The norm of polynomials in large random and deterministic matrices

Introduction and statement of result

For a Hermitian $N\times N$ matrix $H_{N}$, let $\mathcal{L}_{H_{N}}$ denote its empirical eigenvalue distribution, namely

where $\delta_{\lambda}$ is the Dirac mass in $\lambda$ and $\lambda_{1},\ldots,\lambda_{N}$ are the eigenvalues of $H_{N}$. The empirical eigenvalue distribution of large dimensional random matrices has been studied with much interest for a long time. One pioneering result is Wigner’s theorem , from 1958. Let $W_{N}$ be an $N\times N$ Wigner matrix. Then the theorem states that, under appropriate assumptions, the $n$-th moment of $\mathcal{L}_{W_{N}}$ converges in expectation to the $n$-th moment of the semicircular law as $N$ goes to infinity for any integer $n$. This result has been generalized in many directions, notably by Arnold for the almost sure convergence of the moments. The convergence of the empirical eigenvalue distribution for covariance matrices was first shown by Marc̆enko and Pastur in 1967, and has been generalized in the late 1970’s and the early 1980’s by many people, including Grenander and Silverstein , Wachter , Jonsson , Yin and Krishnaiah , Bai, Yin and Krishnaiah and Yin . In 1991, Voiculescu discovered a connection between large random matrices and free probability theory. He showed the so-called asymptotic freeness theorem, which has been generalized for instance in , which implies the almost sure weak convergence of the empirical eigenvalue distribution for Hermitian matrices $H_{N}$ of the form

$P$ is a fixed polynomial in $2p+q$ non commutative indeterminates,

$\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ is a family of independent $N\times N$ matrices of the normalized Gaussian Unitary Ensemble (GUE),

$\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ are $N\times N$ matrices with appropriate assumptions (see Theorem 1.3 below).

Let $\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ be a family of independent, normalized GUE matrices and $\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ be a family of $N\times N$ matrices, possibly random but independent of $\mathbf{X}_{N}$. Assume that for every Hermitian matrix $H_{N}$ of the form

where $P$ is a polynomial in $2q$ non commutative indeterminates, we have with probability one that:

Convergence of the empirical eigenvalue distribution: there exists a compactly supported measure $\mu$ on the real line such that the empirical eigenvalue distribution of $H_{N}$ converges weakly to $\mu$ as $N$ goes to infinity.

Convergence of the spectrum: for any $\varepsilon>0$, almost surely there exists $N_{0}$ such that for all $N\geqslant N_{0}$,

Then almost surely the convergences of the empirical eigenvalue distribution and of the spectrum also hold for all Hermitian matrices $H_{N}=P(\mathbf{X}_{N},\mathbf{Y}_{N},\mathbf{Y}_{N}^{*})$, where $P$ is a polynomial in $p+2q$ non commutative indeterminates.

Theorem 1.1 is a straightforward consequence of Theorem 1.6 below, where the language of free probability is used. Moreover, Theorem 1.6 specifies Theorem 1.1 by giving a description of the limit of the empirical eigenvalue distribution. For readers convenience, we recall some definitions (see and for details).

The elements of $\mathcal{A}$ are called non commutative random variables. We will always assume that $\tau$ is a trace, i.e. that it satisfies $\tau[ab]=\tau[ba]$ for every $a,b\in\mathcal{A}$. The trace $\tau$ is said to be faithful when it satisfies $\tau[a^{*}a]=0$ only if $a=0$.

The non commutative law of a family $\mathbf{a}=(a_{1},\ldots,a_{p})$ of non commutative random variables is defined as the linear functional P\mapsto\tau\big{[}P(\mathbf{a},\mathbf{a}^{*})\ \big{]}, defined on the set of polynomials in $2p$ non commutative indeterminates. The convergence in law is the pointwise convergence relative to this functional.

as soon as $i_{1}\neq i_{2}\neq\ldots\neq i_{K}$ and \tau\big{[}P_{k}(\mathbf{a}_{i_{k}},\mathbf{a}_{i_{k}}^{*})\ \big{]}=0 for $k=1,\ldots,K$.

with $d\sigma(t)=\frac{1}{2\pi}\sqrt{4-t^{2}}\ \mathbf{1}_{|t|\leqslant 2}\ dt$ the semicircle distribution.

Recall first the statement of Voiculescu’s asymptotic freeness theorem.

Let $\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ be a family of independent, normalized GUE matrices and $\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ be a family of $N\times N$ matrices, possibly random but independent of $\mathbf{X}_{N}$. Let $\mathbf{x}=(x_{1},\ldots,x_{p})$ be a free semicircular system in a ∗-probability space $(\mathcal{A},.^{*},\tau)$ and $\mathbf{y}=(y_{1},\ldots,y_{q})$ in $\mathcal{A}^{q}$ be a family of non commutative random variables free from $\mathbf{x}$. Assume the following.

where $\tau_{N}$ denotes the normalized trace of $N\times N$ matrices.

Boundedness of the spectrum: Almost surely, for $j=1,\ldots,q$ one has

where $\|\cdot\|$ denotes the operator norm.

In Haagerup and Thorbjørnsen strengthened the connection between random matrices and free probability. Limits of random matrices have now to be seen in more elaborated structure, called $\mathcal{C}^{*}$-probability space, which is endowed with a norm.

A $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ consists of a ∗-probability space $(\mathcal{A},.^{*},\tau)$ and a norm $\|\cdot\|$ such that $(\mathcal{A},.^{*},\|\cdot\|)$ is a $\mathcal{C}^{*}$-algebra.

By the Gelfand-Naimark-Segal construction, one can always realize $\mathcal{A}$ as a norm-closed $\mathcal{C}^{*}$-subalgebra of the algebra of bounded operators on a Hilbert space. Hence we can use functional calculus on $\mathcal{A}$. Moreover, if $\tau$ is a faithful trace, then the norm $\|\cdot\|$ is uniquely determined by the following formula (see [28, Proposition 3.17]):

Let $X_{1}^{(N)},\ldots,X_{p}^{(N)}$ be independent, normalized $N\times N$ GUE matrices and let $x_{1},\ldots,x_{p}$ be a free semicircular system in a $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ with a faithful trace. Then almost surely, one has: for all polynomials $P$ in $p$ non commutative indeterminates, one has

This article is mainly devoted to the following theorem which is a generalization of Theorem 1.5 in the setting of Theorem 1.3.

Let $\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ be a family of independent, normalized GUE matrices and $\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ be a family of $N\times N$ matrices, possibly random but independent of $\mathbf{X}_{N}$. Let $\mathbf{x}=(x_{1},\ldots,x_{p})$ and $\mathbf{y}=(y_{1},\ldots,y_{q})$ be a family of non commutative random variables in a $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ with a faithful trace, such that $\mathbf{x}$ is a free semicircular system free from $\mathbf{y}$. Assume the following. Strong convergence of $\mathbf{Y}_{N}$: Almost surely, for all polynomials $P$ in $2q$ non commutative indeterminates, one has

Then, almost surely, for all polynomials $P$ in $p+2q$ non commutative indeterminates, one has

The convergence of the normalized traces stated in (1.13) is the content of Voiculescu’s asymptotic freeness theorem and is recalled in order to give a coherent and complete statement. Theorem 1.1 is easily deduced from Theorem 1.6 by applying Hamburger’s theorem for the convergence of the measure and functional calculus for the convergence of the spectrum. Organization of the paper: In Section 2 we give applications of Theorem 1.6 which are proved in Section 9. Sections 3 to 8 are dedicated to the proof of Theorem 1.6. Acknowledgments: The author would like to thank Alice Guionnet for dedicating much time for many discussions to the subjects of this paper and, along with Manjunath Krishnapur and Ofer Zeitouni, for the communication of Lemma 8.2. He is very much obliged to Dimitri Shlyakhtenko for his contribution to this paper. He would like to thank Benoit Collins for pointing out an error in a previous version of Corollary 2.1 and giving the idea to fix it. He also likes to thank Mikael de la Salle for useful discussions.

Applications

The first and the simpler matrix model that may be investigated to play the role of matrices $\mathbf{Y}_{N}$ in Theorem 1.6 consists of deterministic diagonal matrices with real entries and prescribed asymptotic spectral measure.

Let $\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ be a family of independent, normalized GUE matrices and let $\mathbf{D}_{N}=(D_{1}^{(N)},\ldots,D_{q}^{(N)})$ be $N\times N$ deterministic real diagonal matrices, such that for any $j=1,\ldots,q$,

the empirical spectral distribution of $D^{(N)}_{j}$ converges weakly to a compactly supported probability measure $\mu_{j}$,

the diagonal entries of $D_{j}^{(N)}$ are non decreasing:

for all $\varepsilon>0$, there exists $N_{0}$ such that for all $N\geqslant N_{0}$, for all $j=1\dots q$,

Let $v=(v_{1},\ldots,v_{q})$ in $^{q}$. We set \mathbf{D}_{N}^{v}=\big{(}D_{1}^{(N)}(v_{1}),\ldots,D_{q}^{(N)}(v_{q})\big{)}, where for any $j=1,\ldots,q$, one has

Let $\mathbf{x}=(x_{1},\ldots,x_{p})$ and \mathbf{d}^{v}=\big{(}d_{1}(v),\ldots,d_{q}(v)\big{)} be non commutative random variables in a $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ with a faithful trace, such that

$\mathbf{x}$ is a free semicircular system, free from $\mathbf{d}^{v}$,

The variables $d_{1}(v),\ldots,d_{q}(v)$ commute, are selfadjoint and for all polynomials $P$ in $q$ indeterminates, one has

Then, with probability one, for all polynomials $P$ in $p+q$ non commutative indeterminates, one has

for any $v$ in $^{q}$ except in a countable set.

Remark that the non commutative random variables $d_{1},\ldots,d_{q}$ can be realized as classical random variables, $d_{j}$ being $\mu_{j}$-distributed for $j=1,\ldots,q$. The dependence between the random variables is trivial since Formula (2.1) exhibits a deterministic coupling. The convergence of the normalized trace (2.2) actually holds for any $v$. In general, the convergence (2.3) of the norm can fail: the family of matrices $\mathbf{D}=(D_{1}^{(N)},D_{2}^{(N)})$ where

gives a counterexample (consider their difference). Furthermore, let mention that it is clear that we always can take one of the $v_{i}$ to be zero.

2 Non-white Wishart matrices

Theorem 1.6 may be used to deduce the same result for some Wishart matrices as for the GUE matrices. Let $r,s_{1},\ldots,s_{p}\geqslant 1$ be integers. Let $\mathbf{Z}_{N}=(Z_{1}^{(N)},\ldots,Z_{p}^{(N)})$ be a family of independent positive definite Hermitian random matrices such that for $j=1,\ldots,p$ the matrix $Z_{j}^{(N)}$ is of size $s_{j}N\times s_{j}N$. Let $\mathbf{W}_{N}=\mathbf{W}_{N}(\mathbf{Z})=(W_{1}^{(N)},\ldots,W_{p}^{(N)})$ be the family of $rN\times rN$ matrices defined by: for each $j=1,\ldots,p$, $W_{j}^{(N)}=M_{j}^{(N)}\ Z_{j}^{(N)}\ M_{j}^{(N)*}$, where $M_{j}^{(N)}$ is a $rN\times s_{j}N$ matrix whose entries are random variables,

Let $\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ be a family of $rN\times rN$ random matrices, independent of $\mathbf{Z}_{N}$ and $\mathbf{W}_{N}$. Assume that the families of matrices $(Z_{1}^{(N)}),\ldots,(Z_{q}^{(N)}),\mathbf{Y}_{N}$ satisfy separately the assumptions of Theorem 1.6. Then, almost surely, for all polynomials $P$ in $p+2q$ non commutative indeterminates, one has

where $\|\cdot\|$ is given by Formula (1.9) with $\tau$ a faithful trace for which the non commutative random variables $\mathbf{w}=(w_{1},\ldots,w_{p})$ and $\mathbf{y}=(y_{1},\ldots,y_{q})$ are free.

In , motivated by applications in statistics and wireless communications, the authors study the global limiting behavior of the spectrum of the following matrix, referred as separable covariance matrix:

where $X_{n}$ is a $n\times m$ random matrix, $A_{n}^{1/2}$ is a nonnegative definite square root of the nonnegative definite $n\times n$ Hermitian matrix $A_{n}$ and $B_{n}$ is a $m\times m$ diagonal matrix with nonnegative diagonal entries. It is shown in that, for $n$ large enough, almost surely the eigenvalues of $C_{n}$ belong in a small neighborhood of the limiting distribution under the following assumptions:

$m=m(n)$ with $c_{n}:=n/m\underset{n\rightarrow\infty}{\longrightarrow}c>0$.

The entries of $X_{n}$ are independent, identically distributed, standardized complex and with a finite fourth moment.

The empirical eigenvalue distribution $\mathcal{L}_{A_{n}}$ (respectively $\mathcal{L}_{B_{n}}$) of $A_{n}$ (respectively $B_{n}$) converges weakly to a compactly supported probability measure $\nu_{a}$ (respectively $\nu_{b}$) and the operator norms of $A_{n}$ and $B_{n}$ are uniformly bounded.

Now consider the following situation, where Corollary 2.2 may be applied

$n=n(N)=rN$, $m=m(N)=sN$ for fixed positive integers $r$ and $s$,

the entries of $X_{n}$ are independent, identically distributed, standardized complex Gaussian,

the empirical eigenvalue distribution of $A_{n}$ (respectively ${B_{n}}$) converges weakly to a compactly supported probability measure,

for $N$ large enough, the eigenvalues of $A_{n}$ (respectively $B_{n}$) belong in a small neighborhood of its limiting distribution.

Then we obtain by Corollary 2.2 that for $N$ large enough, almost surely the eigenvalues of $C_{n}$ belong in a small neighborhood of the limiting distribution. The advantage of our version is the replacement of assumption 4 by assumption 4’. Replacing assumptions 1’ and 2’ by assumptions 1 and 2 could be an interesting question.

3 Block matrices

It will be shown as a consequence of Theorem 1.6 that the convergence of norms (1.14) also holds for block matrices.

4 Channel matrices

We give a potential application of Theorem 1.6 in the context of communication, where rectangular block random matrices are sometimes investigated for the study of wireless Multiple-input Multiple-Output (MIMO) systems . In the case of Intersymbol-Interference, the channel matrix $H$ reflects the channel effect during a transmission and is of the form

the family of $rN\times rN$ matrices $\mathbf{C}=(C_{1},\ldots,C_{L})$ and the family of $tN\times tN$ matrices $\mathbf{D}=(D_{1},\ldots,D_{L})$ satisfy separately the assumptions of Theorem 1.6,

the families of matrices $\mathbf{M}$, $\mathbf{C}$ and $\mathbf{D}$ are independent.

The strategy of proof

Let $\mathbf{X}_{N}=(X_{1}^{(N)},\ldots,X_{p}^{(N)})$ and $\mathbf{Y}_{N}=(Y_{1}^{(N)},\ldots,Y_{q}^{(N)})$ be as in Theorem 1.6. We start with some remarks in order to simplify the proof.

We can suppose that the matrices of $\mathbf{Y}_{N}$ are Hermitian. Indeed for any $j=1,\ldots,q$, one has $Y_{j}^{(N)}=$ Re $Y_{j}^{(N)}+i$ Im $Y_{j}^{(N)}$, where

It is sufficient to prove that for any polynomial the convergence of the norm in (1.14) holds almost surely (instead of almost surely the convergence holds for all polynomials). Indeed we can switch the words ”for all polynomials with rational coefficients“ and ”almost surely“ and both the left and the right hand side in (1.14) are continuous in $P$.

where the trace $\tau$ is completely defined by:

$\mathbf{x}=(x_{1},\ldots,x_{p})$ is a free semicircular system,

$\mathbf{y}=(y_{1},\ldots,y_{q})$ is the limit in law of $\mathbf{Y}_{N}$,

Since $\tau$ is faithful on the ∗-algebra spanned by $\mathbf{x}$ and $\mathbf{y}$, we can always assume that $\tau$ is a faithful trace on $\mathcal{A}$. Moreover, the matrices $\mathbf{Y}_{N}$ are uniformly bounded in operator norm. If we define $\|\cdot\|$ in $\mathcal{A}$ by Formula (1.9), then $\|y_{j}\|$ is finite for every $j=1,\ldots,q$. Hence, we can assume that $\mathcal{A}$ is a $\mathcal{C}^{*}$-probability space endowed with the norm $\|\cdot\|$. Haagerup and Thorbjørnsen describe in a method to show that for all non commutative polynomials $P$, almost surely one has

We present in this section this method with some modification to fit our situation. First, it is easy to see the following.

For all non commutative polynomials $P$, almost surely one has

In a $\mathcal{C}^{*}$-algebra $(\mathcal{A},.^{*},\|\cdot\|)$, one has $\forall a\in\mathcal{A}$, $\|a\|^{2}=\|a^{*}a\|$. Hence, without loss of generality, we can suppose that $H_{N}:=P(\mathbf{X}_{N},\mathbf{Y}_{N},\mathbf{Y}_{N}^{*})$ is non negative Hermitian and $h:=P(\mathbf{x},\mathbf{y},\mathbf{y}^{*})$ is selfadjoint. Let $\mathcal{L}_{N}$ denote the empirical spectral distribution of $H_{N}$:

It remains to show that the limsup is smaller than the right hand side in (3.3). The method is carried out in many steps.

the families $\mathbf{x}$, $\mathbf{y}$, $\mathbf{Y}_{1}$, $\mathbf{Y}_{2},\dots,\mathbf{Y}_{N},\dots$ are free,

for any polynomials $P$ in $q$ non commutative indeterminates $\tau[P(\mathbf{Y}_{N})]:=\tau_{N}[P(\mathbf{Y}_{N})]$.

The intermediate object $L(\mathbf{x},\mathbf{Y}_{N})$ is therefore well defined as an element of $\mathcal{A}$. We use a theorem about norm convergence, due to D. Shlyakhtenko and stated in Appendix A, to relate the spectrum of $L(\mathbf{x},\mathbf{Y}_{N})$ with the spectrum of $L(\mathbf{x},\mathbf{y})$.

Step 2. An intermediate inclusion of spectrum: for all $\varepsilon>0$ there exists $N_{0}$ such that for all $N\geqslant N_{0}$, one has

for a constant $c$ and with $\|\cdot\|$ denoting the operator norm.

Organization of the proof We tackle the different points of the proof described above in the following order:

Proof of Step 5. The asymptotic subordination property for random matrices is stated in Theorem 5.1 in a more general situation. The matrices $\mathbf{Y}_{N}$ can be random, independent of $\mathbf{X}_{N}$, satisfying a Poincaré inequality, without assumption on their asymptotic properties. This result is based on the Schwinger-Dyson equation and on the Poincaré inequality satisfied by the law of $\mathbf{X}_{N}$.

Proof of Estimate (3.9). The estimate will follow easily from the two previous items.

Proof of Step 3. The method is quite standard once Steps 2 and 4 are established. We use a version due to which is based on the use of local concentration inequalities.

Proof of Step 4: the subordination property for matrix-valued non commutative random variables

This lemma will be used throughout this paper. See [19, Lemma 3.1] for a proof.

The functional $G_{z}$ is well defined by Lemma 4.1 and satifies

The following proposition states the fundamental property of the amalgamated $\mathcal{R}$-transform, namely the subordination property, which is the keystone of our proof of Theorem 1.6.

Suppose that the families $\mathbf{x}$ and $\mathbf{y}$ are free. Then one has

Linearity property: There is a $\gamma$ such that, in the domain $V_{\gamma}$, one has

Subordination property: There is $\delta$ such that, for every $\Lambda$ in $U_{\delta}$, one has

Semicircular case: If $(x_{1},\ldots,x_{p})$ is a free semicircular system, then we get

The linearity property has been shown by Voiculescu in and the $\mathcal{R}$-transform of $s$ has been computed by Lehner in . We deduce easily the subordination property since by Equation (4.3): there exists $\gamma>0$ such that for all $\Lambda\in V_{\gamma}$,

Then there exists a $\delta>0$ such that, with $G_{s+t}(\Lambda)$ instead of $\Lambda$ in the previous equality,

We compose by $G^{(-1)}_{t}$ to obtain the result. ∎

Let $s$ and $t$ be as in Proposition 4.2, with $\mathbf{x}$ a free semicircular system.

where $c=\frac{1}{\varepsilon}\sum_{j=1}^{p}\|a_{j}\|^{2}$.

Proof of Step 5: the asymptotic subordination property for random matrices

We denote by $\mathcal{R}_{s}$ the functional

The proof of Theorem 5.1 is carried out in two steps.

In Section 5.1 we state a mean Schwinger-Dyson equation for random Stieltjes transforms (Proposition 5.2).

In Section 5.2 we deduce from Proposition 5.2 a Schwinger-Dyson equation for mean Stieltjes transforms (Proposition 5.3).

Theorem 5.1 is a direct consequence of Proposition 5.3 as it is shown in Section 5.3.

This induces an analogue formula for independent matrices of the GUE, called the Schwinger-Dyson equation, where the Hermitian symmetry of the matrices plays a key role. For instance, if $P$ is a monomial in $p$ non commutative indeterminates, one has for $i=1,\ldots,p$,

the sum over all decompositions $P=Lx_{i}R$ for $L$ and $R$ monomials being viewed as the partial derivative. This formula has an analogue for analytical maps instead of polynomials. The case of the function $\mathbf{X}_{N}\mapsto(\Lambda\otimes\mathbf{1}_{N}-S_{N})^{-1}$ is investigated in details in [19, Formula (3.9)], our proof is obtained by minor modifications.

We take the partial trace in Equation (5.4) to obtain:

Re-injecting this expression in the left hand side of Equation (5.5), one gets Equation (5.3):

2 Schwinger-Dyson equation for mean Stieltjes transforms

is controlled in operator norm by the following estimate:

with c={k^{9/2}\sigma}\sum_{j=1}^{p}\|a_{j}\|^{2}\Big{(}\sum_{j=1}^{p}\left\|a_{j}\right\|+\sum_{j=1}^{q}\|b_{j}\|\Big{)}^{2}.

The Poincar inequality is compatible with tensor product and then such a formula is still valid when $F$ is a function of the matrices $\bf X_{N}$ and $\bf Y_{N}$ with $v^{2}=\frac{\sigma}{N}$. We will often deal with matrices of size $k\times k$. Since the integer $k$ is fixed, we can use intensively the equivalence of norms, the constants appearing will not modify the order of convergence. For any integer $K$, we denote the Euclidean norm of a $K\times K$ matrix $A=(a_{m,n})_{1\leqslant m,n\leqslant K}$ by

Recall that if $A,B$ are $K\times K$ matrices we have the following inequalities

To estimate the operator norm of $\Theta_{N}$ we use the domination by the infinity norm (5.15) in order to split the contributions due to $M_{N}$ and due to $l_{N}-L_{N}$: we get

where we have denoted the $N\times N$ matrices

Remark that by (5.16), for $u^{\prime},v^{\prime}=1,\ldots,k$,

Then by Cauchy-Schwarz inequality we get:

One is reduced to the study of variances of random variables. To use the Poincar inequality, we write for $u,v,u^{\prime},v^{\prime}=1,\ldots,k$,

The functions and their partial derivatives are bounded (see [19, Lemma 4.6] with minor modifications), so that, since the law of $(\mathbf{X}_{N},\mathbf{Y}_{N})$ satisfies a Poincar inequality with constant $\frac{\sigma}{N}$, one has

We define the set $\mathcal{W}$ of families $(\mathbf{V},\mathbf{W})$ of $N\times N$ Hermitian matrices, with $\mathbf{V}=(V_{1},\ldots,V_{p})$, $\mathbf{W}=(W_{1},\ldots,W_{q})$, of unit Euclidean norm in $\mathbf{R}^{(p+q)N^{2}}$. Then we have

For all $(\mathbf{V},\mathbf{W})$ in $\mathcal{W}$, for all selfadjoint $N\times N$ matrices $\mathbf{A}=(A_{1},\ldots,A_{1})$, $\mathbf{B}=(B_{1},\ldots,B_{1})$:

The Cauchy-Schwarz inequality for $\textrm{Tr}_{k}\otimes\textrm{Tr}_{N}$ (i.e. for $\textrm{Tr}_{kN}$) gives

Using (5.16) to split Euclidean norms into the product of an operator norm and an Euclidean norm, we get:

Remark that, since $(\mathbf{V},\mathbf{W})\in\mathcal{W}$, the norm of the matrices $V_{j}$ and $W_{j}$ is bounded by one. Then we have the following:

We then obtain as desired, by (5.18), (5.19) and (5.20):

where c={k^{9/2}\sigma}\sum_{j=1}^{p}\|a_{j}\|^{2}\Big{(}\sum_{j=1}^{p}\left\|a_{j}\right\|+\sum_{j=1}^{q}\|b_{j}\|\Big{)}^{2}.

3 Proof of Theorem 5.1

where \Theta_{N}(\Lambda)=\Theta_{N}\Big{(}\Lambda,\Lambda-\mathcal{R}_{s}\big{(}G_{S_{N}+T_{N}}(\Lambda)\ \big{)}\ \Big{)} is analytic in $k^{2}$ complex variables. Recall that by (4.7), we have \big{\|}\big{(}\Lambda-\mathcal{R}_{s}\big{(}G_{S_{N}+T_{N}}(\Lambda)\ \big{)}\ \big{)}^{-1}\big{\|}\leqslant\|(\Lambda)^{-1}\|, which gives (when replacing $c$ in (5.13) by $c/2$) the expected estimate of $\Theta_{N}(\Lambda)$.

Proof of Estimate (3.9)

Let $(\mathbf{X}_{N},\mathbf{Y}_{N},\mathbf{x},\mathbf{y})$ be as in Section 3. We assume that \big{(}\mathbf{x},\mathbf{y},(\mathbf{Y}_{N})_{N\geqslant 1}\big{)} are realized in a same $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ with faithful trace, where

the families $\mathbf{x}$, $\mathbf{y}$, $\mathbf{Y}_{1}$, $\mathbf{Y}_{2},\dots,\mathbf{Y}_{N},\dots$ are free,

for any polynomials $P$ in $q$ non commutative indeterminates $\tau[P(\mathbf{Y}_{N})]:=\tau_{N}[P(\mathbf{Y}_{N})]$.

On the other hand, since the matrices of $\mathbf{Y}_{N}$ are deterministic, we can apply Theorem 5.1 with $\sigma=1$

Then for $\eta<1/3$, there exists $N_{0}$ such that for all $N\geqslant N_{0}$ and for any $\Lambda$ in $\Omega_{\eta}^{(N)}$, one has

Then by Proposition 4.3 with $(t,G,\Theta,\Omega,\varepsilon)=(T_{N},G_{S_{N}+T_{N}},\Theta_{N},\Omega^{(N)}_{\eta},1/2)$, one has

where $c$ denotes now the constant c={k^{9/2}}\sum_{j=1}^{p}\|a_{j}\|\Big{(}\sum_{j=1}^{p}\left\|a_{j}\right\|+\sum_{j=1}^{q}\|b_{j}\|\Big{)}^{2}\Big{(}\varepsilon^{-2}+2\sum_{j=1}^{p}\|a_{j}\|^{2}\Big{)}.

Proof of Step 2: An intermediate inclusion of spectrum

For a review on the theory of $\mathcal{C}^{*}$-algebras, we refer the readers to and . Notably, Appendix A of the second reference contains facts about ultrafilters and ultraproducts that are used in this section. Let \big{(}\mathbf{x},\mathbf{y},(\mathbf{Y}_{N})_{N\geqslant 1}\big{)} be as in Section 3. We assume that these non commutative random variables are realized in the same $\mathcal{C}^{*}$-probability space $(\mathcal{A},.^{*},\tau,\|\cdot\|)$ with faithful trace, where

the families $\mathbf{x}$, $\mathbf{y}$, $\mathbf{Y}_{1}$, $\mathbf{Y}_{2},\dots,\mathbf{Y}_{N},\dots$ are free,

for any polynomials $P$ in $q$ non commutative indeterminates $\tau[P(\mathbf{Y}_{N})]:=\tau_{N}[P(\mathbf{Y}_{N})]$.

A consequence of Voiculescu’s theorem and of Shlyakhtenko’s Theorem A.1 in Appendix A is that for all polynomials $P$ in $p+q$ non commutative indeterminates,

Let $\mathcal{A}$ and $\mathcal{B}$ be unital $\mathcal{C}^{*}$-algebra. Let $\pi:\mathcal{A}\rightarrow\mathcal{B}$ be a morphism of unital ∗-algebra. Then $\pi$ is contractive.

It is easy to see that for any $a$ in $\mathcal{A}$, the spectrum of $\pi(a)$ is included in the spectrum of $a$ (since $\lambda\mathbf{1}_{\mathcal{A}}-a$ invertible implies that $\lambda\mathbf{1}_{\mathcal{A}}-\pi(a)$ is also invertible). Hence we get that for all $a$ in $\mathcal{A}$

For the existence we consider the norm given by the spectral radius. The uniqueness follows from Lemma 7.1. ∎

Let $k\geqslant 1$ be an integer. For all $N\geqslant 1$, let $\mathbf{z}_{N}=(z_{1}^{(N)},\ldots,z_{p}^{(N)})$, respectively $\mathbf{z}=(z_{1},\ldots,z_{p})$, be self-adjoint non commutative random variables in a $\mathcal{C}^{*}$- probability space $(\mathcal{A}_{N},.^{*},\tau_{N},\|\cdot\|_{\tau_{N}})$, respectively $(\mathcal{A},.^{*},\tau,\|\cdot\|_{\tau})$. Assume that the traces $\tau_{N}$ and $\tau$ are faithful (hence the notation for the norms) and that for any polynomial $P$ in $p$ non commutative indeterminates,

The algebra $\mathfrak{A}^{(k)}$ is a $\mathcal{C}^{*}$-algebra whose norm $\|\cdot\|_{\mathfrak{A}^{(k)}}$ is given by: for all $a$ in $\mathfrak{A}^{(k)}$, equivalence class of $(a_{N})_{N\geqslant 1}$

for all ultrafilter $\mathcal{U}$. Then the convergence holds when $N$ goes to infinity. ∎

The convergence extends to continuous function on the real line and then, with an appropriate choice of test functions, Step 2 follows. ∎

Proof of Step 3: from Stieltjes transforms to spectra

By Step 2, for all $\varepsilon>0$, there exists $N_{0}\geqslant 1$ such that for all $N\geqslant N_{0}$, one has

With (8.1) and (8.2) established, it is easy to show with minor modifications of [1, Lemma 5.5.5] the following result.

To get an almost sure control of $D_{N}(f)$, we use the fact that the entries of the matrices $\mathbf{X}_{N}$ satisfy a concentration inequality.

With $f$ as in Lemma 8.1, there exists $\kappa>0$ such that, almost surely

Then, for any smooth function $f$, one has

where $\|\cdot\|_{\infty}$ denotes the supremum of the considered function on the set of $kN\times kN$ Hermitian matrices. Hence we get that $|\Psi_{N}^{(f)}|_{\mathcal{L}}\leqslant\rho^{(f)}_{N,\kappa}\leqslant\frac{1}{\sqrt{k}}{N^{-1/2-2\kappa}}$. We fix $f$ a smooth function, non negative, compactly supported and vanishing on a neighborhood of the spectrum of $L({\mathbf{x}},\mathbf{y})$. By the Tchebychev inequality

where we have used Lemma 8.1 ($f^{\prime 2}$ also vanishes in a neighborhood of the spectrum of $L({\mathbf{x}},\mathbf{y})$). Moreover, since $\Psi^{(f)}_{N}$ and $\Phi^{(f)}_{N}$ are equals in $\mathcal{B}^{(f)}_{N,\kappa}$ and $\|\Psi^{(f)}_{N}\|_{\infty}\leqslant\|\Phi^{(f)}_{N}\|_{\infty}$,

Now, by (8.5) applied to $\Psi^{(f)}_{N}$: for all $\delta>0$

By (8.10), (8.11), Lemma 8.1 and the Borel-Cantelli lemma, $D_{N}(f)$ is almost surely of order $N^{1+\kappa}$ at most. ∎

For every $\varepsilon>0$, there exists $N_{0}$ such that for $N\geqslant N_{0}$

Proof of Corollaries 2.1, 2.2 and 2.4

converges weakly to $\mu$. This sequence is tight, since there exists a $B>0$ such that for all $j=1\dots q$, for all $i=1\dots N$, one has $\lambda_{i}(j)\in[-B,B]$. Hence it is sufficient to show the following: for all real numbers $a_{1},\ldots,a_{q}$, for all $\varepsilon>0$, there exists $\eta>0$ such that

\mu_{j_{0}}\big{(}\ ]a_{j_{0}},a_{j_{0}}+\eta]\ \big{)}<\varepsilon/2.

for all $j=1,\ldots,q$, the real numbers $a_{j}+\eta$ and $a_{j_{0}}+\eta$ are points of continuity for $F_{j}$.

By (9.3) with $a=a_{j}+\eta$, there exists $N_{0}\geqslant 1$ such that for all $N\geqslant N_{0}$ and $j=1,\ldots,q$, one has

But $F_{j}(a_{j}+\eta)\geqslant F_{j}(a_{j})\geqslant F_{j_{0}}(a_{j_{0}})$. Then we have

The $\lambda_{i}(j)$ are non decreasing, so we get

On the other hand, by (9.3) with $j=j_{0}$ and $a=a_{j_{0}}+\eta$, there exists $N_{0}\geqslant 1$ such that, for all $N\geqslant N_{0}$, one has

But $F_{j_{0}}(a_{j_{0}}+\eta)\leqslant F_{j_{0}}(a_{j_{0}})+\varepsilon/2$, so that

The $\lambda_{i}(j_{0})$ are non decreasing, then we get

By (9.4) and (9.5) we obtain: for all $N\geqslant N_{0}$

and then (9.2) is satisfied. So the convergence (9.1) holds when $v$ is zero. The convergence of traces, case $v$ in $^{q}$: To deduce the general case we shall need the following lemmas.

In particular, we have the convergence of the quantile of order $v$:

Denote $w=F^{-1}(v)$. Let $\eta\geqslant 0$ be such that $w-\eta$ and $w+\eta$ and points of continuity for $F$. Then, one has

Then, the $\lambda_{i}$ being non decreasing, for any $\varepsilon>0$ there exists $N_{0}$ such that for any $N\geqslant N_{0}$, one has

Since $v$ is a point of continuity for $F^{-1}$, we get that $F(w-\eta)<v$. We chose $\varepsilon<v-F(w-\eta)$. Then, we get $F(w-\eta)+\varepsilon<v$. Hence, there exists $N_{0}$ such that, for any $N\geqslant N_{0}$, one has i_{N}\geqslant\big{(}F(w-\eta)+\varepsilon\big{)}N and so, by (9.6): for any $\eta>0$, there exists $N_{0}$ such that for all $N\geqslant N_{0}$, one has $w-\eta\leqslant\lambda_{i_{N}}$. Hence, we get for all $\eta>0$,

and hence, letting $\eta$ go to zero, we obtain the expected result. ∎

Let $F$ denotes the cumulative distribution function of $\mu$ and $F^{-1}$ its generalized inverse. We set $w_{1}=F^{-1}(v_{1})$, $w_{2}=F^{-1}(v_{2})$, $a_{1}=F(w_{1})-v_{1}$ and $a_{2}=v_{2}-F(w_{2}^{-})$. Then, the empirical eigenvalue distribution of $D_{N}^{(v_{1},v_{2})}$ converges weakly the probability measure proportional to

We only show the lemma for $v_{2}=0$, the general case can be deduce by adapting the reasoning. We then use, for conciseness, the symbols $v,w$ and $a$ instead of $v_{1},w_{1}$ and $a_{1}$ respectively. If $F$ is not continuous in $w$ (i.e. if $\mu(w)\neq 0$) and $v\neq F(w)$, then for any $\alpha$ in $]0,(F(w)-v)/2[$, the map $F^{-1}$ is continuous in $v+\alpha$ and $F(w)-\alpha$. By Lemma 9.1, we get that

Hence, for any continuous function $f$, we get

If $F$ is continuous in $w$, we take $\alpha=0$ in the following. We can always find $\beta>0$, arbitrary small, such that $F(w)+\beta$ is a point of continuity for $F^{-1}$. Remark that we then have

Moreover, we can always find $\gamma$ in ]0,F^{-1}\big{(}F(w)+\beta\big{)}-w[, arbitrary small, such that $w+\gamma$ is a point of continuity for $F$ and $F(w+\gamma)<F(w)+\beta$. Then, by (9.9), we get that, for $N$ large enough

Hence, for any continuous function $f$, we get that for $N$ large enough

Letting $\alpha,\beta,\gamma$ go to zero, we get the result. ∎

Let $v$ in $^{q}$. We now show that, for any polynomial $P$, one has

At the possible price of relabeling the matrices, we assume $v_{1}\geqslant\dots\geqslant v_{q}$ and set

For any $j=1,\ldots,q$, we decompose the matrices $D_{j}^{(N)}(v_{j})$ into

where for any $i=1,\ldots,q$, the matrix $D_{j,i}^{(N)}$ is $N_{i}\times N_{i}$. We set for any $i=1,\ldots,q$, the family $\mathbf{D}_{N}(i)=(D_{1,i}^{(N)},\ldots,D_{q,i}^{(N)})$. For any $i,j=1,\ldots,q$, we denote by $F_{i,j}$ the cumulative distribution function of the measure obtained in Lemma 9.2 with $(D_{N},\mu,v_{1},v_{2})$ replaced by $(D_{j}^{(N)},\mu_{j},v_{i-1},v_{i})$. Then, for any polynomial $P$, one as

By Lemma 9.2 and by the case $v=(0,\ldots,0)$, we deduce that

with the convention $v_{0}=1$. The merge of the different measures gives as expected

It is sufficient then to show that, for any $\eta>0$, there exists $N_{0}\geqslant N$ such that for all $i=1,\ldots,N$, one has

and hence: for all $\varepsilon>0$, there exist $\eta\geqslant 0$ and $N_{0}\geqslant 1$ such that for all $N\geqslant N_{0}$, for all $i=1,\ldots,N$

Suppose that (9.13) is not true: there exist $\eta>0$ and $(N_{k})_{k\geqslant 1}$ an increasing sequence of positive integer such that for all $k\geqslant 1$, there exists $i_{k}$ such that

By compactness, one can always assume that $i_{k}/N_{k}$ converges to $u_{0}$ in $$. For all$j$in$\{1,\ldots,q\}$except a possible$j_{0}$, we have that$u_{0}+v_{j}$is a point of continuity for$F^{-1}_{j}$and so, by Lemma 9.1,$\lambda_{i_{k}+\lfloor v_{j}N_{k}\rfloor}^{(N_{k})}(j)$converges to$F^{-1}_{j}(u_{0}+v_{j})$. Recall that

Then we have, for $N$ large enough and for all $u$ in $$, that$\big{|}\lambda_{i_{k}+\lfloor v_{j_{0}}N_{k}\rfloor}^{(N_{k})}(j_{0})-F_{j_{0}}^{-1}(u+v_{j_{0}})\big{|}>\eta$ i.e.

which is in contradiction with the fact that for $N$ large enough the eigenvalues of $D_{j_{0}}^{(N)}$ belong to a small neighborhood of the support of $\mu_{j_{0}}$.

2 Proof of Corollary 2.2: Wishart matrices

Recall that by definition of the Wishart matrix model for $j=1,\ldots,p$

Let $P$ be a polynomial in $2p+2q+1$ non commutative indeterminates:

where the convergence holds almost surely since each term of the sum converges almost surely. ∎

For all polynomials $P$ in $2p+2q+1$ non commutative indeterminates, almost surely

$\|P(\underbrace{\mathbf{0}_{s_{j}N},\ldots,\mathbf{0}_{s_{j}N}}_{2q+j-1},Z_{j}^{(N)},\underbrace{\mathbf{0}_{s_{j}N},\ldots,\mathbf{0}_{s_{j}N}}_{p},\mathbf{1}_{s_{j}N},\underbrace{\mathbf{0}_{s_{j}N},\ldots,\mathbf{0}_{s_{j}N}}_{p-j})\|$, $j=1,\ldots,p$,

$\|P({\mathbf{y}},{\mathbf{y}}^{*},\underbrace{\mathbf{0},\ldots,\mathbf{0}}_{p},\mathbf{1},\underbrace{\mathbf{0},\ldots,\mathbf{0}}_{p})\|$,

$\|P(\underbrace{\mathbf{0},\ldots,\mathbf{0}}_{2q+j-1},z_{j},\underbrace{\mathbf{0},\ldots,\mathbf{0}}_{p},\mathbf{1},\underbrace{\mathbf{0},\ldots,\mathbf{0}}_{p-1})\|$, $j=1,\ldots,p$.

Define for $j=1,\ldots,p$ the non commutative polynomial $P_{j}$ deduced by the formula

$\mathbf{w}=(w_{1},\ldots,w_{p})$ are free selfadjoint non commutative random variables,

$\mathbf{y}=(y_{1},\ldots,y_{q})$ is the limit in law of $\mathbf{Y}_{N}$,

For any polynomial $P$ in $p+2q$ non commutative indeterminates

where the limits are almost sure. In particular we obtain that, for all polynomials $P$ in $p+2q$ non commutative indeterminates, one has

Together with (9.43), this gives the expected result.

3 Proof of Corollary 2.4: Rectangular band matrices

We only give a sketch of the proof. Details are obtained by minor modification of the proofs of Corollaries 2.2 and 2.3. Let $H$ be as in Corollary 2.4:

We start with the following observation: the operator norm of $H$ is the square root of the operator norm of $H^{*}H$, which is a square block matrix. Its blocks consist of sums of $tN\times tN$ matrices of the form $A_{l}^{*}A_{m}$, $l,m=1\dots L$. By minor modifications of the proof of Corollary 2.2, we get the almost sure convergence of the normalized trace and of the norm for any polynomial in the matrices $\mathbf{A}_{N}=(A_{l}^{*}A_{m})_{l,m=1..L}$ as $N$ goes to the infinity. By Proposition 7.3, we get that the convergences hold for square block matrices and in particular for any polynomial in $H^{*}H$. Hence the result follows by functional calculus.

Appendix A A theorem about norm convergence, by D. ShlyakhtenkoResearch supported by NSF grant DMS-0900776

Lemma Let $(A,\tau)$ be a $C^{*}$-algebra with a faithful trace $\tau$, and consider $B$ to be the universal $C^{*}$-algebra generated by $A$ and elements $L^{(1)},\dots,L^{(n)}$ satisfying $L^{(i)*}xL^{(j)}=\delta_{i=j}\tau(x)$ for all $x\in A$. Moreover, consider the linear functional $\psi$ determined on $*-\textrm{Alg}(A,\{L^{(j)}\}_{j})$ by:

$\psi(x_{0}L^{(i_{1})}x_{1}\cdots x_{k-1}L^{(i_{k})}x_{k}y_{0}L^{(j_{1})*}y_{1}\cdots y_{l-1}L^{(j_{l})*}y_{l})=0$ whenever $x_{1},\dots,x_{k},y_{0},\dots,y_{l}\in A$ and at least one of $k$ and $l$ is nonzero.

Consider the $A,A$-Hilbert bimodule $\mathcal{H}=L^{2}(A,\tau)\otimes A$ with the inner product

and the left and right $A$ actions given by

Let $B$ be the extended Cuntz-Pimsner algebra associated to $\mathcal{H}^{\oplus n}$ (see ), i.e. the universal $C^{*}$-algebra generated by $A$ and operators $L_{h}:h\in\mathcal{H}$ satisfying the relations

It follows from the results of that if we denote by $(\hat{B},\hat{\psi})$ the free product $(A,\tau)*(\mathcal{E},\phi)$, then:

If $h=(\sum_{i}\xi_{i}^{(k)}\otimes a_{i}^{(k)})_{k=1}^{n}\in(A\otimes A)^{\oplus n}\subset\mathcal{H}^{\oplus n}$ is a finite tensor, write

From this we see that (by the universal property of $B$) there exists a $*$-homomorphism $\pi:B\to\hat{B}$, so that $\psi=\hat{\psi}\circ\pi$. Thus all we need to prove is that $\pi$ is injective. But by [30, Prop. 3.3], it follows that $B$ is isomorphic to the Toeplitz algebra $\mathcal{T}$ (since in this case obviously $\langle\mathcal{H}^{\oplus n},\mathcal{H}^{\oplus n}\rangle_{A}=A$) acting on the Fock space $\mathcal{F}=\bigoplus_{k\geqslant 0}(\mathcal{H}^{\oplus n})^{\otimes_{A}k}$. If we denote by $E$ the canonical conditional expectation from $\mathcal{T}$ onto $A$ and consider the state $\theta=\tau\circ E$, then the resulting Hilbert space is the closure of $\mathcal{F}$ in the (faithful) norm $\|\xi\|=\tau(\langle\xi,\xi\rangle_{A})^{1/2}$; from this we see that the GNS representation of $B$ associated to the state $\theta$ on $B$ is faithful. Since $\hat{B}$ is exactly this GNS representation, it follows that $\pi$ is injective. ∎

Then $\mathfrak{A}$ is a $C^{*}$-algebra.

Let now $X_{N}^{(j)}$, $j=1,\dots,n$, $N=1,2,\dots$ be self-adjoint random variables and assume that $X^{(j)}$, $j=1,\dots,n$ are such that for any non-commutative polynomial $P$,

Let $L^{(j)}$, $j=1,\dots,n$ be a family of free creation operators, free from each other and from $\{X_{N}^{(j)}\}_{N,j}\cup\{X^{(j)}\}_{j}$. In other words, they satisfy:

and we denote by $\tau_{N}$ and $\psi_{N}$ the respective states on $A_{N}$ and $B_{N}$ ($\cong(A_{N},\tau_{N})*(\mathcal{E},\phi)$). We denote by $\tau$ and $\psi$ the respective states on $A$ and $B$ ($\cong(A,\tau)*(\mathcal{E},\phi)$).