Central limit theorems for linear statistics of heavy tailed random matrices

Introduction and statement of results

Recall that a Wigner matrix is a symmetric random matrix $A=(a_{i,j})_{i,j=1,\ldots,N}$ such that

the sub-diagonal entries of $A$ are independent and identically distributed (i.i.d.),

the random variables $\sqrt{N}a_{i,j}$ are distributed according to a measure $\mu$ that does not depend on $N$ and have all moments finite.

This model was introduced in 1956 by Wigner who proved the convergence of the moments

when $\mu$ is centered with unit variance. Moments can be easily replaced by bounded continuous functions in the above convergence and this convergence holds almost surely. Assumption 2 can also be weakened to assume only that the second moment is finite. The fluctuations around this limit or around the expectation were first studied by Jonsson in the (slightly different) Wishart model, then by Pastur et al. in , Sinai and Soshnikov with $p\ll N^{1/2}$ possibly going to infinity with $N$. Since then, a long list of further-reaching results have been obtained: the central limit theorem was extended to so-called matrix models where the entries interact via a potential in , the set of test functions was extended and the assumptions on the entries of the Wigner matrices weakened in , a more general model of band matrices was considered in (see also for general covariance matrices), unitary matrices where considered in , and Chatterjee developed a general approach to these questions in , under the condition that the law $\mu$ can be written as a transport of the Gaussian law. Finally, but this is not really our concern here, the fluctuations of the trace of words in several random matrices were studied in . It turns out that in these cases

converges towards a Gaussian variable whose covariance depends on the first four moments of $\mu$. Moments can also be replaced by regular enough functions and Assumption 2 can be weakened to assume that the fourth moment only is finite. The latter condition is however necessary as the covariance for the limiting Gaussian depends on it. The absence of normalization by $\sqrt{N}$ shows that the eigenvalues of $A$ fluctuate very little, as precisely studied by Erdös, Schlein, Yau, Tao, Vu and their co-authors, who analyzed their rigidity in e.g. .

In this article, we extend these results for a variation of the Wigner matrix model where Assumption 2 is removed: some entries of the matrix can be very large, e.g. when $\mu$ does not have any second moment or when it depends on $N$, with moments growing with $N$. Then, Wigner’s convergence theorem (1) does not hold, even when moments are replaced by smooth bounded functions. The analogue of the convergence (1) was studied when the common law $\mu$ of the entries of $A$ belongs to the domain of attraction of an $\alpha$-stable law or $\mu$ depends on $N$ and has moments blowing up with $N$. Although technical, the model introduced in Hypothesis 1.1 below has the advantage of containing these two examples (for $u,v$ some sequences depending implicitly on $N$, $u\ll v$ means that $u/v\longrightarrow 0$ as $N\to\infty$).

Let, for each $N\geq 1$, $A_{N}=[a_{ij}]$ be an $N\times N$ real symmetric random matrix whose sub-diagonal entries are some i.i.d. copies of a random variable $a$ (depending implicitly on $N$) such that: $\bullet$ The random variable $a$ can be decomposed into $a=b+c$ such that as $N\to\infty$,

Moreover, if the $b_{i}$’s are independent copies of $b$,

$\bullet$ For any $\varepsilon>0$ independent of $N$, the random variable $a$ can be decomposed into $a=b_{\varepsilon}+c_{\varepsilon}$ such that

Examples of random matrices satisfying Hypothesis 1.1 are defined as follows.

Let $A=(a_{i,j})_{i,j=1,\ldots,N}$ be a random symmetric matrix with i.i.d. sub-diagonal entries.

We say that $A$ is a Lévy matrix of parameter $\alpha$ in $]0,2[$ when $A=X/a_{N}$ where the entries $x_{ij}$ of $X$ have absolute values in the domain of attraction of $\alpha$-stable distribution, more precisely

We say that $A$ is a Wigner matrix with exploding moments with parameter $(C_{k})_{k\geq 1}$ whenever the entries of $A$ are centered, and for any $k\geq 1$

Both Lévy matrices and Wigner matrices with exploding moments satisfy Hypothesis 1.1. For Lévy matrices, the function $\Phi$ is given by formula

The proof of this lemma, and of Lemmas 1.8 and 1.12, which show that our hypotheses hold for both Lévy matrices and Wigner matrices, are given in Section 6.

One can easily see that our results also apply to complex Hermitian matrices: in this case, one only needs to require Hypothesis 1.1 to be satisfied by the absolute value of non diagonal entries and to have $a_{11}$ going to zero as $N\to\infty$.

A Lévy matrix whose entries are truncated in an appropriate way is a Wigner matrix with exploding moments . The recentered versionThe recentering has in fact asymptotically no effect on the spectral measure $A$ as it is a rank one perturbation. of the adjacency matrix of an Erdös-Rényi graph, i.e. of a matrix $A$ such that

is also an exploding moments Wigner matrix, with $\Phi(\lambda)=p(e^{-i\lambda}-1)$ (the measure $m$ of Lemma 1.3 is $p\delta_{1}$). In this case the fluctuations were already studied in . The method of can be adapted to study the fluctuations of linear statistics of Wigner matrices with exploding moments. Nevertheless, since we actually use Wigner matrices with exploding moments to study of the fluctuations of Lévy matrices, it is worthwhile to study these ensembles together.

The weak convergence of the empirical eigenvalues distribution of a Lévy matrix has been established in (see also ) where it was shown that for any bounded continuous function $f$,

where $\mu_{\alpha}$ is a heavy tailed probability measure which depends only on $\alpha$. Moreover, $\mu_{\alpha}$ converges towards the semicircle law as $\alpha$ goes to $2$.

The convergence in moments, in expectation and in probability, of the empirical eigenvalues distribution of a Wigner matrix with exploding moments has been established by Zakharevich in . In that case, moments are well defined and for any continuous bounded function $f$,

where $\mu_{\mathbf{C}}$ is a probability measure which depends only on the sequence $\mathbf{C}:=(C_{k})_{k\geq 1}$.

We shall first state the fluctuations of moments of a Wigner matrix with exploding moments around their limit, namely prove the following theorem.

Let $A=(a_{i,j})_{i,j=1,\ldots,N}$ be a Wigner matrix with exploding moments with parameter $(C_{k})_{k\geq 1}$. Then the process

converges in distribution to a centered Gaussian process.

This theorem has been established for the slightly more restrictive model of adjacency matrices of weighted Erdös-Rényi graphs in . Our proof is based on the moment method, the covariance of the process is of combinatorial nature, given in Section 2, Formula (40) and Theorem 2.2.

Note that the speed of the central limit theorem is $N^{-1/2}$ as for independent integrable random variables, but differently from what happens for standard Wigner’s matrices. This phenomenon has also already been observed for adjacency matrix of random graphs and we will see below that it also holds for Lévy matrices. It suggests that the repulsive interactions exhibited by the eigenvalues of most models of random matrices with lighter tails than heavy tailed matrices no longer work here.

By the previous results, for both Lévy and exploding moments Wigner matrices, $N^{-1}\operatorname{Tr}G(z)$ converges in probability to a deterministic limit as the parameter $N$ tends to infinity. We study the associated fluctuations.

In fact, even in the case of Wigner matrices with exploding moments, the CLT for moments does not imply a priori the CLT for Stieltjes functions even though concentration inequalities hold on the right scale, see . Indeed, one cannot approximate smooth bounded functions by polynomials for the total variation norm unless one can restrict oneself to compact subsets, a point which is not clear in this heavy tail setting. However, with additional arguments based on martingale technology, we shall prove the following result, valid for both Lévy matrices and Wigner matrices with exploding moments.

Under Hypothesis 1.1, where we assume additionally that for $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such that, for any $k\geq 1$, one has

Under the hypotheses of Theorem 1.5, the process

where $Z$ is a centered gaussian process with covariance $C(z,z^{\prime})$ given in Theorem 1.5 and the function $\Psi_{f}$ is given by

where $\phi$ is a smooth compactly supported function equal to one in the neighborhood of the of the origin.

The function $C(z,z^{\prime})$ in Theorem 1.5 is given by $C(z,z^{\prime})=L(z,z^{\prime})-L(z)L(z^{\prime})$, where

where $G_{k}(z)=(z-A_{k})^{-1}$ for $A_{k}$ the matrix obtained from $A$ by deleting the $k$-th row and column, and $G^{\prime}_{k}(z^{\prime})=(z^{\prime}-A_{k}^{\prime})^{-1}$ for $A_{k}^{\prime}$ a copy of $A_{k}$ where the entries $(i,j)$ for $i$ or $j\geq k$ are independent of $A_{k}$ and the others are those of $A_{k}$. Also, we assume $t\Im z\geq 0$ and $t^{\prime}\Im z^{\prime}\geq 0$ in (19).

The existence of the limit (19) is a consequence of a generalized convergence in moments, namely the convergence in distribution of traffics, of $(A_{k},A^{\prime}_{k})$ stated in , see Lemma 3.2. However, under stronger assumptions, an independent proof of this convergence and an intrinsic characterization of $\rho_{z,z^{\prime}}^{u}$ are provided in Theorem 1.13 below.

Let us first mention that the map $\rho_{z}$ is the almost sure point-wise limit

This fact was known in the Lévy case and is proved in greater generality in Corollary 3.3. Let us first give a characterization of $\rho_{z}$.

The function $\Phi$ of (7) admits the decomposition

where $g(y)$ is a function such that for some constants $K,\gamma>-1,\kappa\geq 0$, we have

Both Lévy matrices and Wigner matrices with exploding moments satisfy Hypothesis 1.7. For Lévy matrices, the function $g$ is $g(y)=C_{\alpha}y^{\frac{\alpha}{2}-1}$, with $C_{\alpha}=-\sigma i^{\alpha/2}$, whereas for Wigner matrices with exploding moments, the function $g$ is

Above, the second point in Hypotheses 1.1 is not required anymore, as it served mainly to prove convergence, which is now insured by the uniqueness of limit points.

Note that the asymptotics of Wigner matrices with bounded moments is also described by (25). In this case $\Phi(\lambda)=-i\lambda$, so $g(y)=-i$ and $\rho_{z}(t)=-it\underset{N\rightarrow\infty}{\lim}\frac{1}{N}\operatorname{Tr}G(z)$, which leads, by formula (21), to the classical quadratic equation

Let us now give a fixed point characterization for the function $\rho_{z,z^{\prime}}?(t,t^{\prime})$ of (19).

The function $\Phi$ of (7) either has the form

or admits the decomposition, for $x,y$ non zero:

Unfortunately $\tau$ does not satisfy (29) as its density blows up at $v=v^{\prime}$: we shall treat both case separately.

The following lemma insures us that our two main examples satisfy Hypothesis 1.10.

Both Lévy matrices and Wigner matrices with exploding moments satisfy Hypothesis 1.10. For Wigner matrices with exploding moments, the measure $\tau$ is given by

Under Hypotheses 1.1, 1.7 and 1.10, the conclusions of Theorem 1.5 and Corollary 1.6 hold and the parameter $\rho_{z,z^{\prime}}$ of (18) is given by

where $\rho_{z,z^{\prime}}^{u,1}(t,s),\rho_{z,z^{\prime}}^{2}(t,s)$ are analytic functions on $\Lambda=\{z,z^{\prime}:t\Im z>0\,,s\Im z^{\prime}>0\}$ and uniformly continuous on compacts in the variables $(t,s)$ (and $\beta$- Hölder for $\beta>\alpha/2$ in the Lévy matrices case, see Lemma 5.1), given by (1.13) as far as $\rho_{z,z^{\prime}}^{u,2}(t,s)$ is concerned and unique solution, among such functions, of the following fixed point equation (1.13) as far as $\rho_{z,z^{\prime}}^{u,1}(t,s)$ is concerned:

with the notations $\operatorname{sgn}_{z}:=\operatorname{sign}(\Im z)$, $\operatorname{sgn}_{z^{\prime}}:=\operatorname{sign}(\Im z^{\prime})$ and the measures $\tau$, $\mu$ defined by Hypothesis 1.10 and Remark 1.11.

Let us conclude this introduction with three remarks.

Let $A=X/a_{N}$ be a Lévy matrix as defined at Definition 1.2 but with $\alpha=2$. Then using Example c) p. 44. of instead of the hypothesis made at Equation (7), one can prove that as $N\to\infty$, the spectral measure of $A$ converges almost surely to the semi-circle law with support $$(see (26)). This result somehow “fills the gap” between heavy-tailed matrices and finite second moment Wigner matrices. It allows for example to state that if$P(|X_{ij}|\geq u)\sim cu^{-2}$, with$c>0$, even though the entries of$X$do not have any second moment, we have that the empirical spectral law of$\frac{X}{\sqrt{cN\log(N)}}$converges almost surely to the semi-circle law with support$$.

Our results also have an application to standard Wigner matrices (i.e. symmetric random matrices of the form $A=X/\sqrt{N}$, with $X$ having centered i.i.d. sub-diagonal entries with variance one and not depending on $N$. In this case, the function $\Phi$ of (7) is linear, which implies that $L(z,z^{\prime})=L(z)L(z^{\prime})$ for all $z,z^{\prime}$, so that the covariance is null, (25) is the self-consistent equation satisfied by the Stieltjes transform of the semi-circle law, namely (26), and Corollary 1.6 only means that for functions $f\in\mathcal{A}$, we have, for the convergence in probability,

This result is new for Wigner matrices whose entries have a second but not a fourth moment, (36) brings new information. Indeed, for such matrices, which could be called “semi heavy-tailed random matrices”, the convergence to the semi circle law holds (see or the remark right above that one) but the largest eigenvalues do not tend to the upper-bound of the support of the semi-circle law, are asymptotically in the scale $N^{\frac{4-\alpha}{2\alpha}}$ (with $\alpha\in(2,4)$ as in Equation (8) when such an exponent exists) and distributed according to a Poisson process (see ), and it is not clear what the rate of convergence to the semi-circle law will be. Equation (36) shows that this rate is $\ll N^{-1/2}$.

but otherwise a non trivial recentering should occur. See the end of Section 2.

CLT for the moments of Wigner matrices with exploding moments

The goal of this section is to prove Theorem 1.4. In order to prove the CLT for the moments of the empirical eigenvalues distribution of $A$, we use a modification of the method of moments inspired by which consists of studying more general functionals of the entries of the matrix (the so-called injective moments) than only its moments. We describe this approach below.

Let $A$ be a Wigner matrix with exploding moments. Let $K\geq 1$ be an integer. The normalized trace of the $K$-th power of $A$ can be expanded in the following way.

where $\mathcal{P}(K)$ is the set of partitions of $\{1,\ldots,K\}$ and $S_{\pi}$ is the set of multi-indices $\mathbf{i}=(i_{1},\ldots,i_{K})$ in $\{1,\ldots,N\}^{K}$ such that $n\sim_{\pi}m\Leftrightarrow i_{n}=i_{m}$.

We interpret $\tau_{N}^{0}$ as a functional on graphs instead of partitions. Let $\pi$ be a partition of $\{1,\ldots,K\}$. Let $T^{\pi}=(V,E)$ be the undirected graph (with possibly multiple edges and loops) whose set of vertices $V$ is $\pi$ and with multi-set of edges $E$ given by: there is one edge between two blocks $V_{i}$ and $V_{j}$ of $\pi$ for each $n$ in $\{1,\ldots,K\}$ such that $n\in V_{i}$ and $n+1\in V_{j}$ (with notation modulo $K$). Then, one has

where $[N]=\{1,\ldots,N\}$ and for any edge $e=\{i,j\}$ we have denoted A\big{(}\phi(e)\big{)}=A\big{(}\phi(i),\phi(j)\big{)}. There is no ambiguity in the previous definition since the matrix $A$ is symmetric.

to a Gaussian process, it is sufficient to prove the convergence of

Before giving the proof of this fact, we recall a result from , namely the convergence of $\tau_{N}^{0}[T^{\pi}]$ for any partition $\pi$. These limits are involved in our computation of the covariance of the limiting process of \big{(}Z_{N}(T^{\pi})\big{)}_{\pi\in\cup_{K}\mathcal{P}(K)}, and this convergence will be useful in the proof of the CLT for Stieltjes transforms latter.

Let $A$ be a Wigner matrix with exploding moments with parameter $(C_{k})_{k\geq 1}$. For any partition $\pi$ in $\cup_{K}\mathcal{P}(K)$, with $\tau_{N}^{0}[T^{\pi}]$ defined in (37),

where a fat tree is a graph that becomes a tree when the multiplicity of the edges is forgotten, and for $T$ such a graph we have denoted $q_{k}$ the number of edges of $T$ with multiplicity $2k$.

Let $A$ be a Wigner matrix with exploding moments. Then, the process \big{(}Z_{N}(T^{\pi})\big{)}_{\pi\in\cup_{K}\mathcal{P}(K)} defined by (39) converges to a centered Gaussian process \big{(}z(T^{\pi})\big{)}_{\pi\in\cup_{K}\mathcal{P}(K)} whose covariance is given by: for any $T^{\pi_{1}},T^{\pi_{2}}$,

where $\tau^{0}[T]$ is given by Proposition 2.1 and $\mathcal{P}_{\sharp}(T^{\pi_{1}},T^{\pi_{2}})$ is the set of graphs obtained by considering disjoint copies of the graphs $T^{\pi_{1}}$ and $T^{\pi_{2}}$ and gluing them by requiring that they have at least one edge (and therefore two “adjacent” vertices) in common.

We show the convergence of joint moments of \big{(}Z_{N}(T^{\pi})\big{)}. Gaussian distribution being characterized by its moments, this will prove the theorem. Let $T_{1}=(V_{1},E_{1}),\ldots,T_{p}=(V_{n},E_{n})$ be finite undirected graphs, each of them being of the form $T^{\pi}$ for a partition $\pi$. We first write

$\mathcal{P}(V_{1},\ldots,V_{n})$ is the set of partitions of the disjoint union of $V_{1},\ldots,V_{n}$ whose blocks contain at most one element of each $V_{j}$,

$S_{\sigma}$ is the set of families of injective maps, $\phi_{j}:V_{j}\to[N],j=1,\ldots,n$, such that for any $v\in V_{j},v^{\prime}\in V_{j^{\prime}}$, one has $\phi_{j}(v)=\phi_{j^{\prime}}(v^{\prime})\Leftrightarrow v\sim_{\sigma}v^{\prime}$.

First, it should be noticed that by invariance in law of $A$ by conjugacy by permutation matrices, for any $\sigma$ in $\mathcal{P}(V_{1},\ldots,V_{n})$ and $(\phi_{1},\ldots,\phi_{n})$ in $S_{\sigma}$, the quantity $\omega_{N}(\phi_{1},\ldots,\phi_{n})$ depends only on $\sigma$. We then denote $\omega_{N}(\sigma)=\omega_{N}(\phi_{1},\ldots,\phi_{n})$. Moreover, choosing a partition in $\mathcal{P}(V_{1},\ldots,V_{n})$ is equivalent to merge certain vertices of different graphs among $T_{1},\ldots,T_{n}$. We equip $\mathcal{P}(V_{1},\ldots,V_{n})$ with the edges of $T_{1},\ldots,T_{n}$ and say that two vertices are adjacent if there is an edge between them. We denote by $\mathcal{P}_{\sharp}(V_{1},\ldots,V_{n})$ the subset of $\mathcal{P}(V_{1},\ldots,V_{n})$ such that any graph has two adjacent vertices that are merged to two adjacent vertices of an other graph. By the independence of the entries of $X$ and the centering of the components in $\omega_{N}$, for any $\sigma$ in $\mathcal{P}(V_{1},\ldots,V_{n})\setminus\mathcal{P}_{\sharp}(V_{1},\ldots,V_{n})$ one has $\omega_{N}(\sigma)=0$. Hence, since the cardinal of $S_{\sigma}$ is $\frac{N!}{(N-|\sigma|)!}$, we get

Let $\sigma$ in $\mathcal{P}_{\sharp}(V_{1},\ldots,V_{n})$. We now analyze the term $\omega_{N}(\sigma)$. We first expand its product.

where $|\bar{E}_{B}|$ is the number of edges of $T_{B}$ once the multiplicity and the orientation of edges are forgotten. Recall assumption (9): for any $k\geq 1$,

By the Cauchy-Schwarz inequality, for any $k\geq 1$,

Hence, since the measure $\mu_{N}$ is centered, the quantity $\delta_{N}(B)$ is bounded. Denote $T_{\sigma}$ the graph obtained by merging the vertices of $T_{1},\ldots,T_{n}$ that belong to a same block of $\sigma$, $|\bar{E}_{\sigma}|$ its number of edges when orientation and multiplicity is forgotten, and by $c_{\sigma}$ its number of components. We obtain from (42) and (43)

A partition $\sigma\in\mathcal{P}_{\sharp}(V_{1},\ldots,V_{n})$ induces a partition $\bar{\sigma}$ of $\{1,\ldots,n\}$: $i\sim_{\bar{\sigma}}j$ if and only if $T_{i}$ and $T_{j}$ belong to a same connected component of $T_{\sigma}$. Denote by $\mathcal{P}_{2}(n)$ the set of pair partitions of $\{1,\ldots,n\}$. One has

Secondly, one has $|\bar{E}_{\sigma}|-|\bar{E}_{B}|-\sum_{j\notin B}|\bar{E}_{\{j\}}|\leq 0$ with equality if and only if $B=\{1,\ldots,n\}$, so that

Moreover, by [28, Lemma 1.1] $|\sigma|-c_{\sigma}-|\bar{E}_{\sigma}|$ is the number of cycles of $\bar{T}_{\sigma}$, the graph obtained from $T_{\sigma}$ by forgetting the multiplicity and the orientation of its edges. Hence,

By (44), (45) and (46), if we denote by $\delta_{N}(\sigma)=\delta_{N}(\{1,\ldots,n\})$ we get

where we have used the independence of the entries of $A$ to split $\delta_{N}$. The case $n=2$ gives

The general case $n\geq 3$ in (48) gives the Wick formula

which characterizes the Gaussian distribution. ∎

and therefore we obtain the same CLT if we recenter with the limit or the expectation, as noticed in the introduction.

CLT for Stieltjes transform and the method of martingales

Hence one can suppose that in Hypothesis 1.1, $b$ is centered.

and their complex transposes converges in distribution to a complex Gaussian variable. Since $\overline{G(z)}=G(\bar{z})$, it is enough to fix a linear combination

Notice first that (50) implies (51). Let us now prove (52). The proof of (50) will then be the main difficulty of the proof of Theorem 1.5.

Hence by (107) of Lemma 7.5 in the appendix, there is $C$ such that for all $N$ and all $k$,

We now pass to the main part of the proof of Theorem 1.5, namely the proof of (50). It is divided into several steps.

We can get rid of the linear combination in (49) and assume $p=1$. As $\overline{\operatorname{Tr}G(z_{i})}=\operatorname{Tr}G(\overline{z_{i}})$, both $X_{k}^{2}$ and $|X_{k}|^{2}$ are linear combinations of terms of the form

2. Removing the off-diagonal terms

In this section, we prove that we can replace ${Y}_{k}$ in (53) by

Then for $C_{N}$ defined as in (53), as $N\to\infty$, we have the convergence

the $a_{ij}^{\prime}$’s such that $i>k$ or $j>k$ are i.i.d. copies of $a_{11}$ (modulo the fact that $A^{\prime}$ is symmetric), independent of $A$,

for all other pairs $(i,j)$, $a_{ij}^{\prime}=a_{ij}$,

where $f_{k}^{\prime\prime}$ is defined out of $A^{\prime}$ in the same way as $f_{k}^{\prime}$ is defined out of $A$ in (60) (note that the $k$-th column is the same in $A$ and in $A^{\prime}$). It follows that

We shall in the sequel prove that as $N$ tends to infinity, regardless to the value of $k$, we have the almost sure convergences

and for any $u\in(0,1)$, as $N,k\longrightarrow\infty$ so that $k/N\longrightarrow u$, we have

where $L(z,z^{\prime})=\int_{0}^{1}\Psi^{u}(z,z^{\prime})du$. The convergences of (64) and (65) are based on an abstract convergence result stated in next section, where we use the convergence of generalized moments of Proposition 2.1. They are stated in Lemmas 3.4 and 3.5 respectively.

so once (64) and (65) proved, we will have the convergence in $L^{2}$

Thus by (63), we will have proved the convergence of (61), hence completing the proof of the theorem.

3. An abstract convergence result

Remember that $A_{k}=(a_{ij})$ is the square matrix of size $N-1$ obtained by removing the $k$-th row and the $k$-th column of $A$, that $G_{k}(z)=(z-A_{k})^{-1}$ and that $A^{\prime}_{k}$ is a copy of $A_{k}$ where the entries $(i,j)$ for $i$ or $j\geq k$ are independent of $A_{k}$ and the other are those of $A_{k}$. We denote $G_{k}^{\prime}(z^{\prime})=(z^{\prime}-A^{\prime}_{k})^{-1}$.

By e.g. Theorem C.8 of , it is enough to prove that for any bounded and Lipschitz function $f$, $\nu^{k,N}_{z,z^{\prime}}(f)$ converges almost surely to $\nu_{z,z^{\prime}}^{u}(f)$. Moreover, adapting the proof of Lemma 7.4, one can easily see that for any such $f$,

converges almost surely to zero. The only modification of the proof is to complete the resolvent identity by noticing that

As $pN$ is small, $(1-p)^{N}=1-Np+O(Np^{2})$,

and the Claim is proved, by Hypothesis (5).

We consider a polynomial $P=x_{1}^{n_{1}}\overline{x_{1}}^{m_{1}}x_{2}^{n_{2}}\overline{x_{2}}^{m_{2}}$ and remark that

where $\circ$ denotes the Hadamard (entry-wise) product of matrices and

Such a family exists by Proposition 7.10. By [38, Proposition 3.10], the couple of random matrices $(A,A^{\prime})$ satisfies the so-called convergence in distribution of traffics, so that the RHS converges. For the reader’s convenience, we give the limiting value of (68), even if we do not use it later. It is obtained by applying the rule of the so-called traffic freeness in .

we have considered $T$ the graph whose edges are labelled by indeterminates $a$ and $a^{\prime}$, obtained by

identifying the vertex $1$ of each of these graphs (we get a connected graph, bouquet of cycles),

we have denoted by $V_{T}$ the set of vertices of $T$, $\mathcal{P}(V_{T})$ is the set of partitions of $V_{T}$ and $T^{\pi}$ denotes the graph obtained by identifying the vertices of $T$ that belong to a same block of $\pi$,

the quantity $\tau^{0}[T^{\pi}]$ is as in Proposition 2.1,

where the sum is over all partitions of the set $V_{\pi}$ of vertices of $T^{\pi}$ and $|E_{i}|$ is the number of edges between adjacent vertices of $V_{i}$, $i=1,2$.

Formula (69) could also be derived by the same techniques than those developed in Section 2. The random variables $Y_{i}$ and $Y^{\prime}_{j}$ are distributed according to the limiting eigenvalues distribution of $A$. Recall that we assume that the sequence $(C_{k})_{k\geq 2}$ defined in (9) satisfies $C_{k}\leq C^{k}$ for a constant $C>0$. Then, following the proof of [49, Proposition 10], the exponential power series of the limiting eigenvalues distribution of $A$ has a positive radius of convergence. So, by a generalization of [12, Theorem 30.1] to the multi-dimensional case, we get that the distribution of $(Y_{i},Y^{\prime}_{j})$ is characterized by its moments. Then, we get that $(Y_{i},Y^{\prime}_{j})$ converges weakly to a family of random variables $(y_{i},y_{j}^{\prime})$. We set $f_{z}(y)=(z-y)^{-1}$. We then obtain the convergence

This convergence could also be proven without Proposition 7.10 but with appropriate bounds on the growth of moments. We have the following Corollary.

We used the exponential decay to switch the integral and the expectation. This also allows to truncate the integral: for any $M\geq 0$,

where $\epsilon_{M,z,N}$ goes to zero as $M$ tends to infinity, uniformly on $N$ and on the randomness. Remember that by assumption (7), we have

where $\varepsilon^{(1)}_{t,z,N}$ converges almost surely to zero as $N$ goes to infinity. By Corollary 3.3,

where $\varepsilon^{(2)}_{t,z,N}$ converges to zero almost surely. Hence, we deduce the almost sure convergence

As $\rho_{z}^{N}$ has non positive real part, we conclude by dominated convergence theorem and by getting rid of the truncation of the integral. ∎

where $\operatorname{sgn}_{z}:=\operatorname{sgn}(\Im z)$ and $\rho_{z}(t)$ is defined in Corollary 3.3.

Hence, by (72), since by (108) the sign of the imaginary part of $\lambda_{N}(z)$ is $\operatorname{sgn}_{z}$, the random variable $f_{k}$ can be written

where $m>0$ and \eta_{m}(z)=\frac{\partial_{z}\lambda_{N}(z)}{\lambda_{N}(z)}\big{(}1-e^{i\operatorname{sgn}_{z}m\lambda_{N}(z)}\big{)}.

We next show that for all $\varepsilon>0$ there exists $m_{0}$ so that for $m<m_{0}$, $N$ large enough

By (110) and since the sign of the imaginary part of $\lambda_{N}(z)$ is $\operatorname{sgn}_{z}$, one has $|\eta_{m}(z)|\leq 4|\Im z|^{-1}$. More precisely, for any $K>0$, we find

By (4), we deduce that for any $\varepsilon>0$ we can choose $m$ small enough so that for $N$ large enough

so that for $M$ large enough and $|\Im z|\geq\delta>0$,

where $\varepsilon_{m,M,z,N}$ is arbitrary small as $M$ is large, uniformly on $N$ and on the randomness. Moreover, by (79) one has

Remind that in (74) we have shown in the proof of the last item of Corollary 3.3

As in the proof of Corollary 3.3, since the left hand side is analytic and uniformly bounded, we deduce by Montel’s theorem that its convergence entails the convergence of its derivatives. We then get by (81), for all $t,z$ so that $t/\Im z>0$, the almost sure convergence

We then obtain by dominated convergence (remember the integrant is uniformly bounded by (80) and (79)),

so we can let $M$ going to infinity to obtain the almost sure convergence

In the Lévy case, one has $\rho_{z}(t)=t^{\frac{\alpha}{2}}\rho_{z}(1)$, and in the exploding moments case, \big{|}\frac{1}{t}\partial_{z}e^{itz+\rho_{z}(t)}\big{|}\leq 1+\frac{C_{2}}{\Im z^{2}}, so that the integral converges at zero and we obtain

where $\rho^{u}_{z,z^{\prime}}$ is defined in Corollary 3.3.

We shall start again by using Formula (3.4) for $f_{k}$, and its analogue for

where $G_{k}^{\prime}(z^{\prime})$ is defined as $G_{k}(z)$, replacing $z$ by $z^{\prime}$ and the matrix $A$ by the matrix $A^{\prime}$ defined by (62), which gives

The upper bound (79) allows us to bound the first term uniformly by $\log m^{-1}$ and to truncate the integrals for $\operatorname{sgn}_{z}t,\operatorname{sgn}_{z^{\prime}}t^{\prime}\leq M$. Therefore, up to a small error $\varepsilon$ uniform for $M$ large, $m$ small and provided $|\Im z|,|\Im z^{\prime}|\geq\delta>0$, we have

As in the previous case, the upper bound (79) allows us to write

By assumption (7) and Corollary 3.3, we have the following almost sure convergence as $N$ goes to infinity and $\frac{k}{N}$ goes to $u$ in $(0,1)$

Hence we have proved the convergences (64) and (65). This completes the proof of Theorem 1.5.

Proof of Corollary 1.6

Applying this to the empirical measure of eigenvalues and its expectation, we deduce that

Note here that $\bar{\partial}\Psi_{f}(x,y)$ is supported in a compact set $[-c_{0},c_{0}]\times[0,c_{0}^{\prime}]$ and is bounded by $c|y|$ for a finite constant $c$. Hence, the integral is well converging. We next show that we can approximate it by

where $\Delta(x)=\lceil 2^{k}x\rceil 2^{-k}$. The random variable $Z_{N}^{\Delta}(f)$ is only a finite sum of $Z_{N}(x+iy)$ and therefore it converges in law by Theorem 1.5, towards

We finally show the convergence in probability of $Z_{N}^{\Delta}(f)-Z_{N}(f)$ and $Z^{\Delta}(f)-Z(f)$ to zero by bounding their $L^{1}$ norms

But, by Lemma 7.3, there exists a finite constant $C$ such that for any $\varepsilon>0$

Taking $\varepsilon\in(0,1)$ we deduce that there exists a finite constant $C_{\varepsilon}$ such that

Fixed point characterizations

In this section, we provide characterizations of the functions $\rho_{z}$ and $\rho_{z,z^{\prime}}^{u}$ involved in the covariance of the limiting process of Theorem 1.5 as fixed points of certain functions. In fact, we also give an independent proof of the existence of such limits, and of Corollary 3.3.

where we have proved that this convergence holds almost surely in Corollary 3.3. Note however that under the assumptions of Theorem 1.9, the arguments below provide another proof of this convergence where we do not have to assume (14).

Then we use Hypothesis made at (22) to get for $\Im z>0$

Using the definition of $\Phi$ and the fact that we assumed that it is bounded on every compact subset (since $\rho^{N}$ has non positive real part we can cut the integral to keep $y$ bounded up to a small error, as in the previous sections), we have

We now find a fixed point system of equations for the non random function of Corollary 3.3. For $\lambda/\Im z+\lambda^{\prime}/\Im z^{\prime}\geq 0$, we set

where we recall that $G_{k}$ and $G^{\prime}_{k}$ are as in (19). To simplify the notations below, as in the previous section, we denote $(G,G^{\prime})$ instead of $(G_{k},G^{\prime}_{k})$, even though their distribution depends on $k$.

In the sequel we fix, as in Section 3.3, a number $u\in(0,1)$ and will give limits in the regime where $N\to\infty$, $k\to\infty$ and $k/N\longrightarrow u$. We shall then prove that, under the hypotheses of Theorem 1.13 that we assume throughout this section, $(\rho^{N,k,1}_{z,z^{\prime}},\rho^{N,k,2}_{z,z^{\prime}})$ converges almost surely and that its limit satisfies a fixed point system of equations which has a unique analytic solution with non positive real part. The convergence could be shown with minor modifications of Lemma 3.2 under assumption (14), but we do not need this since we work with stronger assumptions. Using the concentration lemma 7.4 (note that $\Phi$ is not Lipschitz but can be approximated by Lipschitz functions uniformly on compacts), it is sufficient to prove the fixed point equation for the expectation of these parameters. Moreover, by exchangeability of the $k$ first entries and $N-k$ last entries

by Lemma 7.2 (by continuity of $\Phi$, it can be approximated by Lipschitz functions). Hence we find that the limit points $\rho_{z,z^{\prime}}^{u,1},\rho_{z,z^{\prime}}^{2}$ of $\rho_{z,z^{\prime}}^{N,k,1},\rho_{z,z^{\prime}}^{N,k,2}$ satisfy (1.13) and (1.13). Moreover,

where for $(\alpha,\alpha^{\prime})\in\{\gamma,\kappa\}^{2}$, we have set

We put $I(\alpha,\alpha^{\prime})=I_{1}(\alpha,\alpha^{\prime})$ where $1$ denote the constant function equal to one. We get after integrating both sides

with $S_{+}^{1}=\{s,t\geq 0,s^{2}+t^{2}=1\}$.

Proving that the limit points $(\rho_{z,z^{\prime}}^{u,1},\rho_{z,z^{\prime}}^{u,2})$ of $(\rho^{N,k,1}_{z,z^{\prime}},\rho^{N,k,2}_{z,z^{\prime}})$ are $\beta$-Hölder maps for a $\beta>\frac{\alpha}{2}$ allows to conclude the proof of Theorem 1.13. This is the content of the following lemma.

We next show that for any matrix model so that $\Phi(x)=-\sigma(ix)^{\frac{\alpha}{2}}$, for any $2\kappa\in(0,\alpha/2)$

where we have used Fubini for non negative functions. But the above integral is well converging at infinity and we know that $\phi_{N}$ converges uniformly on $[0,M]$ for all $M$ finite; hence there exists a finite constant $C$ so that for $N$ large enough

Applying this with $x=tG(z)_{jj}+sG^{\prime}(z^{\prime})_{jj}$ and $y=t^{\prime}G(z)_{jj}+s^{\prime}G^{\prime}(z)_{jj}$ gives

so that we deduce that for all $\kappa>0$

Equation (93) completes the proof by taking $2\kappa=2(\beta-\frac{\alpha}{2})<\frac{\alpha}{2}$. ∎

We now prove the uniqueness of the solution of (1.13) on the set of functions described above. After some change of variables, the equation is equivalent to the following:

where we have denoted in short $W=(w,w^{\prime})$, $Z=(z,z^{\prime})$, $T=(s,t)$, $\rho_{z,z^{\prime}}(W)=\rho_{z}(w)+\rho_{z^{\prime}}(w^{\prime})$ and $W.Z,T.Z$ stand for the scalar products.

After the change of variables $w=r\cos(\theta),w^{\prime}=r\sin(\theta)$, $v=rv^{\prime}$, we can rewrite this system of equations as

with, if $T^{\dagger}=(t,s)$ when $T=(s,t)$,

where we have denoted $e_{\theta}=(\cos(\theta),\sin(\theta))$ and

for a constant $C_{\alpha}$. The desired uniqueness follows from the following lemma.

To study the Lipschitz property of $F^{u}_{z,z^{\prime}}$ as a function of $g$ in $\mathcal{C}_{M,\beta}$ for the norm $\beta$, we first set

We next bound, for given $g_{1},g_{2}$ in $\mathcal{C}_{M,\beta}$, $T_{1}$ in $S^{1}_{+}$ and with $T_{2}$ either in $S^{1}_{+}$ or $T_{2}=0$ (which allows to treat in one time two parts of $\|\cdot\|_{\beta}$)

Moreover, notice that for $(s,t)$ in $S^{1}_{+}$, one has $\max(s,t)\geq 1/\sqrt{2}$ and $\max(\cos\theta,\sin\theta)\geq 1/\sqrt{2}$ and thus

so that $|ie_{\theta}.Z+ivT.Z|\;\geq\;\Delta_{z,z^{\prime}}(1+1_{T\in S_{+}^{1}}v)$. At last, with $a_{i}=e_{\theta}+vT_{i}$ for $i=1,2$, straightforward uses of the $\beta$-norm and the inequalities $|a_{i}|\leq(1+v)$, \big{|}|a_{1}|-|a_{2}|\big{|}\leq v|T_{1}-T_{2}|, $|a_{i}|\geq\frac{1}{\sqrt{2}}(v\vee 1)$ if ${T_{i}\in S_{1}^{+}}$, \big{|}\frac{a_{1}}{|a_{1}|}-\frac{a_{2}}{|a_{2}|}\big{|}\leq(v\vee 1)^{-2}|T_{1}-T_{2}|v(1+v), and (94) gives the estimates

where f_{\beta}(v)=v^{\beta}\Big{(}(1+v)^{\frac{\alpha}{2}+\beta}(v\vee 1)^{-2\beta}+1\Big{)}. Using this series of estimates, we find that,

with $C(z,z^{\prime},M)<1$ if $\Im z\wedge\Im z^{\prime}$ is large enough. ∎

Taking two solutions of (1.13) and (1.13) in ${\mathcal{C}}_{M,\beta}$, we deduce that they are equal when $\Im z\wedge\Im z^{\prime}$ is large enough, and thus everywhere by analyticity.

Proofs of Lemmas 1.3, 1.8 and 1.12

Let us first treat the case of Wigner matrices with exploding moments. First and second parts of Hypothesis 1.1, as well as (4), are satisfied for $c=c_{\varepsilon}=0$. Let us then define $\nu_{N}$ to be the law of $a^{2}=a_{11}^{2}$ and $m_{N}$ to be the measure with density $Nx$ with respect to $\nu_{N}$, so that for any test function $f$, we have

Hence (5) is satisfied if $B$ is chosen large enough. For the convergence of the truncated even moments, see [38, Sect. 1.2.1]. Moreover, (7) follows from the results of e.g. Section 8.1.3 of . At last, (4) is satisfied for Lévy matrices by e.g. Section 10 of .

2. Proof of Lemma 1.8

In the case of Lévy matrices, the expression

which is proved in the following way (using (97)):

It follows that for $m_{N}$ the measure introduced in the proof of Lemma 1.3 above, we have

for $f_{y}(x):=-\frac{J_{1}(2\sqrt{xy})}{\sqrt{xy}}$. As $m_{N}$ converges weakly to $m$ and $f_{y}$ is continuous and bounded, we have

3. Proof of Lemma 1.12

The case of Lévy matrices is obvious. To treat the case of Wigner matrices with exploding moment, first note that by (98), writing

Then one concludes as for the proof of Lemma 1.8.

Appendix

This section is mostly a reminder of results from and .

The next lemma is an easy consequence of Cauchy-Weyl interlacing Theorem. It is an ingredient of the proof of Lemma 7.3.

where $\|f\|_{\operatorname{Lip}}:=\sup_{x\neq y}\frac{|f(y)-f(x)|}{|y-x|}$ and $\|f\|_{\infty}:=\sup_{x}|f(x)|$, both supremums running over the elements of $B_{z,z^{\prime},t,t^{\prime}}$.

Moreover, let $M=tG(z)+t^{\prime}G^{\prime}(z^{\prime})$ and note that$A_{1}-\bar{A}_{1}$ and $A_{2}-\bar{A}_{2}$ have rank one so that $M-\bar{M}$ has rank one. On the other hand it is bounded uniformly by $C=C(z,z^{\prime},t,t^{\prime})$. Hence, we can write $M-\bar{M}=cuu^{*}$ with a unit vector $u$ and $c$ bounded by $C$. Therefore, since $f$ is Lipschitz,

Then, for $\lambda/\Im z\geq 0$, $\lambda^{\prime}/\Im z^{\prime}\geq 0$, we have for all $\delta\geq 0$, $s\in\{0,1\}$,

The first point is proved as in [14, Lemma C.3]. We outline the proof of the second point which is very similar to [14, Lemma C.3]. We concentrate on $\rho^{N,k,1}_{\lambda,\lambda^{\prime}}$, the other case being similar. By Azuma-Hoefding inequality, it is sufficient to show that

Let $H_{k}$ be the submatrix of $H$ obtained by removing its $k$-th row and its $k$-th column and set $G_{k}:=(z-H_{k})^{-1}$. Let also $\mathbf{a}_{k}$ be the $k$-th column of $H$ where the $k$-th entry has been removed. Then

For (106), see [4, Th. A.5]. For (107), see Lemma 7.1. ∎

With the notation introduced above the previous lemma, for each $1\leq j\leq N$,

Let us now prove (110). By (108), we know that $\Im z$ and $-\Im G_{jj}$ have the same sign, so

Hence it remains only to prove (110). This is a direct consequence of (108) and (109) which imply the second and last inequality

2. Vanishing of non diagonal terms in certain quadratic sums of random vectors

Let $\|M\|_{\operatorname{op}}$ denote the operator norm of a complex matrix with respect to the canonical Hermitian norms.

For each $N\geq 1$, let $(a_{1},\ldots,a_{N})$ be a family of i.i.d. copies of an random variable $a$ such that $a$ can be decomposed into $a=b+c$ with $b,c$ such that $b$ is centered and (2), (3) of Hypothesis 1.1 are satisfied. Let also $B_{N}$ be a non random $N\times N$ matrix such that $N^{-1}\operatorname{Tr}(B_{N}B_{N}^{*})$ is bounded. Then we have the convergence in probability

3. CLT for martingales

Let $(\mathcal{F}_{k})_{k\geq 0}$ be a filtration such that $\mathcal{F}_{0}=\{\emptyset,\Omega\}$ and let $(M_{k})_{k\geq 0}$ be a square-integrable complex-valued martingale starting at zero with respect to this filtration. For $k\geq 1$, we define the random variables

Let now everything depend on a parameter $N$, so that $\mathcal{F}_{k}=\mathcal{F}_{k}(N),Y_{k}=Y_{k}(N),v=v(N),\tau=\tau(N),L(\varepsilon)=L(\varepsilon,N),\ldots$

Then we have the convergence in distribution

4. Extension of CLTs for random matrices

The following lemma is borrowed from the paper of Shcherbina and Tirozzi , except that we do not require, in the hypotheses here, $V$ to be continuous, which is very useful in our case.

where $u_{N}$ is a sequence of real numbers. We make the following hypotheses :

Then $V$ is continuous on $\mathcal{L}_{1}$, can be (uniquely) continuously extended to $\mathcal{L}$ and (114) is true for any $\varphi\in\mathcal{L}$.

This is exactly Proposition 4 of , except that in , the hypotheses include the continuity of $V$. Let us prove that the hypotheses made here imply that $V$ is continuous on $\mathcal{L}_{1}$. For any $\varphi\in\mathcal{L}_{1}$, $V(\varphi)$ is the second moment of the limit law of $Z_{N}(\varphi)$. Hence

This proves that the quadratic form $V$ is continuous.∎

5. On the Hadamard product of Hermitian matrices

Let $A_{1},\ldots,A_{p}$ be $N$ by $N$ Hermitian random matrices whose entries have all their moments. Then, there exists a family of random variables $(a_{1},\ldots,a_{p})$ whose joint distribution is given by:

where $\circ$ denotes the Hadamard (entry-wise) product.

Step 1. We first assume that the matrices are deterministic and have distinct eigenvalues. By the spectral decomposition, for $j=1,\ldots,p$, we have $A_{j}=\sum_{i=1}^{N}\lambda_{j,i}u_{j,i}u_{j,i}^{*}$ where $\Lambda_{j}=(\lambda_{j,i})_{i=1,\ldots,N}$ is the family of eigenvalues of $A_{j}$ arranged in increasing order, and $U_{j}=(u_{j,i})_{i=1,\ldots,N}$ is the family of associated eigenvectors. For any $n_{1},\ldots,n_{p}\geq 0$, one has

For $j=1,\ldots,p$, we set $\textrm{d}\mu_{\Lambda_{j}}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{j,i}}$ the empirical eigenvalues distributions of $A_{j}$. We denote F_{\Lambda_{j}}(t)=\mu_{\Lambda_{j}}\big{(}(-\infty,t]\big{)} the cumulative function of $\textrm{d}\mu_{\Lambda_{j}}$. Since the eigenvalues of the matrices are distinct, one has $F_{\Lambda_{j}}(\lambda_{j,i})=\frac{i}{N}$ for any $i=1,\ldots,N$, $j=1,\ldots,p$. Hence, we have

where f_{N}\big{(}(\lambda_{j},\Lambda_{j},U_{j})_{j=1,\ldots,p}\big{)}=\bigg{(}N^{p-1}\sum_{k=1}^{N}\prod_{j=1}^{p}\big{|}u_{j,(NF_{\Lambda_{j}}(\lambda_{j}))}(k)\big{|}^{2}\bigg{)}. Hence, a family of random variables $(a_{1},\ldots,a_{p})$ as in the proposition exists and its joint distribution has density f_{N}\big{(}(\cdot,\Lambda_{j},U_{j})_{j=1,\ldots,p}\big{)} with respect to $\mu_{\Lambda_{1}}\otimes\dots\otimes\mu_{\Lambda_{p}}$. Step 2. We now assume that $(A_{1},\ldots,A_{p})$ are random and that their joint distributions have a density with respect to the Lesbegue measure on $\mathcal{H}_{N}^{p}$, where $\mathcal{H}_{N}$ is the space of Hermitian matrices of size $N$. In particular, the matrices have almost surely $N$ distinct eigenvalues, see . The spectral decompositions of the previous step are measurable (see [21, Section 5.3]) and, with the above notations $(\Lambda_{j},U_{j})$ for eigenvalues and eigenvectors of $A_{j}$, we can write the joint distribution of $(A_{1},\ldots,A_{p})$ in the form g_{N}\big{(}(\Lambda_{j},U_{j})_{j=1,\ldots,p}\big{)}\textrm{d}\mu_{\Delta_{N}}(\Lambda_{1})\dots\textrm{d}\mu_{\Delta_{N}}(\Lambda_{p})\textrm{d}\mu_{\mathcal{U}_{N}}(U_{1})\dots\textrm{d}\mu_{\mathcal{U}_{N}}(U_{p}). The symbol $\mu_{\Delta_{N}}$ denotes the Lebesgue measure on $\Delta_{N}=\{(x_{1},\ldots,x_{N})|x_{1}<\dots<x_{N}\}$ and $\mu_{\mathcal{U}_{N}}$ is the Haar measure on the set $\mathcal{U}_{N}$ of unitary matrices of size $N$. For any $n_{1},\ldots,n_{p}\geq 0$, one has

where $f_{N}$ is as in the previous step and

For any $i_{1},\ldots,i_{p}=1,\ldots,N$, we have

where $h_{N}^{(i_{1},\ldots,i_{p})}(\lambda_{1},\ldots,\lambda_{p})$ is obtained by integrating with respect to the variables $\lambda_{k_{1}},\ldots,\lambda_{k_{p}}$ for $k_{1}\neq i_{1},\ldots,k_{p}\neq i_{p}$. We finally obtain