass:atom\], we define the correlations $\rho_k := {\mathbb{E}}[ \xi_{k,1} \xi_{k,2} ]$ for the atom variables $\xi_{k,1}, \xi_{k,2}$. In addition, we let $\tau > 0$ be such that $$\sum_{k=1}^m \left( {\mathbb{E}}|\xi_{k,1}|^{2+\tau} + {\mathbb{E}}|\xi_{k,2}|^{2+\tau} \right) < \infty.$$

Proof of Theorem \[thm:main\]  {#sec:linear}
==============================

A matrix-valued Stieltjes transform {#sec:resolvent}
===================================

Least singular value bound {#sec:singularvalue}
==========================

Completing the argument {#sec:complete}
=======================

Proof of Theorem \[thm:prodWigner\] {#sec:Wigner}
===================================

This section is devoted to the proof of Theorem \[thm:prodWigner\]. As noted above, the proof of Theorem \[thm:prodWigner\] is very similar to the proof of Theorem \[thm:main\]. We define the linearized matrix ${\mathbf}{Z}_N := \frac{1}{\sqrt{N}} {\mathbf}{Y}_N$, where $${\mathbf}{Y}_N := \begin{bmatrix} 0 & {\mathbf}{Y}_{N,1} \\ {\mathbf}{Y}_{N,2} & 0 \end{bmatrix}.$$

As before, the proof of Theorem \[thm:prodWigner\] reduces to showing that the ESD of ${\mathbf}{Z}_N$ converges to the circular law as $N$ tends to infinity.

\[thm:circularWigner\] Under the assumptions of Theorem \[thm:prodWigner\], the ESD of ${\mathbf}{Z}_N := \frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the circular law $F_{\mathrm{circ}}$ as $N \to \infty$.

The proof that Theorem \[thm:prodWigner\] follows from Theorem \[thm:circularWigner\] is identical to the proof given in Section \[sec:linear\].

The proof of Theorem \[thm:circularWigner\] follows the same arguments outlined in Section \[sec:complete\] (taking ${\mathbf}{A}_N = 0$). Indeed, the proof in Section \[sec:complete\] requires three key inputs: Lemma \[lemma:concentrate\], Theorem \[approxthm\], and Theorem \[thm:least-sing-value\]. Thus, in order to complete the proof, we will need versions of these results for the matrix ${\mathbf}{Z}_N$ defined above.

We first observe that both Lemma \[lemma:concentrate\] and Theorem \[approxthm\] hold for the matrix ${\mathbf}{Z}_N$ defined above. Indeed, the matrices ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ (as well as their truncated counterparts) are elliptic random matrices. In fact, the proofs of Lemma \[lemma:concentrate\] and Theorem \[approxthm\] do not require any conditions on the correlations $\rho_k$. Therefore, it only remains to prove the following analogue of Theorem \[thm:least-sing-value\].

\[thm:least-sing-valueWigner\] Under the assumptions of Theorem \[thm:prodWigner\], there exists $A >0$ such that for almost every $z \in \mathbb{C}$, almost surely $$\lim_{N \to \infty} {\ensuremath{\mathbf{1}_{\{\sigma_{2N}( {\mathbf}{Z}_N - z{\mathbf}{I}) \leq N^{-A}\}}}} = 0.$$

In order to prove Theorem \[thm:least-sing-valueWigner\], we will need the following result due to Nguyen [@NgLSV]. The version stated in [@NgLSV] requires that the deterministic matrices ${\mathbf}{F}_{N,k}$ be real symmetric. However, the proof given in [@NgLSV] only requires that ${\mathbf}{F}_{N,k}$ be complex symmetric [@NgPC]. In particular, the proof only uses that the $(i,j)$-entry of ${\mathbf}{F}_{N,k}$ be equal to the $(j,i)$-entry. We present the most general version below.

\[thm:NgLSV\] For each $k=1,2$, let ${\mathbf}{F}_{N,k}$ be a $N \times N$ complex symmetric matrix whose entries are bounded in magnitude by $N^\alpha$, for some $\alpha > 0$. Then, under the assumptions of Theorem \[thm:prodWigner\], for any $B > 0$, there exists $A>0$ (depending on $\alpha, B$) such that $${\mathbb{P}}\left( \min_{k=1,2} \sigma_{N}( {\mathbf}{Y}_{N,k} + {\mathbf}{F}_{N,k}) \leq N^{-A} \right) = O(N^{-B}).$$

We now prove Theorem \[thm:least-sing-valueWigner\]

Let $A$ be a large positive constant to be chosen later. By the Borel-Cantelli lemma, it suffices to show that, for almost every $z \in \mathbb{C}$, $${\mathbb{P}}\left( \left\| ({\mathbf}{Z}_N - z {\mathbf}{I})^{-1} \right\| \geq N^{A} \right) = O(N^{-2}).$$ By , we observe that, for $z \neq 0$, $({\mathbf}{Z}_N - z {\mathbf}{I})^{-1}$ has the form $$\begin{bmatrix} z \left(\frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} & \left(\frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} \\ z^2 \left(\frac{1}{N} {\mathbf}{Y}_{N,2} {\mathbf}{Y}_{N,1} - z^2 {\mathbf}{I} \right)^{-1} \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} & z \left( \frac{1}{N} {\mathbf}{Y}_{N,2} {\mathbf}{Y}_{N,1} - z^2 {\mathbf}{I} \right)^{-1} \end{bmatrix}$$ provided the relevant inverses exist. Thus, it suffices to show that the spectral norm of each block above is $O(N^{A})$ with probability $1-O(N^{-2})$. The treatment of each block is similar; as an illustration, we will show that $${\mathbb{P}}\left( \left\| \left(\frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} \right\| \geq N^{A} \right) = O(N^{-2}).$$

Indeed, since $${\mathbb{P}}\left( \left\| \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} \right\| \geq N^{100} \right) \leq N^{-200} {\mathbb{E}}\left\| \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} \right\|^2_2 = O(N^{-2}),$$ it suffices to show that there exists $A>200$ such that $$\label{eq:show2prodbnd}
    {\mathbb{P}}\left( \left\| \left( \frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \right\| \geq N^{A} \right) = O(N^{-2}).$$

By Theorem \[thm:NgLSV\], there exists $A'>0$ such that the event $$\Omega_N := \left\{ \left\| \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1} \right\| \leq N^{A'} \right\}$$ holds with probability $1-O(N^{-2})$. On this event, we observe that $$\left( \frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} = \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1} \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} - z^2 \left(\frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1} \right)^{-1}.$$ Therefore, we conclude that $$\begin{aligned}
    {\mathbb{P}}&\left( \left\| \left( \frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \right\| \geq N^{A} \right) \\
    &\leq {\mathbb{P}}\left( \left\| \left( \frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \right\| \geq N^{A} \bigg| \Omega_N \right) {\mathbb{P}}(\Omega_N) + {\mathbb{P}}(\Omega_N^C) \\
    &\leq O(N^{-2}) + {\mathbb{P}}\left( \left\| \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} - z^2 \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2}\right)^{-1} \right)^{-1} \right\| \geq N^{A/2} \bigg| \Omega_N \right)\end{aligned}$$ for $A$ sufficiently large.

We now recall that ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ are independent, and, on the event $\Omega_N$, the entries of $\left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1}$ are bounded in magnitude by $N^{A'}$. In addition, we observe that $z^2 \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1}$ is a complex symmetric matrix since ${\mathbf}{Y}_{N,2}$ is a real symmetric matrix. Therefore, by Theorem \[thm:NgLSV\], there exists $A>0$ such that $${\mathbb{P}}\left( \left\| \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} - z^2 \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2}\right)^{-1} \right)^{-1} \right\| \geq N^{A/2} \bigg| \Omega_N \right) = O(N^{-2}).$$ This verifies , and hence the proof of Theorem \[thm:least-sing-valueWigner\] is complete.

Truncation {#sec:truncationproof}
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[^1]: S. O’Rourke has been supported by grant AFOSAR-FA-9550-12-1-0083

[^2]: D. Renfrew is partly supported by NSF grant DMS-0838680

[^3]: A. Soshnikov has been supported in part by NSF grant DMS-1007558

[^4]: V. Vu is supported by research grants DMS-0901216, DMS-1307797, and AFOSAR-FA-9550-12-1-0083.
