On the outlying eigenvalues of a polynomial in large independent random matrices

TL;DR

This paper studies the asymptotic behavior of eigenvalues of polynomials in large independent random matrices, identifying conditions under which outliers occur due to spikes and extending previous results to more general polynomials.

Contribution

It extends the understanding of eigenvalue outliers for polynomials of large random matrices, using free probability tools, beyond the classical sum case.

Findings

Eigenvalues of polynomial random matrices converge to a deterministic measure.

Outliers are characterized in terms of spikes and free probability subordination functions.

Results apply to both Hermitian and Wigner matrices.

Abstract

Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A\_N,B\_N)$, where $A\_N$ and $B\_N$ are independent Hermitian random matrices and the distribution of $B\_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A\_N$ and $B\_N$ converge almost surely to deterministic probability measures $\mu $ and $\nu$, respectively. In addition, the eigenvalues of $A\_N$ and $B\_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A\_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A\_N,B\_N)$ converges to a certain deterministic probability measure $\eta$ (sometimes denoted $\eta=P^\square(\mu,\nu)$) and, when there are no spikes, the eigenvalues of $P(A\_N,B\_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A\_N,B\_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu,\nu,P$, and the spikes of $A\_N$. We establish a similar result when $B\_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

Figures

Equations

$$\frac{1}{n^2}\sum _{1\le i,j\le n}\mathbb{P}\left(|X_{ij}|>x\right)\le K\mathbb{P}\left(|Z|>x\right)x>x_0,n>n_0.$$

$$\sigma (a)=\{\lambda \in \mathbb{C}:\lambda 1-a\text{ is not invertible in }\mathcal{A}\}.$$

$$\underset{N\to \infty }{\lim }{\tau }_N(a_N^k)={\tau }_0(a_0^k),k\in \mathbb{N}.$$

$$\mathrm{supp}({\mu }_{a_N})\subset \mathrm{supp}({\mu }_{a_0})+(-\epsilon ,\epsilon )$$

$$\mathrm{supp}({\mu }_{a_0})\subset \mathrm{supp}({\mu }_{a_N})+(-\epsilon ,\epsilon )$$

$$\underset{N\to \infty }{\lim }\Vert P(a_N)\Vert =\Vert P(a_0)\Vert ,$$

$$(\alpha X_{i_1}{X}_{i_2}\cdots X_{i_n}{)}^{\ast }=\overline{\alpha }{X}_{i_n}\cdots X_{i_2}{X}_{i_1},\alpha \in \mathbb{C},i_1,i_1,\dots ,i_n\in \{1,\dots ,k\}.$$

$$\underset{N\to \infty }{\lim }{\tau }_N(P(a_N))={\tau }_0(P(a_0)),P\in \mathbb{C}⟨X_1,\dots ,X_k⟩.$$

$$\underset{N\to \infty }{\lim }\Vert P(a_N)\Vert =\Vert P(a_0)\Vert ,P\in \mathbb{C}⟨X_1,\dots ,X_k⟩.$$

$$\mathrm{d}{\nu }_{0,1}(t)=\frac{1}{2\pi }\sqrt{4-t^2}{1\mathsf{I}}_{[-2,2]}(t)\mathrm{d}t.$$

$$\sigma (A_N)\setminus \{{\theta }_1,\dots ,{\theta }_p\}\subseteq \mathrm{supp}(\mu )+(-\epsilon ,\epsilon ),N\ge N(\epsilon ).$$

$$Z_N=P(A_N,B_N)$$

$$Z_N=P\left(A_N,\frac{X_N}{\sqrt{N}}\right)$$

$$\underset{N\to \infty }{\lim }{\mu }_{P(A_N,B_N)}={\mu }_{P(a,b)}$$

$$\underset{N\to \infty }{\lim }{\mu }_{P(A_N,X_N/\sqrt{N})}={\mu }_{P(a,b)}$$

$$z\alpha \otimes 1-L,$$

$$L={\gamma }_0\otimes 1+{\gamma }_1\otimes X_1+\cdots +{\gamma }_k\otimes X_k,$$

$$L=\left[\begin{array}{cc}0& u\\ v& Q\end{array}\right],$$

$$L=-\begin{bmatrix}0&0&\cdots&0&X_{i_{1}}\\ 0&0&\cdots&X_{i_{2}}&-1\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\ 0&X_{i_{\ell-1}}&\cdots&0&0\\ X_{i_{\ell}}&-1&\cdots&0&0\end{bmatrix}.$$

$$L_j=\left[\begin{array}{cc}0& u_j\\ v_j& Q_j\end{array}\right]\in M_{n_j}(\mathbb{C}⟨X_1,\dots ,X_k⟩),j=1,2,$$

$$L=\left[\begin{array}{ccc}0& u_1& u_2\\ v_1& Q_1& 0\\ v_2& 0& Q_2\end{array}\right]=\left[\begin{array}{cc}0& u\\ v& Q\end{array}\right]\in M_{n_1+n_2-1}(\mathbb{C}⟨X_1,\dots X_k⟩).$$

$$\left[\begin{array}{cc}0& u_0\\ v_0& Q_0\end{array}\right].$$

$$\left[\begin{array}{ccc}0& u_0& v_0^{\ast }\\ u_0^{\ast }& 0& Q_0^{\ast }\\ v_0& Q_0& 0\end{array}\right]=\left[\begin{array}{cc}0& u\\ u^{\ast }& Q\end{array}\right]$$

$$L=\left[\begin{array}{cc}0& u\\ v& Q\end{array}\right]\in M_n(\mathbb{C}⟨X_1,\dots ,X_k⟩)$$

$$T_{0}=\begin{bmatrix}0&\cdots&1\\ \vdots&\reflectbox{$\ddots$}&\vdots\\ 1&\cdots&0\end{bmatrix},$$

$$L=\left[\begin{array}{cc}0& u\\ v& Q\end{array}\right]\in M_n(\mathbb{C}⟨X_1,\dots ,X_k⟩)$$

$$\det (z{e}_{1,1}\otimes I_N-L(S_1,\dots ,S_k))=\pm \det (z{I}_n-P(S_1,\dots ,S_k)),$$

$$\dim \ker (z{I}_n-P(S_1,\dots ,S_k))=\dim \ker (z{e}_{1,1}\otimes I_N-L(S_1,\dots ,S_k))z\in \mathbb{C}.$$

$$\left[\begin{array}{cc}1& -u{Q}^{-1}\\ 0& 1_{n-1}\end{array}\right]\left[\begin{array}{cc}z& -u\\ -v& -Q\end{array}\right]\left[\begin{array}{cc}1& 0\\ -Q^{-1}v& 1_{n-1}\end{array}\right]=\left[\begin{array}{cc}z-P& 0\\ 0& -Q\end{array}\right],z\in \mathbb{C}.$$

$$L=\left[\begin{array}{cc}0& u^{\ast }\\ u& Q\end{array}\right]\in M_n(\mathbb{C}⟨X_1,\dots ,X_k⟩)$$

Full text

HB was supported by a grant from the National Science Foundation. This work was started while HB was visiting the Institute of Mathematics of Toulouse as Professeur Invité.

1. Introduction

Let $\mu$ and $\nu$ be two Borel probability measures with bounded support on $\mathbb{R}$. Suppose given, for each positive integer $N$, selfadjoint $N\times N$ independent random matrices $A_{N}$ and $B_{N}$, with the following properties:

1. (a)

the distribution of $B_{N}$ is invariant under conjugation by unitary $N\times N$ matrices; 2. (b)

the empirical eigenvalue distributions of $A_{N}$ and $B_{N}$ converge almost surely to $\mu$ and $\nu$, respectively; 3. (c)

the eigenvalues of $A_{N}$ and $B_{N}$ converge uniformly almost surely to the supports of $\mu$ and $\nu$, respectively, with the exception of a fixed number $p$ of spikes, that is, fixed eigenvalues of $A_{N}$ that lie outside the support of $\mu$.

When spikes are absent, that is, when $p=0$, it was shown in [23] that the eigenvalues of $A_{N}+B_{N}$ converge uniformly almost surely to the support of the free additive convolution $\mu\boxplus\nu$. When $p>0$, the eigenvalues of $A_{N}+B_{N}$ also converge uniformly almost surely to a compact set $K\subset\mathbb{R}$ such that $\mathrm{supp}(\mu\boxplus\nu)\subset K$ and $K\setminus\mathrm{supp}(\mu\boxplus\nu)$ has no accumulation points in $\mathbb{R}\setminus\mathrm{supp}(\mu\boxplus\nu)$. Moreover, if $t\in K\setminus\mathrm{supp}(\mu\boxplus\nu)$, then $\omega(t)$ is one of the spikes of $A_{N}$, where $\omega$ is a certain subordination function arising in free probability. The relative position of the eigenvectors corresponding to spikes and outliers is also given in terms of subordination functions. We refer to [11] for this result.

Our purpose is to show that analogous results hold when the sum $A_{N}+B_{N}$ is replaced by an arbitrary selfadjoint polynomial $P(A_{N},B_{N})$. Then, by a comparison procedure to the particular case when $B_{N}$ is a G.U.E. (Gaussian unitary ensemble), we are also able to identify the outliers of an arbitrary selfadjoint polynomial $P(A_{N},\frac{X_{N}}{\sqrt{N}})$ when $X_{N}$ is a Wigner matrix independent from $A_{N}$. This extends an earlier result [22] pertaining to additive deformations of Wigner matrices. More precisely we consider a Hermitian matrix $X_{N}=[X_{ij}]_{i,j=1}^{N}$, where $[X_{ij}]_{i\geq 1,j\geq 1}$ is an infinite array of random variables such that

1. (X0)

$X_{N}$ is independent from $A_{N}$, 2. (X1)

$X_{ii}$, $\sqrt{2}\Re(X_{ij}),i<j$, $\sqrt{2}\Im(X_{ij}),i<j$, are independent, centered with variance 1, 3. (X2)

there exist $K,x_{0}>0$, $n_{0}\in\mathbb{N}$, and a random variable $Z$ with finite fourth moment such that

[TABLE] 4. (X3)

$\sup\{\mathbb{E}(|X_{ij}|^{3}):i,j\in\mathbb{N},i<j\}<+\infty.$

Remark 1.1**.**

The matrix $X_{N}$ is called a G.U.E. if the variables $X_{ii}$, $\sqrt{2}\Re(X_{ij}),{i<j}$, and $\sqrt{2}\Im(X_{ij}),{i<j}$, are independent standard Gaussian. Assumptions (X2) and (X3) obviously hold if these variables are merely independent and identically distributed with a finite fourth moment.

Our result lies in the lineage of recent, and not so recent, works [5, 7, 8, 14, 18, 19, 21, 22, 26, 27, 31, 33, 35, 39, 40, 41] studying the influence of additive or multiplicative perturbations on the extremal eigenvalues of classical random matrix models, the seminal paper being [7], where the so-called BBP phase transition was observed.

We note that Shlyakhtenko [45] considered a framework which makes it possible to understand this kind of result as a manifestation of infinitesimal freeness. In fact, the results of [45] also allow one to detect the presence of spikes from the behaviour of the bulk of the eigenvalues of $P(A_{N},B_{N})$, even when $P(A_{N},B_{N})$ has no outlying eigenvalues. In a related result, Collins, Hasebe and Sakuma [24] study the ‘purely spike’ case in which $\mu=\nu=\delta_{0}$ and the eigenvalues of $A_{N}$ and $B_{N}$ accumulate to given sequences $(a_{k})_{k=1}^{\infty}$ and $(b_{k})_{k=1}^{\infty}$ of real numbers converging to zero.

2. Notation and preliminaries on strong asymptotic freeness

We recall that a $C^{*}$-probability space is a pair $(\mathcal{A},\tau)$, where $\mathcal{A}$ is a $C^{*}$-algebra and $\tau$ is a state on $\mathcal{A}$. We always assume that $\tau$ is faithful. The elements of $\mathcal{A}$ are referred to as random variables.

If $(\Omega,\Sigma,P)$ is a classical probability space, then $(L^{\infty}(\Omega),\mathbb{E})$ is a $C^{*}$-probability space, where $\mathbb{E}$ is the usual expected value. Given $N\in\mathbb{N}$, $(M_{N}(\mathbb{C}),{\rm tr}_{N})$ is a $C^{*}$-probability space, where ${\rm tr}_{N}=\frac{1}{N}{\rm Tr}_{N}$ denotes the normalized trace. More generally, if $(\mathcal{A},\tau)$ is an arbitrary $C^{*}$-probability space and $N\in\mathbb{N}$, then $M_{N}(\mathcal{A})=M_{N}(\mathbb{C})\otimes\mathcal{A}$ becomes a $C^{*}$-probability space with the state ${\rm tr}_{N}\otimes\tau$.

The distribution $\mu_{a}$ of a selfadjoint element $a$ in a $C^{*}$-probability space $(\mathcal{A},\tau)$ is a compactly supported probability measure on $\mathbb{R}$, uniquely determined by the requirement that $\int_{\mathbb{R}}t^{n}\,d\mu_{a}(t)=\tau(a^{n})$, $n\in\mathbb{N}$. The spectrum of an element $a\in\mathcal{A}$ is

[TABLE]

For instance, if $A\in M_{N}(\mathbb{C})$ is a selfadjoint matrix, then the distribution of $A$ relative to ${\rm tr}_{N}$ is the measure $\mu_{A}=\frac{1}{N}\sum_{j=1}^{N}\delta_{\lambda_{j}(A)}$, where $\{\lambda_{1}(A),\dots,\lambda_{N}(A)\}$ is the list of the eigenvalues of $A$, repeated according to multiplicity. As usual, the support $\text{supp}(\mu)$ of a Borel probability measure $\mu$ on $\mathbb{R}$ is the smallest closed set $F\subset\mathbb{R}$ with the property that $\mu(F)=1$. It is known that if $a=a^{*}\in\mathcal{A}$ and $\tau$ is faithful, then $\sigma(a)=\mathrm{supp}(\mu_{a}).$ In the following, we assume that $\tau$ is a tracial state, that is, $\tau(ab)=\tau(ba),a,b\in\mathcal{A}$.

Suppose that we are given $C^{*}$-probability spaces $\{(\mathcal{A}_{N},\tau_{N})\}_{N=0}^{\infty}$ and selfadjoint elements $a_{N}\in\mathcal{A}_{N}$, $N\geq 0$. We say that $\{a_{N}\}_{N=1}^{\infty}$ converges in distribution to $a_{0}$ if

[TABLE]

We say that $\{a_{N}\}_{N=1}^{\infty}$ converges strongly in distribution to $a_{0}$ (or to $\mu_{a_{0}}$) if, in addition to (2.1), the sequence $\{\mathrm{supp}(\mu_{a_{N}})\}_{N=1}^{\infty}$ converges to $\mathrm{supp}(\mu_{a_{0}})$ in the Hausdorff metric. This condition simply means that for every $\varepsilon>0$ there exists $N(\varepsilon)\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

for every $N\geq N(\varepsilon)$. If all the traces $\tau_{N}$ are faithful, strong convergence can be reformulated as follows:

[TABLE]

for every polynomial $P$ with complex coefficients. This observation allows us to extend the concept of (strong) convergence in distribution to $k$-tuples of random variables, $k\in\mathbb{N}$. For every $k\in\mathbb{N}$, we denote by $\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$ the algebra of polynomials with complex coefficients in $k$ noncommuting indeterminates $X_{1},\dots,X_{k}$. This is a $*$-algebra with the adjoint operation determined by

[TABLE]

Suppose that $\{(\mathcal{A}_{N},\tau_{N})\}_{N=0}^{\infty}$ is a sequence of $C^{*}$-probability spaces, $k\in\mathbb{N}$, and $\{a_{N}\}_{N=0}^{\infty}$ is a sequence of $k$-tuples $a_{N}=(a_{N,1},\dots,a_{N,k})\in\mathcal{A}_{N}^{k}$ of selfadjoint elements. We say that $\{a_{N}\}_{N=1}^{\infty}$ converges in distribution to $a_{0}$ if

[TABLE]

We say that $\{a_{N}\}_{N=1}^{\infty}$ converges strongly in distribution to $a_{0}$ if, in addition to (2.2), we have

[TABLE]

The above concepts extend to $k$-tuples $a_{N}=(a_{N,1},\dots,a_{N,k})\in\mathcal{A}_{N}^{k}$ which do not necessarily consist of selfadjoint elements. The only change is that one must use polynomials in the variables $a_{N,j}$ and their adjoints $a_{N,j}^{*}$, $j=1,\dots,k$.

Remark 2.1**.**

Suppose that all the states $\tau_{N},N\in\mathbb{N}$, are faithful. As seen in [23, Proposition 2.1], $\{a_{N}\}_{N=1}^{\infty}$ converges strongly in distribution to $a_{0}$ if and only if $\{P(a_{N})\}_{N=1}^{\infty}$ converges strongly in distribution to $P(a_{0})$ for every selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle.$ Moreover, strong convergence in distribution also implies strong convergence at the matricial level. The following result is [36, Proposition 7.3].

Proposition 2.2**.**

Let $\{(\mathcal{A}_{N},\tau_{N})\}_{N=0}^{\infty}$ be $C^{*}$-probability spaces with faithful states $\{\tau_{N}\}_{N=0}^{\infty}$, let $k\in\mathbb{N}$, and let $\{a_{N}\}_{N=0}^{\infty}$ be a sequence of $k$-tuples of selfadjoint elements $a_{N}\in\mathcal{A}_{N}^{k}$. Suppose that $\{a_{N}\}_{N=1}^{\infty}$ converges strongly in distribution to $a_{0}$. Then $\lim_{N\to\infty}\|P(a_{N})\|=\|P(a_{0})\|$ for every $n\in\mathbb{N}$ and every matrix polynomial $P\in M_{n}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$.

A special case of strong convergence in distribution arises from the consideration of random matrices in $M_{N}(\mathbb{C})$. The following result follows from [23, Theorem 1.4] and [12, Theorem 1.2].

Theorem 2.3**.**

Let $(\mathcal{A}_{N},\tau_{N})$ denote the space $(M_{N}(\mathbb{C}),\rm tr_{N})$, $N\in\mathbb{N}$. Suppose that $k_{1},k_{2},k_{3}\in\mathbb{N}$ are fixed, $u_{N}=(U_{N,1},\dots,U_{N,k_{1}})$, $x_{N}=(X_{N,1},\dots,X_{N,k_{2}})$ and $a_{N}=(A_{N,1},\dots,A_{N,k_{3}})$ are mutually independent random tuples of matrices in some classical probability space such that:

• (i)

$U_{N,1},\dots,U_{N,k_{1}}$* are independent unitaries distributed according to the Haar measure on the unitary group ${\rm U}(N),N\in\mathbb{N}$.*

• (ii)

$X_{N,1},\dots,X_{N,k_{2}}$* are independent Hermitian matrices, each satisfying assumptions $(X1),(X2),$ and $(X3)$ in the introduction.*

• (iii)

$a_{N}$* is a vector of $N\times N$ selfadjoint matrices such that the sequence $\{a_{N}\}_{N=1}^{\infty}$ converges strongly almost surely in distribution to some deterministic $k_{3}$-tuple in a $C^{*}$-probability space.*

Then there exist a $C^{*}$-probability space $(\mathcal{A},\tau)$, a free family $u=(u_{1},\dots,u_{k_{1}})\in\mathcal{A}^{k_{1}}$ of Haar unitaries, a semicircular system $x=(x_{1},\dots,x_{k_{2}})\in\mathcal{A}^{k_{2}}$ and $a=(a_{1},\dots,a_{k_{3}})\in\mathcal{A}^{k_{3}}$, such that, $u,x,$ and $a$ are free and $\{(u_{N},x_{N},a_{N})\}_{N=1}^{\infty}$ converges strongly almost surely in distribution to $(u,x,a)$.

We recall that a tuple $(x_{1},\dots,x_{k})$ of elements in a ${C}^{*}$-probability space $\left({\mathcal{A}},\tau\right)$ is called a semicircular system if $\{x_{1},\dots,x_{k}\}$ is a free family of selfadjoint random variables, and for every $i=1,\ldots,k$, $\mu_{x_{i}}$ is the standard semicircular distribution $\nu_{0,1}$ defined by

[TABLE]

An element $u\in\mathcal{A}$ is called a Haar unitary if $u^{*}=u^{-1}$ and $\tau(u^{n})=0$ for all $n\in\mathbb{Z}\setminus\{0\}$. Note that Theorem 1.2 in [12] deals with deterministic $a_{N}$ but the random case readily follows as pointed out by assertion 2 in [36, Section 3]. The point of Theorem 2.3 is, of course, that the resulting convergence is strong. Convergence in distribution was established earlier (see [49], [25], [3, Theorem 5.4.5]).

We also need a simple coupling result from [23, Lemma 5.1].

Lemma 2.4**.**

Suppose given selfadjoint matrices $C_{N},D_{N}\in M_{N}(\mathbb{C})$, $N\in\mathbb{N}$, such that the sequences $\{C_{N}\}_{N\in\mathbb{N}}$ and $\{D_{N}\}_{N\in\mathbb{N}}$ converge strongly in distribution. Then there exist diagonal matrices $\widetilde{C}_{N},\widetilde{D}_{N}\in M_{N}(\mathbb{C})$, $N\geq 1$, such that $\mu_{\widetilde{C}_{N}}=\mu_{C_{N}}$, $\mu_{\widetilde{D}_{N}}=\mu_{D_{N}}$, and the sequence $\{(\widetilde{C}_{N},\widetilde{D}_{N})\}_{N\in\mathbb{N}}$ converges strongly in distribution.

3. Description of the models

In order to describe in detail our matrix models, we need two compactly supported probability measures $\mu$ and $\nu$ on $\mathbb{R}$, a positive integer $p$, and a sequence of fixed real numbers $\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{p}$ in $\mathbb{R}\setminus\text{supp}(\mu)$. The matrix $A_{N}\in M_{N}(\mathbb{C})$ is random selfadjoint for all $N\in\mathbb{N},N\geq 1$ and satisfies the following conditions:

• (A1)

almost surely, the sequence $\{A_{N}\}_{N=1}^{\infty}$ converges in distribution to $\mu$,

• (A2)

$\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{p}$ are $p$ eigenvalues of $A_{N}$, and

• (A3)

the other eigenvalues of $A_{N}$, which may be random, converge uniformly almost surely to ${\rm supp}(\mu)$: almost surely, for every $\varepsilon>0$ there exists $N(\varepsilon)\in\mathbb{N}$ such that

[TABLE]

In other words, only the $p$ eigenvalues $\theta_{1},\ldots,\theta_{p}$ prevent $\{A_{N}\}_{N=1}^{\infty}$ from converging strongly in distribution to $\mu$.

We investigate two polynomial matricial models, both involving $A_{N}$. The first model involves a sequence $\{B_{N}\}_{N=1}^{\infty}$ of random Hermitian matrices such that

• (B0)

$B_{N}$ is independent from $A_{N}$,

• (B1)

$B_{N}$ converges strongly in distribution to the compactly supported probability measure $\nu$ on $\mathbb{R}$,

• (B2)

for each $N$, the distribution of $B_{N}$ is invariant under conjugation by arbitrary $N\times N$ unitary matrices.

The matricial model is

[TABLE]

for an arbitrary selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},X_{2}\rangle$.

The second model deals with $N\times N$ random Hermitian Wigner matrices $X_{N}=[{X}_{ij}]_{i,j=1}^{N}$, where $[X_{ij}]_{i\geq 1,j\geq 1}$ is an infinite array of random variables satifying conditions $(X0)-(X3)$ in the introduction. The matricial model is

[TABLE]

for an arbitrary selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},X_{2}\rangle$.

In the discussion of the first model, we use results of Voiculescu [49] (see also [54]), who showed that there exist a free pair $(a,b)$ of selfadjoint elements in a II1-factor $(\mathcal{A},\tau)$ such that, almost surely, the sequence $\{(A_{N},B_{N})\}_{N=1}^{\infty}$ converges in distribution to $(a,b)$. Thus, $\mu=\mu_{a},\nu=\mu_{b}$, and the sequence $\{P(A_{N},B_{N})\}_{N=1}^{\infty}$ converges in distribution to $P(a,b)$ (that is,

[TABLE]

in the weak∗ topology) for every selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},X_{2}\rangle$. When $p=0$, Lemma 2.4, Theorem 2.3 and Remark 2.1, show that, almost surely, this convergence is strong (see the proof of Corollary 2.2 in [23]).

For the second model we use [12, Proposition 2.2] and [3, Theorem 5.4.5], where it is seen that for every selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},X_{2}\rangle$ we have

[TABLE]

almost surely in the weak∗ topology, where $a$ and $b$ are freely independent selfadjoint noncommutative random variables, $\mu_{a}=\mu$, and $\mu_{b}=\nu_{0,1}$. As in the first model, Theorem 2.3 and Remark 2.1 show that, almost surely, the sequence $\{P(A_{N},{X_{N}}/{\sqrt{N}})\}_{N=1}^{\infty}$ converges strongly in distribution to $P(a,x)$ provided that $p=0$.

Our main result applies, of course, to the case in which $p>0$. Let $Y_{N}$ be either $B_{N}$ or ${X_{N}}/{\sqrt{N}}$. The set of outliers of $P(A_{N},Y_{N})$ is calculated from the spikes $\theta_{1},\dots,\theta_{p}$ using Voiculescu’s matrix subordination function [52]. When $Y_{N}=B_{N}$, we also show that the eigenvectors associated to these outlying eigenvalues have projections of computable size onto the eigenspaces of $A_{N}$ corresponding to the spikes. The precise statements are Theorems 6.1 and 6.3. Sections 4 and 5 contain the necessary tools from operator-valued noncommutative probability theory while Sections 7–10 are dedicated to the proofs of the main results.

4. Linearization

As in [4, 13], we use linearization to reduce a problem about a polynomial in freely independent, or asymptotically freely independent, random variables, to a problem about the addition of matrices having these random variables as entries. Suppose that $P\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$. For our purposes, a linearization of $P$ is a linear polynomial of the form

[TABLE]

where $z$ is a complex variable, and

[TABLE]

with $\alpha,\gamma_{0},\dots,\gamma_{k}\in M_{n}(\mathbb{C})$ for some $n\in\mathbb{N}$, and the following property is satisfied: given $z\in\mathbb{C}$ and elements $a_{1},\dots,a_{k}$ in a $C^{*}$-algebra $\mathcal{A}$, $z-P(a_{1},\dots,a_{k})$ is invertible in $\mathcal{A}$ if and only if $z\alpha\otimes 1-L(a_{1},\dots,a_{k})$ is invertible in $M_{n}(\mathcal{A})$. Usually, this is achieved by ensuring that $(z\alpha\otimes 1-L)^{-1}$ exists as an element of $M_{n}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle\langle(z-P)^{-1}\rangle)$ and $(z-P)^{-1}$ is one of the entries of the $(z\alpha\otimes 1-L)^{-1}$. It is known (see, for instance, [42]) that every polynomial has a linearization. See [29] for earlier uses of linearization in free probability.

In the following we also say, more concisely, that $L$ is a linearization of $P$. We also suppress the unit of the algebra $\mathcal{A}$ when there is no risk of confusion. For instance, we may write $z\alpha-L$ in place of $z\alpha\otimes 1-L.$

We describe in some detail a linearization procedure from [4] (see also [34]) that has several advantages. In this procedure, we always have $\alpha=e_{1,1}$, where $e_{1,1}$ denotes the matrix whose only nonzero entry equals $1$ and occurs in the first row and first column. Given $P\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$, we produce an integer $n\in\mathbb{N}$ and a linear polynomial $L\in M_{n}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$ of the form

[TABLE]

such that $u\in M_{1\times(n-1)}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$, $v\in M_{(n-1)\times 1}(\mathbb{C}\langle X_{1}\dots,X_{k}\rangle)$, $Q$ is an invertible matrix in $M_{n-1}(\mathbb{C}\langle X_{1},\dots X_{k}\rangle)$ whose inverse is a polynomial of degree less than or equal to the degree of $P$, and $uQ^{-1}v=-P$. Moreover, if $P=P^{*}$, the coefficients of $L$ can be chosen to be selfadjoint matrices in $M_{n}(\mathbb{C})$.

The construction proceeds by induction on the number of monomials in the given polynomial. If $P$ is a monomial of degree [math] or $1$, we set $n=1$ and $L=P$. If $P=X_{i_{1}}X_{i_{2}}X_{i_{3}}\cdots X_{i_{\ell-1}}X_{i_{\ell}}$, where $\ell\geq 2$ and $i_{1},\dots,i_{\ell}\in\{1,\dots,k\}$, we set $n=\ell$ and

[TABLE]

As noted in [34], the lower right $(\ell-1)\times(\ell-1)$ corner of this matrix has an inverse of degree $\ell-2$ in the algebra $M_{\ell-1}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$. (The constant term in this inverse is a selfadjoint matrix and its spectrum is contained in $\{-1,1\}$.) Suppose now that $p=P_{1}+P_{2}$, where $P_{1},P_{2}\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$, and that linear polynomials

[TABLE]

with the desired properties have been found for $P_{1}$ and $P_{2}$. Then we set $n=n_{1}+n_{2}-1$ and observe that the matrix

[TABLE]

is a linearization of $P_{1}+P_{2}$ with the desired properties. The construction of a linearization is now easily completed for an arbitrary polynomial. Suppose now that $P$ is a selfadjoint polynomial, so $P=P_{0}+P_{0}^{*}$ for some other polynomial $P_{0}$. Suppose that the matrix

[TABLE]

of size $n_{0}$ is a linearization of $P_{0}$. Then we set $n=2n_{0}-1$ and observe that the selfadjoint linear polynomial

[TABLE]

linearizes $P$. It is easy to verify inductively that this construction produces a matrix $Q$ such that the constant term of $Q^{-1}$ has spectrum contained in $\{1,-1\}$. These properties of $Q$ [34], and particularly the following observation, facilitate our analysis.

Lemma 4.1**.**

Let $P\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$, and let

[TABLE]

be a linearization of $P$ as constructed above. There exist a permutation matrix $T\in M_{n-1}$ and a strictly lower triangular matrix $N\in M_{n-1}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$ such that $Q^{-1}=T(1_{n-1}+N)$.

Proof.

We show that there exist a permutation matrix $T_{0}\in M_{n-1}$ and a strictly lower triangular matrix permutation matrices $N_{0}\in M_{n-1}(\mathbb{C}\langle X_{1},\dots,X_{k}\rangle)$ such that $Q=(1_{n-1}-N_{0})T_{0}$. Then we can define $T=T_{0}^{-1}$ and $N=\sum_{j=1}^{n-2}N_{0}^{j}$. The existence of $T_{0}$ and $N_{0}$ is proved by following inductively the construction of $L$. If $P=X_{i_{1}}\cdots X_{i_{\ell}}$, $\ell\geq 2$, we define

[TABLE]

and let the only nonzero entries of $N_{0}$ be $X_{i_{2}},\dots,X_{i_{\ell}}$ just below the main diagonal. If $P=P_{1}+P_{2}$, and linearizations for $P_{1}$ and $P_{2}$ have been found, then the desired matrices are obtained simply by taking direct sums of the matrices corresponding to $P_{1}$ and $P_{2}$. The case in which $P=P_{0}+P_{0}^{*}$ is treated similarly (different factorizations must be used for $Q_{0}$ and $Q_{0}^{*}$). ∎

Lemma 4.2**.**

Suppose that $P\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$, and let

[TABLE]

be a linearization of $P$ with the properties outlined above. Then for every $N\in\mathbb{N}$, and for every $S_{1},\dots,S_{k}\in M_{N}(\mathbb{C})$, we have

[TABLE]

where the sign is $\det(Q(S_{1},\dots,S_{k}))$. Moreover,

[TABLE]

Proof.

Suppressing the variables $S_{1},\dots,S_{k}$, we have

[TABLE]

Lemma 4.1 implies that $\det Q(S_{1},\dots,S_{k})$ is $\pm 1$ and the determinant identity follows immediately. The dimension of the kernel of a square matrix does not change if the matrix is multiplied by some other invertible matrices. Also, since $Q$ is invertible, the kernel of the matrix on the right hand side of the last equality is easily identified with $\ker(z-P)$. The last assertion follows from these observations. ∎

In the case of selfadjoint polynomials, applied to selfadjoint matrices, we can estimate how far $ze_{1,1}-L$ is from not being invertible.

Lemma 4.3**.**

Suppose that $P=P^{*}\in\mathbb{C}\langle X_{1},\dots,X_{k}\rangle$, and let

[TABLE]

be a linearization of $P$ with the properties outlined above. There exist polynomials $T_{1},T_{2}\in\mathbb{C}[X_{1},\dots,X_{k}]$ with nonnegative coefficients with the following property: given arbitrary selfadjoint elements $S_{1},\dots,S_{k}$ in a unital $C^{*}$-algebra $\mathcal{A}$, and given $z_{0}\in\mathbb{C}$ such that $z_{0}-P(S)$ is invertible, we have

[TABLE]

In particular, given two real constants $C,\delta>0$, there exists $\varepsilon>0$ such that ${\mathrm{dist}}(z_{0},\sigma(P(S)))\geq\delta$ and $\|S_{1}\|+\cdots+\|S_{k}\|\leq C$ imply $\mathrm{dist}(0,\sigma(z_{0}e_{1,1}-L(S)))\geq\varepsilon$.

Proof.

For every element $a$ of a $C^{*}$-algebra, we have ${\rm{dist}}(0,\sigma(a))\geq 1/\|a^{-1}\|$. Equality is achieved, for instance, if $a=a^{*}$. A matrix calculation (in which we suppress the variables $S$) shows that

[TABLE]

The lemma follows now because the entries of $u(S)$, $u^{*}(S)$, and $Q(S)^{-1}$ are polynomials in $S$, and

[TABLE]

because $P(S)$ is selfadjoint. ∎

The dependence on $L$ in the above lemma is given via the norms of $Q^{-1}$ and of $u$. Since $\lim_{z\to\infty}\|(ze_{1,1}-L(S))^{-1}\|\neq 0$, we see that $T_{2}\neq 0$.

5. Subordination

Consider a von Neumann algebra $\mathcal{M}$ endowed with a normal faithful tracial state $\tau$, let $\mathcal{B}\subset\mathcal{N}\subset\mathcal{M}$ be unital von Neumann subalgebras, and denote by $E_{\mathcal{N}}:\mathcal{M}\to\mathcal{N}$ the unique trace-preserving conditional expectation of $\mathcal{M}$ onto $\mathcal{N}$ (see [46, Proposition V.2.36]). Denote by $\mathbb{H}^{+}(\mathcal{M})$ the operator upper-half plane of $\mathcal{M}$: $\mathbb{H}^{+}(\mathcal{M})=\{x\in\mathcal{M}\colon\Im x:=(x-x^{*})/2i>0\}$. Given two arbitrary selfadjoint elements $c,d\in\mathcal{M}$, we define the open set $\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}$ to consist of those elements $\beta\in\mathcal{B}$ such that $\beta-(c+d)$ is invertible and $E_{\mathcal{N}}((\beta-(c+d))^{-1})$ is invertible as well. Then the function

[TABLE]

defined by

[TABLE]

is analytic. This equation can also be written as

[TABLE]

Properties (1), (2), and (3) in the following lemma are easy observations, while (4) follows as in [10, Remark 2.5].

Lemma 5.1**.**

Fix $\mathcal{B}\subset\mathcal{N}\subset\mathcal{M}$ and $c,d\in\mathcal{M}$ as above. Then:

1. (1)

The set $\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}$ is selfadjoint. 2. (2)

$\omega_{c,d,\mathcal{B},\mathcal{N}}(\beta^{*})=\omega_{c,d,\mathcal{B},\mathcal{N}}(\beta)^{*}$, $\beta\in\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}.$ 3. (3)

$\mathbb{H}^{+}(\mathcal{B})\subset\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}$* and $\omega_{c,d,\mathcal{B},\mathcal{N}}(\mathbb{H}^{+}(\mathcal{B}))\subset\mathbb{H}^{+}(\mathcal{N})$.* 4. (4)

$\Im(\omega_{c,d,\mathcal{B},\mathcal{N}}(\beta))\geq\Im(\beta)$, $\beta\in\mathbb{H}^{+}(\mathcal{B})$.

There is one important case in which $\omega_{c,d,\mathcal{B},\mathcal{N}}$ takes values in $\mathcal{B}$, and thus (5.2) allows us to view $\omega_{c,d,\mathcal{B},\mathcal{N}}|\mathbb{H}^{+}(\mathcal{B})$ as a subordination function in the sense of Littlewood. Denote by $\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}^{0}$ the connected component of $\mathcal{G}_{c,d,\mathcal{B},\mathcal{N}}$ that contains $\mathbb{H}^{+}(B)$. The following basic result is from [48].

Theorem 5.2**.**

With the above notation, suppose that $c$ and $d$ are free over $\mathcal{B}$ and $\mathcal{N}=\mathcal{B}\langle c\rangle$ is the unital von Neumann generated by $\mathcal{B}$ and $c$. Then

[TABLE]

In our applications, the algebra $\mathcal{B}$ is (isomorphic to) $M_{n}(\mathbb{C})$ for some $n\in\mathbb{N}$. More precisely, let $\mathcal{M}$ be a von Neumann algebra endowed with a normal faithful tracial state $\tau$, and let $n\in\mathbb{N}$. Then $M_{n}(\mathbb{C})$ can be identified with the subalgebra $M_{n}(\mathbb{C})\otimes 1$ of $M_{n}({\mathcal{M}})=M_{n}(\mathbb{C})\otimes\mathcal{M}$. Moreover, $M_{n}({\mathcal{M}})$ is endowed with the faithful normal tracial state $\mathrm{tr}_{n}\otimes\tau=(1/n)\mathrm{Tr}_{n}\otimes\tau$, and $\mathrm{Id}_{M_{n}(\mathbb{C})}\otimes\tau$ is the trace-preserving conditional expectation from $M_{n}({\mathcal{M}})$ to $M_{n}(\mathbb{C})$. The following result is from [37].

Proposition 5.3**.**

Let $\mathcal{M}$ be a von Neumann algebra endowed with a normal faithful tracial state $\tau$, let $c,d\in\mathcal{M}$ be freely independent, let $n$ be a positive integer, and let $\gamma_{1},\gamma_{2}\in M_{n}(\mathbb{C})$. Then $\gamma_{1}\otimes c$ and $\gamma_{2}\otimes d$ are free over $M_{n}(\mathbb{C})$.

We show next how the spectrum of $P(c,d)$ relates with the functions $\omega$ defined above. Thus, we fix $P=P^{*}\in\mathbb{C}\langle X_{1},X_{2}\rangle$ and a linearization $L=\gamma_{0}\otimes 1+\gamma_{1}\otimes X_{1}+\gamma_{2}\otimes X_{2}$ of $P$ as constructed in Section 4. Thus, $\gamma_{0},\gamma_{1},\gamma_{2}\in M_{n}(\mathbb{C})$ are selfadjoint matrices for some $n\in\mathbb{N}$. (Clearly $\gamma_{1}\neq 0$ unless $P\in\mathbb{C}\langle X_{2}\rangle$.) Then we consider the random variables $\gamma_{1}\otimes c$ and $\gamma_{2}\otimes d$ in $M_{n}(\mathcal{M})$, the algebra $\mathcal{B}=M_{n}(\mathbb{C})\subset M_{n}(\mathcal{M})$, and $\mathcal{N}=M_{n}(\mathbb{C}\langle c\rangle)$; clearly $\mathcal{B}\subset\mathcal{N}\subset M_{n}(\mathcal{M})$. We set

[TABLE]

and

[TABLE]

Thus,

[TABLE]

The left hand side of this equation is defined if $\beta=ze_{11}-\gamma_{0}$ for some $z\in\mathbb{C}\setminus\sigma(P(c,d))$, so it would be desirable that $ze_{11}-\gamma_{0}\in\mathcal{G}$ for such values of $z$. This is not true except for special cases. (One such case applies to $P=X_{1}+X_{2}$ if $d$ is a semicircular variable free from $c$ [16, 9].) The following lemma offers a partial result.

Lemma 5.4**.**

With the notation above, there exists $k>0$ depending only on $L$, $\|c\|$, and $\|d\|$ such that $ze_{1,1}-\gamma_{0}\in\mathcal{G}$ if $|z|>k$. The analytic function $u(z)=\omega(ze_{1,1}-\gamma_{0})$ satisfies the equation $u(\overline{z})=u(z)^{*}$ for $|z|>k$.

Proof.

Define an analytic function $F:\mathbb{C}\setminus\sigma(P(c,d))\to\mathcal{N}$ by

[TABLE]

We show that $F(z)$ is invertible if $|z|$ is sufficiently large. Suppressing the variables $c$ and $d$ from the notation, it follows from the factorization used in the proof of Lemma 4.2 that

[TABLE]

Moreover, because of the matrix structure of $\mathcal{N}$, this matrix can be obtained by applying $E_{\mathbb{C}\langle c\rangle}$ entrywise. According to the Schur complement formula, a matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$ is invertible if both $A$ and $D-BA^{-1}C$ are invertible. For our matrix, we have $A=A(z)=E_{\mathbb{C}\langle c\rangle}((z-P)^{-1})$. The fact that $\|zA(z)-1\|<1$ for $|z|>2\|P\|$ implies that $A(z)$ is invertible. Next, we see that $\|(z-P)^{-1}\|$ and $\|A(z)^{-1}\|$ are comparable to $1/|z|$ and $|z|$, respectively. Using these estimates, one sees also that $\lim_{|z|\to\infty}\|B(z)A(z)^{-1}C(z)\|=0$, so the invertibility of $F(z)$ would follow from the invertibility of $D(z)$ for large $|z|$. Since

[TABLE]

we only need to verify that $E_{M_{n-1}({\mathbb{C}\langle c\rangle})}(Q^{-1})$ is invertible. Write $Q^{-1}=T(1_{n-1}+N)$ as in Lemma 4.1. We have

[TABLE]

and $E_{M_{n\!-\!1}({\mathbb{C}\langle c\rangle})}(N)$ is strictly lower triangular. The invertibility of $E_{M_{n\!-\!1}({\mathbb{C}\langle c\rangle})}(Q^{-1})$ follows. The quantities $\|D(z)+E_{M_{n-1}({\mathbb{C}\langle c\rangle})}(Q^{-1})\|$ and $\|Q^{-1}\|$ can be estimated using only $L$, $\|c\|$, and $\|d\|$, and this shows that $k$ can be chosen as a function of these objects. The last assertion of the lemma is immediate. ∎

The estimates in the preceding proof apply, by virtue of continuity, to nearby points in $M_{n}(\mathbb{C})$. We record the result for later use.

Corollary 5.5**.**

Let $c,d$ and $k$ be as in Lemma 5.4 and let $z\in\mathbb{C}\setminus[-k,k]$. Then there exist a constant $k^{\prime}>0$ and a neighborhood $W$ of $ze_{1,1}-\gamma_{0}$, depending only on $\|c\|,\|d\|$, and $L$, such that $V\subset\mathcal{G}$ and $\|\omega(\beta)\|\leq k^{\prime}$ for $\beta\in W$.

In some cases of interest, the analytic function $u$ extends to the entire upper and lower half-planes. We recall that a function $v$ defined in a domain $G\subset\mathbb{C}$ with values in a Banach space $\mathcal{X}$ is said to be meromorphic if, for every $z_{0}\in G$, the function $(z-z_{0})^{n}v(z)$ is analytic in a neighborhood of $z_{0}$ for sufficiently large $n$. For instance, if $\mathcal{X}$ is a finite dimensional Banach algebra and $h:G\to\mathcal{X}$ is an analytic function such that $h(z)$ is invertible for some $z\in G$, then the function $v(z)=h(z)^{-1}$ is meromorphic in $G$. This fact follows easily once we identify $\mathcal{X}$ with an algebra of matrices, so the inverse can be calculated using determinants.

Lemma 5.6**.**

The function $u$ defined in Lemma 5.4 is meromorphic in $\mathbb{C}\setminus\sigma(P(c,d))$.

Proof.

The lemma follow immediately from the observation preceding the statement applied to the function $F$ defined in (5.3) which is analytic in $\mathbb{C}\setminus\sigma(P(c,d))$ since, by hypothesis and by Theorem 5.2, $u$ takes values in a finite dimensional algebra. ∎

The conclusion of the preceding lemma applies, for instance, when $\mathcal{M}=M_{N}\otimes L^{\infty}(\Omega)$ with the usual trace $\mathrm{tr}_{N}\otimes\mathbb{E}$. This situation arises in the study of random matrices. The function $u$ is also meromorphic provided that $c$ and $d$ are free random variables and $\mathcal{N}=M_{n}(\mathbb{C})\langle\gamma_{1}\otimes c\rangle$. Since $\gamma_{1}\neq 0$, we have $\mathcal{N}=M_{n}(\mathbb{C}\langle c\rangle)$.

Lemma 5.7**.**

If $c$ and $d$ are free and $\mathcal{N}=M_{n}(\mathbb{C})\langle\gamma_{1}\otimes c\rangle$, then the function $u$ defined in Lemma 5.4 is meromorphic in $\mathbb{C}\setminus\sigma(P(c,d))$ with values in $M_{n}(\mathbb{C})$. Moreover, given an arbitrary $\lambda\in\sigma(c)$, the function $(u-\lambda\gamma_{1})^{-1}$ extends analytically to $\mathbb{C}\setminus\sigma(P(c,d))$.

Proof.

Theorem 5.2 shows that $\omega$ takes values in $M_{n}(\mathbb{C})$. We have established that the domain $\mathcal{G}$ of $\omega$ contains $z\otimes e_{11}-\gamma_{0}$ for sufficiently large $|z|$. Fix a character $\chi$ of the commutative $C^{*}$-algebra $\mathbb{C}\langle c\rangle$ and denote by $\chi_{n}:M_{n}(\mathbb{C})\langle\gamma_{1}\otimes c\rangle\to M_{n}(\mathbb{C})$ the algebra homomorphism obtained by applying $\chi$ to each entry. Using the notation (5.3), we have

[TABLE]

for sufficiently large $|z|$. It follows immediately that the function

[TABLE]

is a meromorphic continuation of $u$ to $\mathbb{C}\setminus\sigma(P(c,d))$. Moreover, the equation

[TABLE]

holds for large values $|z|$ and hence it holds on the entire domain of analyticity of $u_{1}$. It follows that $\mathcal{G}$ contains $z\otimes e_{11}-\gamma_{0}$ whenever $z\in\mathbb{C}\setminus\sigma(P(c,d))$ is not a pole of $u_{1}$, and $\omega(z\otimes e_{11}-\gamma_{0})=u_{1}(z)$ for such values of $z$. To verify the last assertion, choose $\chi$ such that $\chi(c)=\lambda$ and apply $\chi_{n}$ to (5.4) to obtain

[TABLE]

Thus $\chi_{n}\circ F$ is an analytic extension of $(u-\lambda\gamma_{1})^{-1}$ to $\mathbb{C}\setminus\sigma(P(c,d))$. ∎

6. Main results and example

Fix a polynomial $P=P^{*}\in\mathbb{C}\langle X_{1},X_{2}\rangle$ and choose, as in Section 4, a linearization of $P$ of the form $ze_{1,1}-L$, where $L=\gamma_{0}\otimes 1+\gamma_{1}\otimes X_{1}+\gamma_{2}\otimes X_{2}\in M_{n}(\mathbb{C}\langle X_{1},X_{2}\rangle)$. In particular, $\gamma_{0},\gamma_{1},\gamma_{2}\in M_{n}(\mathbb{C})$ are selfadjoint matrices.

Suppose that $\{A_{N}\}_{N\in\mathbb{N}}$ and $\{B_{N}\}_{N\in\mathbb{N}}$ are two sequences of selfadjoint random matrices satisfying the hypotheses (A1)–(A3) and (B0)–(B2) of Section 3. As noted earlier, the pairs $(A_{N},B_{N})$ in $M_{N}(\mathbb{C})$ converge almost surely in distribution to a pair $(a,b)$ of freely independent selfadjoint random variables in a $C^{*}$-probability space $(\mathcal{A},\tau)$ such that $\mu_{a}=\mu$ and $\mu_{b}=\nu$. By Theorem 5.2, there exists a selfadjoint open set $\mathcal{G}\subset M_{n}(\mathbb{C})$, and an analytic function $\omega\colon\mathcal{G}\to M_{n}(\mathbb{C})$ such that

[TABLE]

As shown in Lemma 5.7, the map

[TABLE]

is meromorphic on $\mathbb{C}\setminus\sigma(P(a,b))$. Define a new function

[TABLE]

It follows from Lemma 5.1 that $u_{0}$ continues analytically to a neighbourhood of $\mathbb{R}\setminus\sigma(P(a,b))$. (Indeed, $u_{0}$ is bounded near every pole of $u$.) Define

[TABLE]

and denote by $m_{j}(t)$ the order of $t$ as a zero of $H_{j}(z)$ at $z=t$. Also set $m(t)=m_{1}(t)+\cdots+m_{p}(t)$ for $t\in\mathbb{R}\setminus\sigma(P(a,b))$, and note that $\{t:m(t)\neq 0\}$ is an isolated set in $\mathbb{R}\setminus\sigma(P(a,b))$. With this notation, we are ready to state our first main result. The notation $E_{A_{N}}$ indicates the spectral measure of the matrix $A_{N}$, that is, $E_{A_{N}}(S)$ is the orthogonal projection onto the linear span of the eigenvectors of $A$ corresponding to eigenvalues in the Borel set $S$.

Theorem 6.1**.**

(1)* Suppose that $t\in\mathbb{R}\setminus\sigma(P(a,b))$. Then there exists $\delta_{0}>0$ such that for every $\delta\in(0,\delta_{0})$, almost surely for large $N$, the random matrix $P(A_{N},B_{N})$ has exactly $m(t)$ eigenvalues in the interval $(t-\delta,t+\delta)$, counting multiplicity.*

(2)* Suppose in addition that the spikes of $A_{N}$ are distinct and $\det H_{i_{0}}(t)=0$. Then, for $\varepsilon$ small enough, almost surely*

[TABLE]

where $\mathcal{C}_{i}(t)=\lim_{z\to t}(z-t)\left[(u(z)-\theta_{i}\gamma_{1})^{-1}\right]_{1,1}$ is the residue of the meromorphic function $\left[(u(z)-\theta_{i}\gamma_{1})^{-1}\right]_{1,1}$ at $z=t$.

Remark 6.2**.**

If we know in addition that $\omega$ is analytic at the point $\beta=te_{1,1}-\gamma_{0}$, then the function $H_{j}(z)$ can be replaced by $z\mapsto\det[\theta_{j}\gamma_{1}-u(z)].$ In that case, $m(t)$ is equal to the multiplicity of $t$ as a zero of

[TABLE]

This situation arises, for instance, when $b$ is a semicircular variable and it is relevant when $B_{N}$ is replaced by a Wigner matrix $X_{N}/\sqrt{N}$. Under the hypotheses $(X0)-(X3)$ of Section 3, we obtain the following result. Note that the subordination function $\omega$ has the more explicit form

[TABLE]

Theorem 6.3**.**

Let $a$ and $b$ be free selfadjoint elements in a ${C}^{*}$-probability space $({\mathcal{A}},\tau)$ with distribution $\mu$ and $\nu_{0,1}$ respectively $($see (2.3)$)$, $t\in\mathbb{R}\setminus\sigma({P(a,b)})$, and let $m(t)$ be defined as in Remark 6.2. Then, for sufficiently small $\varepsilon$, almost surely for large $N$, there are exactly $m(t)$ eigenvalues of $P(A_{N},{X_{N}}/{\sqrt{N}})$ in an $\varepsilon$-neighborhood of $t$.

Remark 6.4**.**

The subordination function can be calculated more explicitly if $\mu=\delta_{0}$ (and hence $a=0$). In this case,

[TABLE]

As an illustration, consider the random matrix

[TABLE]

where $X_{N}$ is a standard G.U.E. matrix of size $N$ (thus, each entry of $X_{N}$ has unit norm in $L^{2}(\Omega)$) and

[TABLE]

In this case, $A_{N}$ has rank one, and thus $\mu=\delta_{0}$. It follows that the limit spectral measure $\rho$ of $M$ is the same as the limit spectral measure of $X_{N}^{2}/{N}$. Thus, $\eta$ is the Marchenko-Pastur distribution $\rho$ with parameter 1:

[TABLE]

The polynomial $P$ is $P(X_{1},X_{2})=X_{1}X_{2}+X_{2}X_{1}+X_{2}^{2}$, $\mu=\delta_{0}$, and $\nu$ is the standard semi-circular distribution. An economical linearization of $P$ is provided by $L=\gamma_{0}\otimes 1+\gamma_{1}\otimes X_{1}+\gamma_{2}\otimes X_{2}$, where

[TABLE]

Denote by

[TABLE]

the Cauchy transform of the measure $\eta$. (The branch of the square root is chosen so $\sqrt{z^{2}-4z}>0$ for $z>4$.) This function satisfies the quadratic equation $zG_{\eta}(z)^{2}-zG_{\eta}(z)+1=0$. Suppose now that $x\notin[0,4]$. Denoting by $E={\rm Id}_{M_{3}(\mathbb{C})}\otimes\tau\colon M_{3}(\mathcal{A})\to M_{3}(\mathbb{C})$ the usual expectation and using Remark 6.4, we have

[TABLE]

The inverse of $(xe_{1,1}-\gamma_{0})\otimes 1-\gamma_{2}\otimes b$ is then calculated explicitly and application of the expected value to its entries yields eventually

[TABLE]

After calculation, the equation $\det[\gamma_{1}\theta-\omega(xe_{11}-\gamma_{0})]=0$ reduces to

[TABLE]

This equation has two solutions, namely

[TABLE]

one of which is negative. The positive solution belongs to $[4,+\infty)$ precisely when $|\theta|>\sqrt{2}$. Thus, the matrix $M_{N}$ exhibits one (negative) outlier when $0<|\theta|\leq\sqrt{2}$ and two outliers (one negative and one $>4$) when $|\theta|>\sqrt{2}$. The second situation is illustrated by the simulation presented in Figure 1.

7. Outline of the proofs

We consider first the matricial model (3.1), that is, $Z_{N}=P(A_{N},B_{N})$, where $A_{N}$ and $B_{N}$ are independent and the distribution of $B_{N}$ is invariant under unitary conjugation. As seen in [23, Proposition 6.1], $B_{N}$ can be written as $B_{N}=U_{N}D_{N}U_{N}^{*}$ almost surely, where $U_{N}$ is distributed according to the Haar measure on the unitary group ${\rm U}(N)$, $D_{N}$ is a diagonal random matrix, and $U_{N}$ is independent from $D_{N}$. As pointed out in [36, Section 3, Assertion 2], it suffices to prove Theorem 6.1 under the assumption that $A_{N}$ and $D_{N}$ are constant selfadjoint matrices that can be taken to be diagonal in the standard basis. Thus, we work with

[TABLE]

and

[TABLE]

where $\lambda_{j}(A_{N})=\theta_{j},1\leq j\leq p$, and $U_{N}$ is uniformly distributed in ${\rm U}(N)$.

Similarly, the proof for the second model $Z_{N}=P(A_{N},X_{N}/\sqrt{N})$ reduces to the special case in which $A_{N}$ is a constant matrix.

Choose a linearization $L$ of $P$ as in Section 4. In the spirit of [14], the first step in the proofs of Theorems 6.1 and 6.3 consists of reducing the problem to the convergence of random matrix function $F_{N}$ of fixed size $np$, involving the generalized resolvent of the linearization applied to $Z_{N}$. For the first model, this convergence is established in Section 8 by extending the arguments of [11] and making use of the properties of the operator-valued subordination function described in Section 5. For the second model, the convergence of $F_{N}$ is obtained in Section 10 via a comparison with the G.U.E. case. The case in which $X_{N}$ is a G.U.E. is, of course, a particular case of the unitarily invariant model.

8. Expectations of matrix-valued random analytic maps

As seen earlier in this paper, it suffices to prove Theorem 6.1 in the special case in which the matrix $A_{N}$ is constant and $B_{N}$ is a random unitary conjugate of another constant matrix. In this section, we establish some useful ingredients specific to this situation. We fix sequences $\{C_{N}\}_{N\in\mathbb{N}}$ and $\{D_{N}\}_{N\in\mathbb{N}}$, where $C_{N},D_{N}\in M_{N}(\mathbb{C})$, and a sequence $\{U_{N}\}_{N\in\mathbb{N}}$ of random matrices such that $U_{N}$ is uniformly distributed in the unitary group $\mathrm{U}(N)$. We also fix a selfadjoint polynomial $P\in\mathbb{C}\langle X_{1},X_{2}\rangle$ and a selfadjoint linearization

[TABLE]

of $P$ as in Section 4. The random variables $c_{N}=C_{N}\otimes 1_{\Omega}$ and $d_{N}=U_{N}D_{N}U_{N}^{*}$ are viewed as elements of the noncommutative probability space $(\mathcal{M}_{N},\tau_{N})$, where $\mathcal{M}_{N}=M_{N}(\mathbb{C})\otimes L^{\infty}(\Omega)$ and $\tau_{N}=\mathrm{tr}_{N}\otimes\mathbb{E}$. For every $N\in\mathbb{N}$ we consider the elements $\gamma_{1}\otimes c_{N},\gamma_{2}\otimes d_{N}\in M_{n}(\mathbb{C})\otimes\mathcal{M}_{N}$, and the algebras $\mathcal{B}_{N}=M_{n}(\mathbb{C})\otimes I_{N}$ and $\mathcal{N}_{N}=M_{n}(\mathbb{C})\otimes M_{N}(\mathbb{C})$, both identified with subalgebras of $M_{n}(\mathbb{C})\otimes\mathcal{M}_{N}$. The conditional expectation $E_{\mathcal{N}_{N}}:M_{n}(\mathbb{C})\otimes\mathcal{M}_{N}\to\mathcal{N}_{N}$ is simply the expected value and it is accordingly denoted $\mathbb{E}$. We use the notation

[TABLE]

Recall that $\mathcal{G}_{N}$ consists of those matrices $\beta\in M_{n}(\mathbb{C})$ with the property that $\beta-\gamma_{1}\otimes c_{N}-\gamma_{2}\otimes d_{N}$ is invertible in $M_{n}(\mathbb{C})\otimes\mathcal{M}_{N}$ and $\mathbb{E}((\beta-\gamma_{1}\otimes c_{N}-\gamma_{2}\otimes d_{N})^{-1})$ is invertible in $\mathcal{N}_{N}$. In particular, if $\beta\in\mathcal{G}_{N}$ then the matrix $\beta-\gamma_{1}\otimes C_{N}-\gamma_{2}\otimes VD_{N}V^{*}$ is invertible for every $V\in{\mathrm{U}}(N)$. The set $\mathcal{G}_{N}$ is open and it contains $\mathbb{H}^{+}(M_{n}(\mathbb{C}))$. According to Lemma 5.6 and the remarks following it, the function $u_{N}(z)=\omega_{N}(ze_{1,1}-\gamma_{0})$ is meromorphic in $\mathbb{C}\setminus\sigma(P(c_{N},d_{N}))$.

For simplicity of notation, we write

[TABLE]

and observe that $R_{N}(\beta)$ is an element of $M_{n}(\mathbb{C})\otimes M_{N}(\mathbb{C})\otimes L^{\infty}(\Omega)$, that is, a random matrix of size $nN$. We also write

[TABLE]

for sample values of this random variable. The function $\omega_{N}$ is given by

[TABLE]

We start by showing that the matrix $\omega_{N}(\beta)$ has a block diagonal form, thus extending [11, Lemma 4.7]. We recall that the commutant and double commutant of a set $S\subset M_{m}(\mathbb{C})$ are denoted by $S^{\prime}$ and $S^{\prime\prime}$, respectively. We use the fact that $M_{n}(\mathbb{C})\otimes S^{\prime\prime}=(I_{n}\otimes S^{\prime})^{\prime}$. If $S=\{C_{N}\}$ then $\{C_{N}\}^{\prime\prime}$ is the linear span of the matrices $\{I_{N},C_{N},\dots,C_{N}^{N-1}\}$. In particular, every eigenvector of $C_{N}$ is a common eigenvector for the elements of $\{C_{N}\}^{\prime\prime}$.

For each $N\in\mathbb{N}$, we select an eigenbasis $\{f_{N}^{(1)},\dots,f_{N}^{(N)}\}$ for the operator $C_{N}$ and denote by $\lambda_{N}^{(j)}$ the corresponding eigenvalues, that is, $C_{N}f_{N}^{(j)}=\lambda_{N}^{(j)}f_{N}^{(j)}$. We write $P_{N}^{(j)}=f_{N}^{(j)}\otimes f_{N}^{(j)*}\in M_{N}(\mathbb{C})$ for the orthogonal projection onto the space generated by $f_{N}^{(j)}$, $j=1,\dots,N$. Thus, the double commutant $\{C_{N}\}^{\prime\prime}$ is contained in the linear span of $\{P_{N}^{(j)}:j=1,\dots,n\}$.

We write ${\textbf{[}}x,y{\textbf{]}}=xy-yx$ for the commutator of two elements $x,y$ in an algebra.

Lemma 8.1**.**

For every $\beta\in\mathcal{G}_{N}$ we have:

• (1)

$\mathbb{E}(R_{N}(\beta))\in M_{n}(\mathbb{C})\otimes\{C_{N}\}^{\prime\prime}$. In particular, there exist analytic functions $\omega_{N}^{(j)}:\mathcal{G}_{N}\to M_{n}(\mathbb{C})$, $j=1,\dots,N$, such that

[TABLE]

• (2)

For every $Z\in M_{N}(\mathbb{C})$,

[TABLE]

Proof.

The first assertion in (1) follows from an application of (2) to an arbitrary matrix $Z\in\{C_{N}\}^{\prime}$ and from the fact that $M_{n}(\mathbb{C})\otimes\{C_{N}\}^{\prime\prime}=(I_{n}\otimes\{C_{N}\}^{\prime})^{\prime}$. The second assertion follows because we also have $\gamma_{1}\otimes C_{N}\in M_{n}(\mathbb{C})\otimes\{C_{N}\}^{\prime\prime}$. To prove (2), observe that the analytic map

[TABLE]

is defined for $W$ in an open neighbourhood of the set of selfadjoint matrices in $M_{N}(\mathbb{C})$. The unitary invariance of $d_{N}$ implies that $H(W)=\mathbb{E}(R(\beta))$ if $W$ is selfadjoint. Since the selfadjoint matrices form a uniqueness set for analytic functions, we conclude that $H$ is constant on an open subset of $M_{N}(\mathbb{C})$ containing the selfadjoint matrices. Given an arbitrary $Z\in M_{N}(\mathbb{C})$, we conclude that the function $\varepsilon\mapsto H(\varepsilon Z)$ is defined and constant for $\varepsilon\in\mathbb{C}$ with $|\varepsilon|$ sufficiently small. Differentiate with respect to $\varepsilon$ and set $\varepsilon=0$, we obtain

[TABLE]

The equality

[TABLE]

applied in the relation above, yields (2). ∎

The following result is simply a reformulation of Lemma 8.1 that emphasizes the fact that the functions $\beta\mapsto(\omega_{N}^{(j)}(\beta)-\lambda_{N}^{(j)}\gamma_{1})^{-1}$ extend holomorphically to the open set $\{\beta\in\mathbb{M}_{N}(\mathbb{C}):\beta-\gamma_{1}\otimes c_{N}-\gamma_{2}\otimes d_{N}\text{ is invertible}\}.$

Corollary 8.2**.**

We have

[TABLE]

for every $\beta\in\mathcal{G}_{N}$ such that $\omega_{N}{(j)}(\beta)-\lambda_{N}^{(j)}\gamma_{1}$ is invertible.

It is useful to rewrite assertion (2) of Lemma 8.1 as follows:

[TABLE]

This is analogous to [11, (4.10)] and the derivation is practically identical. Relation (8.4) allows us to estimate the differences between the matrices $\omega_{N}^{(j)}(\beta)$ once we control the differences $R_{N}(\beta)-\mathbb{E}(R_{N}(\beta))$. For this purpose, we use the concentration of measure result in [3, Corollary 4.4.28]. This requires estimating the Lipschitz constant of the map $V\mapsto R_{N}(V,\beta)$ (see (8.2)) in the Hilbert-Schmidt norm. We use the notation $\|T\|_{2}=\textrm{Tr}_{m}(T^{*}T)$ for the Hilbert-Schmidt norm of an arbitrary matrix $T\in M_{m}(\mathbb{C})$.

Lemma 8.3**.**

Suppose that $N\in\mathbb{N}$ and $\beta\in\mathcal{G}_{N}$. Then

[TABLE]

where $r=\|R_{N}(\beta)\|_{M_{n}(\mathbb{C})\otimes\mathcal{M}_{N}}$.

Proof.

A simple calculation shows that

[TABLE]

Next we see that

[TABLE]

and thus

[TABLE]

Use now the equality $\|T\otimes S\|_{2}=\|T\|_{2}\|S\|_{2}$ to deduce that

[TABLE]

The lemma follows from this estimate. ∎

Proposition 8.4**.**

Suppose that $\sup\{\|D_{N}\|:N\in\mathbb{N}\}<+\infty$, $\beta\in\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$, and moreover,

[TABLE]

Let $X_{N},Y_{N}\in M_{n}\otimes M_{N}$ be matrices of norm one and rank uniformly bounded by $m\in\mathbb{N}$. Then:

• (1)

Almost surely,

[TABLE]

• (2)

There exists $k>0$ such that

[TABLE]

In particular, there exists a dense countable subset $\Lambda\subset\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$ such that almost surely, (8.5) holds for any $\beta\in\Lambda$.

Proof.

An arbitrary operator of rank $m$ can be written as a sum of $m$ operators of rank one (with the same or smaller norm). Thus we may, and do, restrict ourselves to the case in which the operators $X_{N}$ and $Y_{N}$ are projections of rank $1$. In this case, $X_{N}R_{N}(V,\beta)Y_{N}$ is a scalar multiple of a fixed operator of rank one and Lemma 8.3 shows that this function satisfies a Lipschitz estimate. This estimate, combined with [3, Corollary 4.4.28], shows that

[TABLE]

for every $\varepsilon>0$ and every $\alpha\in(0,1/2)$. The hypothesis implies that the last denominator has a bound independent of $N$. Part (1) of the lemma follows from this inequality, while (2) follows from the formula $\mathbb{E}(|Z|)=\int_{0}^{+\infty}\mathbb{P}(|Z|>t)\,dt$, valid for arbitrary random variables $Z$. ∎

Remark 8.5**.**

While Proposition 8.4 was formulated for $\beta\in\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$, the hypothesis $r<+\infty$ (and therefore the conclusion of the proposition) is also satisfied in the following cases:

• (1)

$\Im\beta>0$ with $r\leq\|(\Im\beta)^{-1}\|$;

• (2)

$\beta=ze_{1,1}-\gamma_{0}$ with $z\in\mathbb{C}^{+}\cup\mathbb{C}^{-}$, by an estimate provided by Lemma 4.3;

• (3)

$\beta=xe_{1,1}-\gamma_{0}$ with $x\in\mathbb{R}$, by the same estimate provided that

[TABLE]

Corollary 8.6**.**

Under the assumptions of Proposition 8.4 suppose that we also have $\sup\{\|C_{N}\|:N\in\mathbb{N}\}<+\infty,$ and set

[TABLE]

Let $K\subset\mathbb{C}\setminus[-t,t]$ be a compact set. Then, almost surely, the functions

[TABLE]

converge to zero uniformly on $K$ as $N\to+\infty$.

Proof.

According to Proposition 8.4 and Remark 8.5(2) and (3), almost surely, for every $z\in\mathbb{C}\setminus[-t,t]$ such that $\Im z\in\mathbb{Q}$ and $\Re z\in\mathbb{Q}$, we have pointwise convergence to zero. A second application of Remark 8.5 yields uniform bounds on $K$ for all of these functions, and this implies uniform convergence because the resolvents involved are analytic in $z$. ∎

We apply the concentration results just proved to operators $X_{N}$ and $Y_{N}$, of the form $I_{n}\otimes P_{N}^{(j)}$, where $\{P_{N}^{(j)}:j=1,\dots,N\}$ are the projections used in Lemma 8.1. The rank of $I_{n}\otimes P_{N}^{(j)}$ is equal to $n$.

Proposition 8.7**.**

Suppose that $\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}<+\infty,$ and let $\beta\in\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$ be such that

[TABLE]

and

[TABLE]

Then $\sup_{N\in\mathbb{N}}N\|\omega_{N}(\beta)-\omega_{N}^{(1)}(\beta)\otimes I_{N}\|<+\infty$.

Proof.

The conclusion of the propostion is equivalent to

[TABLE]

as $N\to+\infty$. Fix $j\in\{2,\dots,N\}$ and set $Z_{N}=f_{j}\otimes f_{1}^{*}\in M_{N}(\mathbb{C}^{N})$. Thus, $Z_{N}$ is an operator of rank one such that $Z_{N}h=\langle h,f_{1}\rangle f_{j}$ for every $h\in\mathbb{C}^{N}$. We have

[TABLE]

and, similarly,

[TABLE]

Next, we apply (8.4) and use the fact that $I\otimes P_{N}^{(j)}$ commutes with $\mathbb{E}(R_{N}(U_{N},\beta))$. Setting $X_{N}=I_{n}\otimes P_{N}^{(k)}$, $Y_{N}=I_{n}\otimes P_{N}^{(1)}$, we obtain

[TABLE]

Since $\|Z_{N}\|=1$, an application of the Cauchy-Schwarz inequality leads to the following estimate:

[TABLE]

We have $|\lambda_{N}^{(j)}-\lambda_{N}^{(1)}|\leq 2\|C_{N}\|$ and the product of the last two factors above is estimated via (8.6) by $kr^{4}/N$ with $k$ independent of $N$. Thus,

[TABLE]

with $k^{\prime}$ independent of $j$ and $N$. The lemma follows. ∎

In the probability model we consider, the sequences $\{C_{N}\}_{N\in\mathbb{N}}$ and $\{D_{N}\}_{N\in\mathbb{N}}$ are uniformly bounded in norm and, in addition, the sequences $\{\mu_{C_{N}}\}_{N\in\mathbb{N}}$ and $\{\mu_{D_{N}}\}_{N\in\mathbb{N}}$ converge weakly to $\mu$ and $\nu$, respectively. We denote by $(a,b)$ a pair of free random variables in some $C^{*}$-probability space $(\mathcal{M},\tau)$ such that the sequence $\{(c_{N},d_{N})\}_{N\in\mathbb{N}}$ converges in distribution to $(a,b)$. We also set

[TABLE]

In other words, $\omega$ is the usual matrix subordination function associated to the pair $(\gamma_{1}\otimes a,\gamma_{2}\otimes b)$.

Proposition 8.8**.**

Suppose that $\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}<+\infty$ and that the sequence $\{(c_{N},d_{N})\}_{N\in\mathbb{N}}$ converges to $(a,b)$ in distribution. Let $\mathcal{D}\subset\mathcal{G}\cap\bigcap_{n\in\mathbb{N}}\mathcal{G}_{N}$ be a connected open set containing $\mathbb{H}^{+}(M_{n}(\mathbb{C}))$ and such that the sequence of functions $\{\|\omega_{N}\|\}_{N\in\mathbb{N}}$ is locally uniformly bounded on $\mathcal{D}$. Then

[TABLE]

Proof.

By hypothesis, the analytic functions $\{\omega_{N}^{(1)}\}_{N\in\mathbb{N}}$ form a normal family on $\mathcal{D}$. By Proposition 8.7, it suffices to prove that every subsequential limit of this sequence equals $\omega$. Suppose that $\{\omega_{N_{k}}^{(1)}\}_{k\in\mathbb{N}}$ converges on $\mathcal{D}$ to a function $\widetilde{\omega}$. Fix $\beta\in H^{+}(M_{n}(\mathbb{C}))$ such that $\Im\beta>\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}$. Then

[TABLE]

and thus

[TABLE]

Setting $N=N_{k}$, letting $k\to+\infty$, and observing that the series on the right is uniformly dominated, we conclude that

[TABLE]

The fact that the pairs $(C_{N},D_{N})$ converge to $(a,b)$ implies that

[TABLE]

Thus, we obtain the equality

[TABLE]

and this easily yields $\widetilde{\omega}(\beta)=\omega(\beta)$. Since $\mathcal{D}$ is connected, we must have $\widetilde{\omega}=\omega$, thus concluding the proof. ∎

Corollary 5.5 implies the following result.

Corollary 8.9**.**

Suppose that $\mathfrak{s}:=\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}<+\infty$ and the sequence $\{(c_{N},d_{N})\}_{N\in\mathbb{N}}$ converges to $(a,b)$ in distribution. Let $\mathfrak{k}=\max\{k,\mathfrak{s}\},$ where $k$ is the constant provided by Lemma 5.4. Then, for every $z\in\mathbb{C}\setminus[-\mathfrak{k},\mathfrak{k}]$, we have $ze_{1,1}-\gamma_{0}\in\mathcal{G}\cap\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$ and

[TABLE]

The preceding results combine to yield convergence results for sample resolvents. For the following statement, it is useful to identify $\mathbb{C}^{m}$ with a subspace of $\mathbb{C}^{N}$ if $m<N$ and to denote by $\{f_{1},\dots,f_{N}\}$ the standard basis in $\mathbb{C}^{N}$. Thus, we have $f_{j}\in\mathbb{C}^{m}$ provided that $j\leq m$.

Proposition 8.10**.**

Suppose that $\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}<+\infty$ and that the sequence $\{(c_{N},d_{N})\}_{N\in\mathbb{N}}$ converges to $(a,b)$ in distribution. Suppose also that $C_{N}$ is diagonal in the standard basis, that is, $C_{N}f_{j}=\lambda_{N}^{(j)}f_{j}$, $j=1,\dots N$, and that $p\in\mathbb{N}$ is such that the limits $\lambda^{(j)}=\lim_{N\to\infty}\lambda_{N}^{(j)}$ exist for $j=1,\dots,p$. Denote by $P_{N}:\mathbb{C}^{N}\to\mathbb{C}^{p}$ the orthogonal projection onto $\mathbb{C}^{p}$, $N\geq p$. Let $\mathcal{D}\subset\mathcal{G}\cap\bigcap_{N\in\mathbb{N}}\mathcal{G}_{N}$ be a connected open set containing $\mathbb{H}^{+}(M_{n}(\mathbb{C}))$ and such that the sequence of functions $\{\|\omega_{N}\|\}_{N\in\mathbb{N}}$ is locally uniformly bounded on $\mathcal{D}$. Then almost surely

[TABLE]

in the norm topology for every $\beta\in\mathcal{D}$.

Proof.

By Proposition 8.4, it suffices to show that the conclusion holds with $\mathbb{E}(R_{N}(\beta))$ in place of $R_{N}(\beta)$. We observe that

[TABLE]

so the desired conclusion follows from Proposition 8.8. ∎

We observe for use in the following result that there exists a domain $\mathcal{D}$ as in the above statement such that $ze_{1,1}-\gamma_{0}\in\mathcal{D}$ for every $z\in\mathbb{C}^{+}$.

When the convergence of $(C_{N},U_{N}D_{N}U_{N}^{*})$ to $(a,b)$ is strong, the preceding result extends beyond $\mathbb{H}^{+}(M_{n}(\mathbb{C}))$. In this case, $\sigma(P(C_{N},U_{N}D_{N}U_{N}^{*}))$ converges almost surely to $\sigma(P(a,b))$ and thus the sample resolvent $R_{N}(U_{N},ze_{1,1}-\gamma_{0})$ is defined almost surely for large $N$ even for $z\in\mathbb{R}\setminus\sigma(P(a,b))$. We also recall that the function

[TABLE]

extends analytically to $\mathbb{C}\setminus\sigma(P(a,b))$ if $\lambda\in\sigma(a)$. These analytic extensions are used in the following statement.

Proposition 8.11**.**

Under the hypothesis of Proposition 8.10, suppose that the pairs $\{(C_{N},U_{N}D_{N}U_{N}^{*})\}_{N\in\mathbb{N}}$ converge strongly to $(a,b)$. Then almost surely

[TABLE]

for $\beta=ze_{1,1}-\gamma_{0}$, $z\in\mathbb{C}\setminus\sigma(a,b)$. The convergence is uniform on compact subsets of $\mathbb{C}\setminus\sigma(P(a,b))$.

Proof.

Strong convergence implies that $\lambda^{(j)}\in\sigma(a)$ for $j=1,\dots,p$, so the functions $(\omega(ze_{1,1}-\gamma_{0})-\lambda^{(j)}\gamma_{1})^{-1}$ extend analytically to $\mathbb{R}\setminus\sigma(P(a,b))$. Let $\mathcal{O}\subset\mathbb{C}\setminus\sigma(P(a,b))$ be a connected open set containing $\{z\}\cup\mathbb{C}^{+}$ which is at a strictly positive distance from $\mathbb{C}\setminus\sigma(P(a,b))$. We prove that the conclusion of the proposition holds for $U_{N}(\xi)$ provided that $\xi\in\Omega$ is such that the conclusion of Corollary 8.6 holds and $\sigma(P(C_{N},VD_{N}V^{*}))\subset\mathbb{R}\setminus\mathcal{O}$ for $V=U_{N}(\xi)$ and sufficiently large $N$. By hypothesis, the collection of such points $\xi\in\Omega$ has probability $1$. Lemma 4.3 shows that the family of functions $\|R_{N}(U_{N}(\xi),ze_{1,1}-\gamma_{0})\|$ is locally uniformly bounded on $\mathcal{O}$ for large $N$. By Montel’s theorem, we can conclude the proof by verifying the conclusions of the proposition for $\beta=ze_{1,1}-\gamma_{0}$ with $z\in\mathbb{C}^{+}$. For such values of $z$, the result follows from Proposition 8.10. ∎

9. The unitarily invariant model

In this section we prove Theorem 6.1 under the additional condition that $A_{N}$ is a constant matrix and $B_{N}$ is a random unitary conjugate of another constant matrix. We may, and do, assume that for $N\geq p$, $A_{N}$ is diagonal in the standard basis $\{f_{1},\dots,f_{N}\}$ with eigenvalues $\theta_{1},\dots,\theta_{p},\lambda_{N}^{(p+1)},\dots\lambda_{N}^{(N)}$ and, as before, $B_{N}=U_{N}D_{N}U_{N}^{*}$. Then the random matrices $a_{N}=A_{N}\otimes 1_{\Omega}$ and $d_{N}=U_{N}D_{N}U_{N}^{*}$ are viewed as elements of the noncommutative probability space $M_{N}(\mathbb{C})\otimes L^{\infty}(\Omega)$. We fix free selfadjoint random variables $(a,b)$ in some tracial $W^{*}$-probability space such that the pairs $(a_{N},d_{N})$ converge in distribution to $(a,b)$ as $N\to\infty$. This convergence is not strong because of the spikes $\theta_{1},\dots,\theta_{p}$. As in [11], we consider closely related pairs $(c_{N},d_{N})$ that do converge strongly to $(a,b)$. Namely, we set $c_{N}=C_{N}\otimes 1_{\Omega}$, where $C_{N}$ is diagonal in the standard basis of $\mathbb{C}^{N}$ with eigenvalues $\lambda_{N}^{(1)},\dots,\lambda_{N}^{(N)}$ that coincide with those of $A_{N}$ except that $\lambda_{N}^{(1)}=\cdots=\lambda_{N}^{(p)}=s$ is an arbitrary (but fixed for the remainder of this section) element of $\mathrm{supp}(\mu)$. For $N\geq p$, the difference $\Delta_{N}=A_{N}-C_{N}$ can then be written as $\Delta_{N}=P_{N}^{*}TP_{N}$, where $T\in M_{p}(\mathbb{C})$ is the diagonal matrix with eigenvalues $\theta_{1}-s\dots,\theta_{p}-s$ and $P_{N}\colon\mathbb{C}^{N}\to\mathbb{C}^{p}$ is the orthogonal projection.

According to Lemma 4.2, $\ker(tI_{N}-P(A_{N},B_{N}))$ and $\ker(te_{1,1}\otimes I_{N}-L(A_{N},B_{N})))$ have the same dimension for every $t\in\mathbb{R}$. Setting $\beta=ze_{1,1}-\gamma_{0}$, strong convergence of the pairs $c_{N},d_{N}$ implies that almost surely the sample resolvent $R_{N}(U_{N},\beta)$ is defined for sufficiently large $N$ if $z\in\mathbb{C}\setminus\sigma(P(c,d))$. (We continue using the notation introduced in (8.1), (8.2), and (8.3).) We need to consider the matrix

[TABLE]

since the order of $t\in\mathbb{R}$ as a zero of its determinant equals $\dim\ker(tI_{N}-P(A_{N},B_{N}))$, and hence the number of eigenvalues of $P(A_{N},B_{N})$ in a neighborhood $V$ of a given $t\in\mathbb{R}\setminus\sigma(P(a,b))$ is the number of zeros of this determinant in $V$. Thus, we need to consider the zeros in $V$ of

[TABLE]

Using Sylvester’s identity ($\det(I_{r}-XY)=\det(I_{p}-YX)$ if $X$ is an $r\times p$ matrix and $Y$ is a $p\times r$ matrix) and the fact that $\Delta_{N}=P_{N}^{*}TP_{N}$, this determinant can be rewritten as

[TABLE]

At this point, we observe that the hypothesis of Proposition 8.11 are satisfied with $\lambda^{(1)}=\cdots=\lambda^{(p)}=s$. We conclude that almost surely

[TABLE]

where

[TABLE]

and the convergence is uniform on compact sets. The limit $F(ze_{1,1}-\gamma_{0})$ is a (deterministic) analytic function on $\mathbb{C}\setminus\sigma(P(a,b))$. An application of Hurwitz’s theorem on zeros of analytic functions (see [1, Theorem 5.2]) yields the following result.

Proposition 9.1**.**

Suppose that $t_{1},t_{2}\in\mathbb{R}$, $t_{1}<t_{2}$, $[t_{1},t_{2}]\subset\mathbb{R}\setminus\sigma(P(a,b))$, $F(t_{j}e_{1,1}-\gamma_{0})\neq 0$ for $j=1,2$, and the function $F(ze_{1,1}-\gamma_{0})$ has at most one zero $t$ in the interval $(t_{1},t_{2})$. Then, almost surely for large $N$, the matrix $P(A_{N},B_{N})$ has exactly $m$ eigenvalues in the interval $[t_{1},t_{2}]$, where $m$ is the order of $t$ as a zero of $F(ze_{1,1}-\gamma_{0})$ and $m=0$ if this function does not vanish on $[t_{1},t_{2}]$.

Part(1) of Theorem 6.1 is a reformulation of Proposition 9.1. To see that this is the case, we observe that $T$ is a diagonal matrix and thus the matrix

[TABLE]

is block diagonal with diagonal blocks

[TABLE]

If $\det(G_{j,s}(z))$ has a zero of order $m_{j}$ at $t$ then the number $m$ in the statement is $m_{1}+\cdots+m_{p}$. We recall that $G_{j,s}$ is analytic on $\mathbb{C}\setminus\sigma(P(a,b))$ but $\omega(ze_{1,1}-\gamma_{0})$ is only meromorphic. It is not immediately apparent that the number $m_{j}$ does not depend on $s$ but this is a consequence of the following result.

Lemma 9.2**.**

Suppose that $\alpha_{1},\alpha_{2}\in M_{n}(\mathbb{C})$ are such that $(\omega(ze_{1,1}-\gamma_{0})-\alpha_{k})^{-1}$ extends analytically to $t$ for $k=1,2$. Then the order of $t$ as a zero of

[TABLE]

does not depend on $k$.

Proof.

An easy calculation shows that

[TABLE]

The desired conclusion follows if we prove that the function

[TABLE]

is analytic and invertible at $z=t$. We have

[TABLE]

and

[TABLE]

so the analyticity and invertibility of $H$ follow from the hypothesis. ∎

We proceed now to Part(2) of Theorem 6.1. Thus, assumptions (A1–A3) and (B0–B2) are in force and, in addition, the spikes $\theta_{1},\dots,\theta_{p}$ are distinct. In particular, $\sup\{\|C_{N}\|+\|D_{N}\|:N\in\mathbb{N}\}<+\infty$.

If $S\subset\mathbb{R}$ is a Borel set and $A\in M_{N}(\mathbb{C})$ is a selfadjoint operator, then $E_{A}(S)$ denotes the orthogonal projection onto the linear span of the eigenvectors of $A$ corresponding to eigenvalues in $S$. For instance, under the hypotheses of Part(2) of Theorem 6.1, $E_{A_{N}}(\{\theta_{j}\})$ is a projection of rank one for $j=1,\dots,p$. If $h\colon\mathbb{R}\to\mathbb{C}$ is a continuous function, then $h(A)$ denotes the usual functional calculus for selfadjoint matrices. Thus, if $Ax=tx$ for some $t\in\mathbb{R}$ and $x\in\mathbb{C}^{N}$, then $h(A)x=h(t)x$.

Fix $t\in\mathbb{R}\setminus\sigma(P(a,b))$ and $\varepsilon>0$ small enough. We need to show that, almost surely,

[TABLE]

Choose $\delta>0$ and $N_{0}\in\mathbb{N}$ such that $[\theta_{j}-\delta,\theta_{j}+\delta]\cap\sigma(A_{N})=\{\theta_{j}\}$, for $N\geq N_{0}$ and $j=1,\dots,p$. Pick infinitely differentiable functions $f_{j}:\mathbb{R}\to[0,1]$ supported in $[\theta_{j}-\delta,\theta_{j}+\delta]$ such that $f_{j}(\theta_{j})=1$, $j=1,\dots,p$. Also pick an infinitely differentiable function $h:\mathbb{R}\to[0,1]$ supported in $(t-\varepsilon,t+\varepsilon)$ such that $h$ is identically $1$ on $[t-\varepsilon/2,t+\varepsilon/2]$. Then part (1) of Theorem 6.1 implies that, almost surely for large $N$, we have

[TABLE]

In anticipation of a concentration inequality, we prove a Lipschitz estimate for the functions $g_{N,j}\colon{\mathrm{U}}(N)\to M_{N}(\mathbb{C})$ defined by

[TABLE]

Lemma 9.3**.**

There exists $k>0$, independent of $N$, such that

[TABLE]

Proof.

Given a Lipschitz function $u:{\mathrm{U}}(N)\to M_{N}(\mathbb{C})$, we denote by $\mathrm{Lip}(u)$ the smallest constant $c$ such that

[TABLE]

and we set $\|u\|_{\infty}=\sup\{\|u(V)\|:v\in{\mathrm{U}}(N)\}$. If $u_{1},u_{2}:{\mathrm{U}}(N)\to M_{N}(\mathbb{C})$ are two Lipschitz functions, then ${\mathrm{Lip}}(u_{1}+u_{2})\leq{\mathrm{Lip}}(u_{2})+{\mathrm{Lip}}(u_{1})$ and

[TABLE]

Since the functions $V\mapsto V$ and $V\mapsto V^{*}$ are Lipschitz with constant $1$, we deduce immediately that the map $V\mapsto P(A_{N},VD_{N}V^{*})$ is Lipschitz with constant bounded independently of $N$. It is well-known that a Lipschitz function $f:\mathbb{R}\to\mathbb{R}$ is also Lipschitz, with the same constant, when viewed as a map on the selfadjoint matrices with the Hilbert-Schmidt norm (see for instance, [18, Lemma A.2]). The function $h$ is infinitely differentiable with compact support, hence Lipschitz. We deduce that the map $V\mapsto h(P(A_{N},VD_{N}V^{*}))$ is Lipschitz with constant bounded independently of $N$. Finally, we have

[TABLE]

and the lemma follows because $\|f_{j}(A_{N})\|_{2}=1$. ∎

An application of [3, Corollary 4.4.28] yields

[TABLE]

and the Borel-Cantelli lemma shows that, almost surely,

[TABLE]

The expected value in (9.2) is estimated using [18, Lemma 6.3] and the fact that $f_{j}(A_{N})$ is the projection onto the $j$th coordinate. If we set $r_{N}(z)=(zI_{N}-P(A_{N},B_{N}))^{-1}$ and $\widetilde{R}_{N}(ze_{1,1}-\gamma_{0})=((ze_{1,1}-\gamma_{0})\otimes I_{N}-L(A_{N},B_{N}))^{-1}$, $z\in\mathbb{C}^{+}$, then

[TABLE]

The construction of the linearization $L$ (Section 4) is such that the matrix $\widetilde{R}_{N}(ze_{1,1}-\gamma_{0})$, viewed as an $n\times n$ block matrix, has $r_{N}(z)$ as its $(1,1)$ entry. By Propositions 8.4(1) and 8.10 (with $A_{N}$ in place of $C_{N}$) and the unitary invariance of the distribution of $B_{N}$, for $z\in\mathbb{C}^{+}$ the matrices

[TABLE]

converge as $N\to\infty$ to the block diagonal matrix with diagonal entries

[TABLE]

It follows that

[TABLE]

We intend to let $N\to\infty$ in (9.3) using (9.4), so we consider the differences

[TABLE]

By Corollary 5.5, these functions are defined on $\mathbb{C}\setminus[-k,k]$ for some $k>0$ and satisfy $\Delta_{j,N}(\overline{z})=\overline{\Delta_{j,N}(z)}$. We claim that there exists a sequence $\{v_{N}\}_{N\in\mathbb{N}}\subset(0,+\infty)$ such that $\lim_{N\to\infty}v_{N}=0$ and

[TABLE]

To verify this claim, we observe first [2] that the function $\mathbb{E}(r_{N}(z)_{j,j})$ is the Cauchy-Stieltjes transform of a Borel probability measure $\sigma_{N,j}$ on $\mathbb{R}$. Since

[TABLE]

the measures $\{\sigma_{N,j}\}_{N\in\mathbb{N}}$ have uniformly bounded supports. Now, (9.4) shows that the Cauchy-Stieltjes transform of any accumulation point of this sequence of measures is equal to $((\omega(ze_{1,1}-\beta_{0})-\theta_{j}\beta_{1})^{-1})_{1,1}$. It follows that this sequence has a weak limit $\sigma_{j}$ that is a Borel probability measure with compact support. The existence of the sequence $\{v_{N}\}_{N\in\mathbb{N}}$ follows from [11, Lemma 4.1] applied to the signed measures $\rho_{N}=\sigma_{N,i}-\sigma_{i}$.

We use (9.5), (9.3), and the Lemma from [20, Appendix] to obtain

[TABLE]

The choice of $h$, and the fact that

[TABLE]

is analytic and real-valued on the intervals $[t-\varepsilon,t-\varepsilon/2]$ and $[t+\varepsilon/2,t+\varepsilon]$, imply that the last line in (9.6) can be rewritten as

[TABLE]

Recall (see, for instance, [1, Chapter 4]) that if $f$ is an analytic function on a simply connected domain $D$, except for an isolated singularity $a$, then $\frac{1}{2\pi i}\int_{\gamma}f(z)\,\mathrm{d}z=n(\gamma,a)\mathrm{Res}_{z=a}f(z)$. Here $\gamma$ is a closed Jordan path in $D$ not containing $a$, $n(\gamma,a)$ is the winding number of $\gamma$ with respect to $a$, and $\mathrm{Res}_{z=a}f(z)$ is that number $R$ which satisfies the condition that $f(z)-\frac{R}{z-a}$ has vanishing period (called the residue of $f$ at $a$). Denote by $\Gamma_{y}$ the rectangle with corners $t\pm(\varepsilon/2)\pm iy$ and let $\gamma_{y}$ be the boundary of $\Gamma_{y}$ oriented counterclockwise. The expression in (9.7) represents the integral of $u$ on the horizontal segments in $\gamma_{y}$. It is clear that the integral of $u$ on the vertical segments is $O(y)$, and thus (9.6) implies the equality

[TABLE]

The alternative formula in Theorem 6.1 follows from the fact that $u$ has a simple pole at $t$ because $u$ maps $\mathbb{C}^{+}$ to $\mathbb{C}^{-}$.

10. The Wigner model

We proceed now to the proof of Theorem 6.3. The matrices $A_{N}$ are subject to the hypotheses (A1–A3), while $X_{N}/\sqrt{N}$ and $X_{N}$ satisfies conditions (X0–X3). By [36, Section 3, Assertion 2], it suffices to proceed under the additional hypothesis that each $A_{N}$ is a constant matrix. The free variables $a$ and $b$ are such that $b$ has standard semicircular distribution $\nu_{0,1}$.

One consequence of the fact that $b$ is a semicircular variable is that the function $\omega$ is analytic on the entire set

[TABLE]

This justifies the comment from Remark 6.2. We recall that in this special case the subordination function is given by

[TABLE]

Since the distribution of the random matrix $X_{N}$ is not usually invariant under unitary conjugation, we can no longer assume that $A_{N}$ is diagonal in the standard basis $\{f_{1},\dots,f_{N}\}$. There is however a (constant) unitary matrix $V_{N}\in M_{N}(\mathbb{C})$ such that $A_{N}$ is diagonal in the basis $\{V_{N}f_{1},\dots,V_{N}f_{N}\}$ with eigenvalues $\theta_{1},\dots,\theta_{p},\lambda_{N}^{(p+1)},\dots\lambda_{N}^{(N)}$, $N\geq p$. Viewing each realization of the random matrices $A_{N}$ and $X_{N}/\sqrt{N}$ as elements of the noncommutative probability space $\left(M_{N}(\mathbb{C}),\mathrm{tr}_{N}\right)$, almost surely the pairs $(A_{N},X_{N}/\sqrt{N})$ converge in distribution, but not strongly, to $(a,b)$ as $N\to\infty$. A modification of $A_{N}$ provides almost surely strongly convergent pairs $(C_{N},X_{N}/\sqrt{N})$. Thus, let $C_{N}$ be diagonal in the basis $\{V_{N}f_{1},\dots,V_{N}f_{N}\}$ with eigenvalues $\lambda_{N}^{(1)},\dots,\lambda_{N}^{(N)}$ that coincide with those of $A_{N}$ except that $\lambda_{N}^{(1)}=\cdots=\lambda_{N}^{(p)}=s$ is an arbitrary (but fixed for the remainder of this section) element of $\mathrm{supp}(\mu)$. For $N\geq p$, the difference $\Delta_{N}=A_{N}-C_{N}$ can then be written as $\Delta_{N}=V_{N}P_{N}^{*}TP_{N}V_{N}^{*}$, where $T\in M_{p}(\mathbb{C})$ is the diagonal matrix with eigenvalues $\theta_{1}-s\dots,\theta_{p}-s$ and $P_{N}:\mathbb{C}^{N}\to\mathbb{C}^{p}$ is the orthogonal projection. Almost surely, the pairs $(C_{N},X_{N}/\sqrt{N})$ converge strongly to $(a,b)$ as shown in [12, Theorem 1.2] and [23, Proposition 2.1]. We continue using the notation introduced in (8.1) and (8.3) with $c_{N}=C_{N}$ and $d_{N}=\frac{X_{N}}{\sqrt{N}}$. The calculation in Section 9 show that, almost surely, for $N$ large enough, the number of eigenvalues of $P(A_{N},X_{N}/\sqrt{N})$ in a small enough neighborhood of $t\in\mathbb{R}\setminus\sigma(P(a,b))$ is equal to the number of zeros of $F_{N}(ze_{1,1}-\gamma_{0})$ in this neighborhood, where

[TABLE]

is a random analytic function. We focus on the study of the large $N$ behavior of the matrix function

[TABLE]

We start with the special case in which $X_{N}$ is replaced by a standard G.U.E.. The following proposition is a consequence of the results in Section 9. Thus, suppose that $(X^{g}_{N})_{N\in\mathbb{N}}$ is a sequence of standard G.U.E. ensembles and we set ${R}_{N}^{g}(\beta)=(\beta\otimes I_{N}-L(C_{N},X^{g}_{N}/\sqrt{N}))^{-1},$ and

[TABLE]

Proposition 10.1**.**

We have

[TABLE]

for every $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$.

Proof.

Since G.U.E. ensembles are invariant under unitary conjugation we may, and do, assume that $V_{N}=I_{N}$ for every $N\in\mathbb{N}$. For every $\beta\in\mathbb{H}^{+}(M_{N}(\mathbb{C}))$ we have

[TABLE]

and the second term is at most $\|(\Im\beta)^{-1}\|\mathbb{P}(\|X^{g}_{N}/\sqrt{N}\|>3)$. As shown by Bai and Yin [6], this number tends to zero as $N\to\infty$. To estimate the first term, we recall that $X^{g}_{N}/\sqrt{N}=U_{N}D_{N}U_{N}^{*}$, where $U_{N}$ is a random matrix uniformly distributed in $\mathrm{U}(N)$ and $D_{N}$ is a random diagonal matrix, independent from $U_{N}$, whose empirical spectral measure converges almost surely to $\nu_{0,1}$ as $N\to\infty$. Thus, we can write

$\mathbb{E}(\mathcal{F}_{N}^{g}(\beta){\mathbf{1}}_{\|X^{g}_{N}/\sqrt{N}\|\leq 3})$

[TABLE]

[TABLE]

Proposition 8.10 can be applied for almost every $\xi_{2}$ and it shows that

[TABLE]

converges to $(\omega(\beta)-s\gamma_{1})^{-1}\otimes I_{p}$. The proposition follows now from an application of the dominated convergence theorem. ∎

Passing to arbitrary Wigner matrices requires an approximation procedure from [12, Section 2]. For every $\varepsilon>0$, there exist random selfadjoint matrices $X_{N}(\varepsilon)=[(X(\varepsilon))_{ij}]_{1\leq i,j\leq N}$ such that

• (H1)

the variables $\sqrt{2}\Re X_{ij}(\varepsilon)$, $\sqrt{2}\Im X_{ij}(\varepsilon)$, $X_{ii}(\varepsilon)$, $i,j\in\mathbb{N}$, $i<j$, are independent, centered with variance $1$ and satisfy a Poincaré inequality with common constant $C_{PI}(\epsilon)$,

• (H2)

for every $m\in\mathbb{N}$,

[TABLE]

and almost surely for large $N$,

[TABLE]

Set

[TABLE]

and

[TABLE]

for $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}^{N}))$. It readily follows that, almost surely for large $N$,

[TABLE]

Properties (H1) and (H2) imply that, for every $\varepsilon>0$,

[TABLE]

[TABLE]

and for any $m\in\mathbb{N}\setminus\{0\}$,

[TABLE]

where for $i\neq j$, $(\kappa_{m}^{i,j,\varepsilon})_{m\geq 1}$ and $(\widetilde{\kappa}_{m}^{i,j,\varepsilon})_{m\geq 1}$ denote the classical cumulants of $\sqrt{2}\Re X_{ij}(\varepsilon)$ and $\sqrt{2}\Im X_{ij}(\varepsilon)$ respectively, and $(\kappa_{m}^{i,i,\varepsilon})_{m\geq 1}$ denote the classical cumulants of $X_{ii}(\varepsilon)$ (we set $(\widetilde{\kappa}_{m}^{i,i,\varepsilon})_{m\geq 1}\equiv 0$).

We use the following notation for an arbitrary matrix $M\in M_{n}(\mathbb{C})\otimes M_{N}(\mathbb{C})$:

[TABLE]

and

[TABLE]

where $e_{j,i}$ (resp. $\hat{e}_{j,i}$) denotes the $n\times n$ (resp. $N\times N$) matrix whose unique nonzero entry equals 1 and occurs in row $j$ and column $i$.

Proposition 10.2**.**

There exists a polynomial $P_{\varepsilon}$ in one variable with nonnegative coefficients such that for all large $N$, for every $v,u\in\{1,\ldots,N\}$, for every $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$, and for every deterministic $B_{N}^{(1)},B_{N}^{(2)}\in M_{n}(\mathbb{C})\otimes M_{N}(\mathbb{C})$ such that $\|B_{N}^{(1)}\|\leq 1$ and $\|B_{N}^{(2)}\|\leq 1$, we have

[TABLE]

and

[TABLE]

The proof uses a well-known lemma.

Lemma 10.3**.**

Let $Z$ be a real-valued random variable such that $\mathbb{E}(|Z|^{p+2})<\infty$. Let $\phi\colon\mathbb{R}\to\mathbb{C}$ be a function whose first $p+1$ derivatives are continuous and bounded. Then,

[TABLE]

where $\kappa_{a}$ are the cumulants of $Z$, $|\eta|\leq C\sup_{t}|\phi^{(p+1)}(t)|\mathbb{E}(|Z|^{p+2})$, and $C$ only depends on $p$.

Proof of Proposition 10.2.

Following the approach of [38, Ch. 18 and 19] we introduce the interpolation matrix $X_{\varepsilon}(\alpha)=\cos\alpha X_{N}(\varepsilon)+\sin\alpha Y_{N}$, $\alpha\in[0,{\pi}/{2}]$, and the corresponding resolvent

[TABLE]

We have

[TABLE]

[TABLE]

Define a basis of the real vector space of selfadjoint matrices in $M_{N}(\mathbb{C})$ as follows:

[TABLE]

[TABLE]

[TABLE]

In the following calculation, we write simply $R_{N}^{\varepsilon,R}$ in place of $R_{N}^{\varepsilon,\alpha}(\beta)$ and $X^{g}$ in place of $X^{g}_{N}$:

[TABLE]

Next, we apply Lemma 10.3 with $p=3$ for $1\leq k\leq N$, $j<k$, to each random variable $Z$ in the set

[TABLE]

and to each $\phi$ in the set

[TABLE]

Setting now $B=B_{N}^{(2)}e_{q,l}\otimes\hat{e}_{u,v}B_{N}^{(1)},$ we have:

[TABLE]

where

[TABLE]

for some $C_{\varepsilon}\geq 0$, while $C(\alpha)$ and $\widetilde{C}(\alpha)$ are polynomials in $\cos\alpha$ and $\sin\alpha$. In the following, $C_{\varepsilon}$ may vary from line to line. It is clear that

[TABLE]

Next, $I_{2}$ and $I_{3}$ are a finite linear combinations of terms of the form

[TABLE]

where ${\mathcal{E}}$ is some subset of $\{1,\ldots,N\}^{2}$, $C^{j,k,\varepsilon}\in\{\kappa_{3}^{j,k,\varepsilon},\widetilde{\kappa}_{3}^{j,k,\varepsilon}\}$, and $(p_{1},\ldots,p_{6})$ is a permutation of $(k,k,k,j,j,j)$. The two following cases hold:

• •

$p_{5}=p_{6}$, in which case Lemma 11.1 yields

[TABLE]

• •

$p_{5}\neq p_{6}$, in which case Lemma 11.1 yields

[TABLE]

We see now that $|I_{j}|\leq C_{\varepsilon}{\|(\Im\beta)^{-1}\|^{4}}/{\sqrt{N}}$ for $j=2,3$. Finally, $I_{5}$ and $I_{6}$ are finite linear combinations of terms of the form

[TABLE]

where ${\mathcal{E}}$ is some subset of $\{1,\ldots,N\}^{2}$, $C^{j,k,\varepsilon}\in\{\kappa_{4}^{j,k,\varepsilon},\widetilde{\kappa}_{4}^{j,k,\varepsilon}\}$ and $(p_{1},\ldots,p_{6})$ is a permutation of $(k,k,k,k,j,j,j,j)$. Lemma 11.1 shows that the norm of such a term can be estimated by

[TABLE]

It follows that $\left|I_{j}\right|\leq C_{\varepsilon}{\left\|(\Im w)^{-1}\right\|^{5}}/{\sqrt{N}}$ for $j=5,6$. The proposition follows. ∎

We show next that $\mathcal{F}_{N}^{\varepsilon}(\beta)$ is close to its expected value. This result uses concentration inequalities in the presence of a Poincaré inequality. We recall that if the law of a random variable $X$ satisfies the Poincaré inequality with constant $C$ and $\alpha\in\mathbb{R}\setminus\{0\}$, then the law of $\alpha X$ satisfies the Poincaré inequality with constant $\alpha^{2}C$. Moreover, suppose that the probability measures $\mu_{1},\ldots,\mu_{r}$ on $\mathbb{R}$ satisfy the Poincaré inequality with constants $C_{1},\ldots,C_{r}$ respectively. Then the product measure $\mu_{1}\otimes\cdots\otimes\mu_{r}$ on $\mathbb{R}^{r}$ satisfies the Poincaré inequality with constant $C=\max\{C_{1},\dots,C_{r}\}$. That is, if $f:\mathbb{C}^{r}\to\mathbb{R}$ is an arbitrary differentiable function such that $f$ and its gradient ${\rm grad}f$ are square integrable relative to $\mu_{1}\otimes\cdots\otimes\mu_{r}$, then

[TABLE]

Here $\mathbb{V}(f)=\int|f-\int f\,{\rm d}\mu_{1}\otimes\cdots\otimes\mu_{r}|^{2}\,{\rm d}\mu_{1}\otimes\cdots\otimes\mu_{r}$ (see [28, Theorem 2.5]).

We use the following concentration result (see [3, Lemma 4.4.3 and Exercise 4.4.5] or [32, Chapter 3]).

Lemma 10.4**.**

Let $\mathbb{P}$ be a probability measure on $\mathbb{R}^{r}$ which satisfies a Poincaré inequality with constant $C$. Then there exist $K_{1}>0$ and $K_{2}>0$ such that, for every Lipschitz function $F$ on $\mathbb{R}^{r}$ with Lipschitz constant $|F|_{\text{\rm Lip}}$, and for every $\varepsilon>0$, we have

[TABLE]

The following result is similar to Proposition 8.4(1).

Proposition 10.5**.**

Suppose that $T_{N},S_{N}\in M_{N}(\mathbb{C})$ are contractions of uniformly bounded rank. Given $\varepsilon>0$ and $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$, almost surely

[TABLE]

Proof.

As in the proof of Proposition 8.4, it suffices to consider the case in which $T_{N}$ and $S_{N}$ are contractions of rank $1$. Write $I_{n}$ as a sum $Q_{1}+\cdots+Q_{n}$ of rank $1$ projections. Then the norm in the statement is at most equal to

[TABLE]

where each $Q_{j,k}$ is a contraction of rank $1$. Given a selfadjoint matrix $Z_{N}\in M_{N}(\mathbb{C})$, we set $R(Z,\beta)=(\beta\otimes I_{N}-L(C_{N},Z))^{-1}$ and $f_{N,j,k}(Z)=\mathrm{Tr}R(Z,\beta)Q_{j,k}$. We have

[TABLE]

and thus

[TABLE]

An application of Lemma 10.4 and of the comment preceding it yield

[TABLE]

for every $\delta>0$, with a constant $C$ that does not depend in $N,j,$ or $k$. The proposition follows by an application of the Borel-Cantelli lemma. ∎

Corollary 10.6**.**

For every $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$ and every $\varepsilon>0$ we have, almost surely,

[TABLE]

Observe now that

[TABLE]

We let $N\to\infty$ and then $\varepsilon\to 0$ and apply (10.6), (10.14), Proposition 10.2, and Proposition 10.1 to obtain the following result.

Theorem 10.7**.**

For every $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$ we have, almost surely, when $N$ goes to infinity, $\lim_{N\to\infty}\mathcal{F}_{N}(\beta)=(\omega(\beta)-s\gamma_{1})^{-1}\otimes I_{p}$.

Everything is now in place for completing the argument.

Proof of Theorem 6.3.

We noted earlier that $\omega(ze_{1,1}-\gamma_{0})$ is analytic in $\mathbb{C}\setminus\sigma(P(a,b))$. For fixed $t\in\mathbb{R}\setminus\sigma(P(a,b))$, set $\Psi(\beta)=\beta\otimes 1-L(a,b)$ and $\Psi_{N}(\beta)=\beta\otimes 1-L(C_{N},X_{N}/\sqrt{N})$. According to Lemma 4.2, $\Psi(te_{1,1}-\gamma_{0})$ is invertible, and thus there exists $\delta>0$ such that

[TABLE]

Theorem 2.3 and Proposition 2.2 imply that, almost surely, for every complex polynomial $Q$ in one variable we have

[TABLE]

Asymptotic freeness implies that almost surely, for every complex polynomial $Q$ in one variable,

[TABLE]

Thus, denoting the Hausdorff distance by $d_{H}$, we deduce that almost surely for $N$ large enough,

[TABLE]

Note that $\Psi_{N}(te_{1,1}-\gamma_{0})$ is selfadjoint. For an arbitrary $\beta\in M_{n}(\mathbb{C})$, we have

[TABLE]

It follows from (10.15) that, almost surely for all large $N$, if $\|\beta-(te_{1,1}-\gamma_{0})\|<\delta/4$, then

[TABLE]

Moreover, denoting by $s_{1}(M)$ the smallest singular value of an arbitrary matrix $M$,

[TABLE]

Thus, almost surely for all large $N$, provided that $\|\beta-(te_{1,1}-\gamma_{0})\|<\delta/4$, we have $\|\mathcal{F}_{N}(\beta)\|=\|(\Psi_{N}(\beta))^{-1}\|\leq 4/\delta.$ In other words, almost surely, the family $\{\mathcal{F}_{N}\}_{N\in\mathbb{N}}$ is normal in a neighborhood of $te_{1,1}-\gamma_{0}$. According to Theorem 10.7, for any $\beta\in\mathbb{H}^{+}(M_{n}(\mathbb{C}))$, almost surely $\mathcal{F}_{N}$ converges towards $(\omega(\beta)-s\gamma_{1})^{-1}\otimes I_{p}$. Set

[TABLE]

Almost surely for any $w\in\Lambda$ such that $\Im w\in M_{n}(\mathbb{Q})$ and $\Re w\in M_{n}(\mathbb{Q})$, $\mathcal{F}_{N}(w)$ converges towards $(\omega(w)-s\gamma_{1})^{-1}\otimes I_{p}$. The Vitali-Montel convergence theorem implies that that almost surely $\mathcal{F}_{N}$ converges towards a holomorphic function on $\{w\in M_{n}(\mathbb{C}),\|w-(te_{1,1}-\gamma_{0})\|<\delta/4\},$ and, in particular, $\mathcal{F}_{N}(ze_{1,1}-\beta_{0})$ converges for any $z\in\mathbb{C}$ such that $|z-t|<\delta/4$ towards $(\omega((ze_{1,1}-\gamma_{0})-s\gamma_{1})^{-1}\otimes I_{p}$.

Now, the Hurwitz theorem on zeros of analytic functions implies that, almost surely for large $N$, the function $F_{N}(ze_{1,1}-\gamma_{0})=\det(I_{n}\otimes I_{p}-\mathcal{F}_{N}(ze_{1,1}-\gamma_{0}))$ has as many zeros in a neighborhood of $t$ as the function

[TABLE]

Now, note that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

The theorem follows. ∎

11. Appendix

The following result is [12, Lemma 8.1].

Lemma 11.1**.**

For any matrix $M\in M_{n}(\mathbb{C})\otimes M_{N}(\mathbb{C})$,

[TABLE]

and for any fixed $k$,

[TABLE]

and

[TABLE]

where $M^{kl}$ is defined by (10.8).