The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations

Introduction

This paper lies in the lineage of recent works studying the influence of some perturbations on the asymptotic spectrum of classical random matrix models. Such questions come from statistics (cf. John) and appeared in the framework of empirical covariance matrices, also called nonwhite Wishart matrices or spiked population models, considered by Baik, Ben Arous and Péché BBP and by Baik and Silverstein BS3. The work BBP deals with random sample covariance matrices $(S_{N})_{N}$ defined by

where $Y_{N}$ is a $p\times N$ complex matrix whose sample column vectors are i.i.d., centered, Gaussian and of covariance matrix a deterministic Hermitian matrix ${\Sigma}_{p}$ having all but finitely many eigenvalues equal to 1. Besides, the size of the samples $N$ and the size of the population $p=p_{N}$ are assumed of the same order (as $N\to\infty$). The authors of BBP first noticed that, as in the classical case (known as the Wishart model) where $\Sigma_{p}=I_{p}$ is the identity matrix, the global limiting behavior of the spectrum of $S_{N}$ is not affected by the matrix $\Sigma_{p}$. Thus, the limiting spectral measure is the well-known Marchenko–Pastur law. On the other hand, they pointed out a phase transition phenomenon for the fluctuations of the largest eigenvalue according to the value of the largest eigenvalue(s) of $\Sigma_{p}$. The approach of BBP does not extend to the real Gaussian setting and the whole analogue of their result is still an open question. Nevertheless, Paul was able to establish in Pa the Gaussian fluctuations of the largest eigenvalue of the real Gaussian matrix $S_{N}$ when the largest eigenvalue of $\Sigma_{p}$ is simple and sufficiently larger than 1. More recently, Baik and Silverstein investigated in BS3 the almost sure limiting behavior of the extremal eigenvalues of complex or real nonnecessarily Gaussian matrices. Under assumptions on the first four moments of the entries of $Y_{N}$, they showed in particular that when exactly $k$ eigenvalues of ${\Sigma}_{p}$ are far from 1, the $k$ first eigenvalues of $S_{N}$ are almost surely outside the limiting Marchenko–Pastur support. Fluctuations of the eigenvalues that jump are universal and have been recently found by Bai and Yao in BY2 (we refer the reader to BY2 for the precise restrictions made on the definition of the covariance matrix $\Sigma_{p}$). Note that the problem of the fluctuations in the very general setting of BS3 is still open.

Our purpose here is to investigate the asymptotic behavior of the first extremal eigenvalues of some complex or real Deformed Wigner matrices. These models can be seen as the additive analogue of the spiked population models and are defined by a sequence $(M_{N})_{N}$ given by

where $W_{N}$ is a Wigner matrix such that the common distribution of its entries satisfies some technical conditions [given in (i) below] and $A_{N}$ is a deterministic matrix of finite rank. We establish the analogue of the main result of BS3, namely that, once $A_{N}$ has exactly $k$ (fixed) eigenvalues far enough from zero, the $k$ first eigenvalues of $M_{N}$ jump almost surely outside the limiting semicircle support. This result is universal (as the one of BS3) since the corresponding limits only involve the variance of the entries of $W_{N}$. On the other hand, at the level of the fluctuations, we exhibit a striking phenomenon in the particular case where $A_{N}$ is diagonal with a sole simple nonnull eigenvalue large enough. Indeed, we find that in this case, the fluctuations of the largest eigenvalue of $M_{N}$ are not universal and strongly depend on the particular law of the entries of $W_{N}$. More precisely, we prove that the limiting distribution of the (properly rescaled) largest eigenvalue of $M_{N}$ is the convolution of the distribution of the entries of $W_{N}$ with a Gaussian law. In particular, if the entries of $W_{N}$ are not Gaussian, the fluctuations of the largest eigenvalue of $M_{N}$ are not Gaussian.

In the following section, we first give the precise definition of the Deformed Wigner matrices (2) considered in this paper and we recall the known results on their asymptotic spectrum. Then, we present our results and sketch the proofs. We also outline the organization of the paper.

Model and results

Throughout this paper, we consider complex or real Deformed Wigner matrices $(M_{N})_{N}$ of the form (2) where the matrices $W_{N}$ and $A_{N}$ are defined as follows:

$W_{N}$ is an $N\times N$ Wigner Hermitian (resp., symmetric) matrix such that the $N^{2}$ random variables $(W_{N})_{ii}$, $\sqrt{2}\Re e((W_{N})_{ij})_{i<j}$, $\sqrt{2}\Im m((W_{N})_{ij})_{i<j}$ [resp., the $N(N+1)/2$ random variables $\frac{1}{\sqrt{2}}(W_{N})_{ii}$, $(W_{N})_{ij}$, $i<j$] are independent identically distributed with a symmetric distribution $\mu$ of variance $\sigma^{2}$ and satisfying a Poincaré inequality (see Section 3).

$A_{N}$ is a deterministic Hermitian (resp., symmetric) matrix of fixed finite rank $r$ and built from a family of $J$ fixed real numbers $\theta_{1}>\cdots>\theta_{J}$ independent of $N$ with some $j_{0}$ such that $\theta_{j_{0}}=0$. We assume that the nonnull eigenvalues $\theta_{j}$ of $A_{N}$ are of fixed multiplicity $k_{j}$ (with $\sum_{j\not=j_{0}}k_{j}=r$), that is, $A_{N}$ is similar to the diagonal matrix

Furthermore, note that this condition implies that $\mu$ has moments of any order (cf. Corollary 3.2 and Proposition 1.10 in L).

Let us now introduce some notations. When the entries of $W_{N}$ are further assumed to be Gaussian, that is, in the complex (resp., real) setting when $W_{N}$ is of the so-called GUE (resp., GOE), we will write $W_{N}^{G}$ instead of $W_{N}$. Then $X_{N}^{G}:=W_{N}^{G}/\sqrt{N}$ will be said to be of the GU(O)E($N,\frac{\sigma^{2}}{N}$) and we will let $M_{N}^{G}=X_{N}^{G}+A_{N}$ be the corresponding Deformed GU(O)E model.

In the following, given an arbitrary Hermitian matrix $B$ of order $N$, we will denote by $\lambda_{1}(B)\geq\cdots\geq\lambda_{N}(B)$ its $N$ ordered eigenvalues and by $\mu_{B}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}(B)}$ its empirical measure. $\operatorname{Spect}(B)$ will denote the spectrum of $B$. For notational convenience, we will also set $\lambda_{0}(B)=+\infty$ and $\lambda_{N+1}(B)=-\infty$.

The Deformed Wigner model is built in such a way that the Wigner theorem is still satisfied. Thus, as in the classical Wigner model ($A_{N}\equiv 0$), the spectral measure $(\mu_{M_{N}})$ converges a.s. toward the semicircle law $\mu_{sc}$ whose density is given by

This result follows from Lemma 2.2 of Bai. Note that it only relies on the two first moment assumptions on the entries of $W_{N}$ and the fact that the $A_{N}$’s are of finite rank.

On the other hand, the asymptotic behavior of the extremal eigenvalues may be affected by the perturbation $A_{N}$. Recently, Péché studied in Pe the Deformed GUE under a finite rank perturbation $A_{N}$ defined by (ii). Following the method of BBP, she highlighted the effects of the nonnull eigenvalues of $A_{N}$ at the level of the fluctuations of the largest eigenvalue of $M_{N}^{G}$. To explain this in more detail, let us recall that when $A_{N}\equiv 0$, it was established in TW that as $N\rightarrow\infty$,

where $F_{2}$ is the well-known GUE Tracy–Widom distribution (see TW for the precise definition). Dealing with the Deformed GUE $M_{N}^{G}$, it appears that this result is modified as soon as the first largest eigenvalue(s) of $A_{N}$ is (are) quite far from zero. In the particular case of a rank-1 perturbation $A_{N}$ having a fixed nonnull eigenvalue $\theta>0$, Pe proved that the fluctuations of the largest eigenvalue of $M_{N}^{G}$ are still given by (5) when $\theta$ is small enough and precisely when $\theta<\sigma$. The limiting law is changed when $\theta=\sigma$. As soon as $\theta>\sigma$, Pe established that the largest eigenvalue $\lambda_{1}(M_{N}^{G})$ fluctuates around

(which is $>2\sigma$ since $\theta>\sigma$) as

Similar results are conjectured for the Deformed GOE but Péché emphasized that her approach fails in the real framework. Indeed, it is based on the explicit Fredholm determinantal representation for the distribution of the largest eigenvalue(s) that is specific to the complex setting. Nevertheless, Maïda Ma obtained a large deviation principle for the largest eigenvalue of the Deformed GOE $M_{N}^{G}$ under a rank-1 deformation $A_{N}$; from this result she could deduce the almost sure limit with respect to the nonnull eigenvalue of $A_{N}$. Thus, under a rank-1 perturbation $A_{N}$ such that $D_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ where $\theta>0$, Ma showed that

Note that the approach of Ma extends with minor modifications to the Deformed GUE. Following the investigations of BS3 in the context of general spiked population models, one can conjecture that such a phenomenon holds in a more general and nonnecessarily Gaussian setting. The first result of our paper, namely the following Theorem 2.1, is related to this question. Before being more explicit, let us recall that when $A_{N}\equiv 0$, the whole spectrum of the rescaled complex or real Wigner matrix $X_{N}=W_{N}/{\sqrt{N}}$ belongs almost surely to the semicircle support $[-2\sigma,2\sigma]$ as $N$ goes to infinity and that (cf. BYi or Theorem 2.12 in Bai)

Note that this last result holds true in a more general setting than the one considered here (see BYi for details) and in particular only requires the finiteness of the fourth moment of the law $\mu$. Moreover, one can readily extend the previous limits to the first extremal eigenvalues of $X_{N}$, that is,

Here, we prove that, under the assumptions (i)–(ii), (12) fails when some of the $\theta_{j}$’s are sufficiently far from zero: as soon as some of the first largest (resp., last smallest) nonnull eigenvalues $\theta_{j}$ of $A_{N}$ are taken strictly larger than $\sigma$ (resp., strictly smaller than $-\sigma$), the same part of the spectrum of $M_{N}$ almost surely exits the semicircle support $[-2\sigma,2\sigma]$ as $N\to\infty$ and the new limits are the $\rho_{\theta_{j}}$’s defined by

Observe that $\rho_{\theta_{j}}$ is $>2\sigma$ (resp., $<-2\sigma$) when $\theta_{j}>\sigma$ (resp., $<-\sigma$) (and $\rho_{\theta_{j}}=\pm 2\sigma$ if $\theta_{j}=\pm\sigma$).

Here is the precise formulation of our result. For definiteness, we set $k_{1}+\cdots+k_{j-1}:=0$ if $j=1$.

Let $J_{+\sigma}$ (resp., $J_{-\sigma}$) be the number of j’s such that $\theta_{j}>\sigma$ (resp., $\theta_{j}<-\sigma$).

$\forall 1\leq j\leq J_{+\sigma},\forall 1\leq i\leq k_{j},\lambda_{k_{1}+\cdots+k_{j-1}+i}(M_{N})\longrightarrow\rho_{\theta_{j}}\mbox{ a.s.}$

$\lambda_{k_{1}+\cdots+k_{J_{+\sigma}}+1}(M_{N})\longrightarrow 2\sigma\mbox{ a.s.}$

$\lambda_{k_{1}+\cdots+k_{J-J_{-\sigma}}}(M_{N})\longrightarrow-2\sigma\mbox{ a.s.}$

$\forall j\geq J-J_{-\sigma}+1,\forall 1\leq i\leq k_{j},\lambda_{k_{1}+\cdots+k_{j-1}+i}(M_{N})\longrightarrow\rho_{\theta_{j}}\mbox{ a.s}.$

Following BS3, one can expect that this theorem holds true in a more general setting than the one considered here, namely one that would only require four first moment conditions on the law $\mu$ of the Wigner entries. As we will explain in the following, the assumption that $\mu$ satisfies a Poincaré inequality is actually fundamental in our reasoning since we will need several variance estimates.

This theorem will be proved in Section 4. The second part of this work is devoted to the study of the particular rank-1 diagonal deformation $A_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ such that $\theta>\sigma$. We investigate the fluctuations of the largest eigenvalue of any real or complex Deformed model $M_{N}$ satisfying (i) around its limit $\rho_{\theta}$. We obtain the following result.

Let $A_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ with $\theta>\sigma$. Define

where $t=4$ (resp., $t=2$) when $W_{N}$ is real (resp., complex) and $m_{4}:=\int x^{4}\,d\mu(x)$. Then

Note that when $m_{4}=3\sigma^{4}$ as in the Gaussian case, the variance of the limiting distribution of $\sqrt{N}(\lambda_{1}(M_{N})-\rho_{\theta})$ is equal to $\sigma_{\theta}^{2}$ (resp., $2\sigma_{\theta}^{2}$) in the complex (resp., real) setting [with $\sigma_{\theta}$ given by (8)].

Since $\mu$ is symmetric, it readily follows from Theorem 2.2 that when $A_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ and $\theta<-\sigma$, the smallest eigenvalue of $M_{N}$ fluctuates as $\sqrt{N}(1-\sigma^{2}/\theta^{2})^{-1}(\lambda_{N}(M_{N})-\rho_{\theta})\stackrel{{\scriptstyle\mathcal{L}}}{{\longrightarrow}}\mu\ast\mathcal{N}(0,v_{\theta}).$

In particular, one derives the analogue of (7) for the Deformed GOE:

Let $A_{N}$ be an arbitrary deterministic symmetric matrix of rank 1 having a nonnull eigenvalue $\theta$ such that $\theta>\sigma$. Then the largest eigenvalue of the Deformed GOE fluctuates as

Obviously, thanks to the orthogonal invariance of the GOE, this result is a direct consequence of Theorem 2.2.

It is worth noticing that, according to the Cramér–Lévy theorem (cf. Fel, Theorem 1, page 525), the limiting distribution (15) is not Gaussian if $\mu$ is not Gaussian. Thus, (15) depends on the particular law $\mu$ of the entries of the Wigner matrix $W_{N}$ which implies the nonuniversality of the fluctuations of the largest eigenvalue of rank-1 diagonal deformation of symmetric or Hermitian Wigner matrices (as conjectured in Remark 1.7 of FePe).

The latter also shows that in the non-Gaussian setting, the fluctuations of the largest eigenvalue depend, not only on the spectrum of the deformation $A_{N}$, but also on the particular definition of the matrix $A_{N}$. Indeed, in collaboration with S. Péché, the third author of the present article has recently stated in FePe the universality of the fluctuations of some Deformed Wigner models under a full deformation $A_{N}$ defined by $(A_{N})_{ij}={\theta}/{N}$ for all $1\leq i,j\leq N$ (see also FK). Before giving some details on this work, we have to specify that FePe considered Deformed models such that the entries of the Wigner matrix $W_{N}$ have sub-Gaussian moments. Nevertheless, thanks to the analysis made in Ru, one can observe that the assumptions of FePe can be reduced and that it is, for example, sufficient to assume that the $W_{i,j}$’s have moments of any order. Thus, the conclusions of FePe apply to the setting considered in our paper. The main result of FePe establishes the universality of the fluctuations of the largest eigenvalue of the complex Deformed model $M_{N}$ associated to a full deformation $A_{N}$ and for any value of the parameter $\theta$. In particular, when $\theta>\sigma$, it is proved therein the universality of the Gaussian fluctuations (7). The approach of FePe is mainly based on a combinatorial method inspired by the work So (which handles the non-Deformed Wigner model) and some results of Pe on the Deformed GUE. The combinatorial arguments of FePe also work (with minor modifications) in the real framework and yield the universality of the fluctuations if $\theta<\sigma$. In the case where $\theta>\sigma$ which is of particular interest here, the analysis made in FePe reduces the universality problem in the real setting to the knowledge of the particular Deformed GOE model (this remark is also valid in the case where $\theta=\sigma$). Here, we will prove the needed results on the Deformed GOE which, thanks to the analysis of FePe and Ru, allow us to claim the following universality.

Then the largest eigenvalue of the Deformed model $M_{N}$ has the Gaussian fluctuations (16).

To be complete, let us notice that the previous result still holds when we allow the distribution $\nu$ of the diagonal entries of $W_{N}$ to be different from $\mu$ provided that $\nu$ is symmetric and has moments of any order.

and the Stieltjes transform of the expectation of the empirical measure of the eigenvalues of $M_{N}$ by

the Stieltjes transform Note that in some papers to which we make reference, the Stieltjes transform is defined with the opposite sign. of a variable $s$ with semicircular distribution $\mu_{sc}$.

Theorem 2.1 is the analogue of the main statement of BS3 established in the context of general spiked population models. The conclusion of BS3 requires numerous results obtained previously by Silverstein and co-authors in CS, BS1 and BS2 (a summary of all this literature can be found in Bai, pages 671–675). From very clever and tedious manipulations of some Stieltjes transforms and the use of the matricial representation (1), these works highlight a very close link between the spectra of the Wishart matrices and the covariance matrix (for quite general covariance matrix which includes the spiked population model). Our approach mimics the one of BS3. Thus, using the fact that the Deformed Wigner model is the additive analogue of the spiked population model, several arguments can be quite easily adapted here (this point has been explained in Chapter 4 of the Ph.D. thesis Fe). Actually, the main point in the proof consists in establishing that for any $\varepsilon>0$, almost surely,

Dealing with the particular diagonal perturbation $A_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ such that $\theta>\sigma$, we obtain the fluctuations of the largest eigenvalue $\lambda_{1}(M_{N})$ (Theorem 2.2) by an approach close to the one of Pa and the ideas of BBPbis. The reasoning relies on the writing of the rescaled variable $\sqrt{N}(\lambda_{1}(M_{N})-\rho_{\theta})$ in terms of the resolvent of a non-Deformed Wigner matrix. Then, to complete the analysis of FePe and justify Theorem 2.4, we focus on the particular Deformed GOE model and improve the previous convergence at the level of Laplace transform.

The paper is organized as follows. In Section 3, we introduce preliminary lemmas which will be of basic use later on. Section 4 is devoted to the proof of Theorem 2.1. We first establish an equation (called master equation or master inequality) satisfied by $g_{N}$ up to some correction of order $\frac{1}{N^{2}}$ (see

Section 4.1). Then we explain how this master equation gives rise to an estimation of type (18) and thus to the inclusion (17) of the spectrum of $M_{N}$ in $K^{\varepsilon}_{\sigma}(\theta_{1},\ldots,\theta_{J})$ (see Sections 4.2 and 4.3). In Section 4.4, we use this inclusion to relate the asymptotic spectra of $A_{N}$ and $M_{N}$ and then deduce Theorem 2.1. Section 5 deals with the fluctuations results. The proof of Theorem 2.2 is given in Section 5.2; Theorem 2.4 is justified in Section 5.3.

Basic lemmas

Note that even if the result in CD is stated in the Hermitian case, the proof is valid and the result still holds in the symmetric case. Now (19) follows putting $g(x_{ij};i\leq j):=f(x_{ij}+(A_{N})_{ij};i\leq j)$ in (20) and noticing that the $(A_{N})_{ij}$ are uniformly bounded in $i,j,N$.

This lemma will be useful to estimate many variances. Now, we recall some useful properties of the resolvent (see KKP; CD).

$\|G(z)\|\leq|\Im m(z)|^{-1}$ where $\|\cdot\|$ denotes the operator norm.

$|G(z)_{ij}|\leq|\Im m(z)|^{-1}$ for all $i,j=1,\ldots,N$.

The derivative with respect to $M$ of the resolvent $G(z)$ satisfies

We just mention that (v) comes readily noticing that the eigenvalues of the normal matrix $G(z)$ are the $\frac{1}{z-\lambda_{i}(M)}$, $i=1,\ldots,N.$

We will also need the following estimations on the Stieltjes transform $g_{\sigma}$ of the semicircular distribution $\mu_{sc}$.

For (21), we refer the reader to Section 3.1 of Bai. Equation (25) is a consequence of $\Im m(g_{\sigma}(z))\Im m(z)<0$. Other inequalities derive from (21) and the definition of $g_{\sigma}.$

Almost sure convergence of the first extremal eigenvalues

for some $k$ and some $l$ and we give the precise majoration in the statements of the theorems or propositions.

Section 4.4 explains how to deduce Theorem 2.1 from the inclusion (17).

The goal of Sections 4.1 and 4.2 is to establish Proposition 4.4 below which is fundamental in the proof of the inclusion (17). Before describing rigorously the different ideas of these two sections, let us help the reader’s intuition by a heuristic understanding of the approach. Assume that we can establish that $g_{N}(z)$ satisfied the rough quadratic equation (also called master inequality):

Then, for any suitable $z$, divided by $g_{N}(z)$ the last approximation would provide us an estimation of $\Lambda_{N}(z)-z$ where $\Lambda_{N}(z)=z_{\sigma}(g_{N}(z))$ with $z_{\sigma}(g)=\frac{1}{g}+\sigma^{2}g$ being the inverse function of $g_{\sigma}$ (see Lemma 4.4 below). Then, intuitively, a Taylor expansion of $g_{\sigma}$ between $\Lambda_{N}(z)$ and $z$ would lead to an estimation of the type

This intuitive process may throw light on the expression (48) of $L_{\sigma}(z)$ in Proposition 4.4 below.

Let us recall the integration by parts formula for the Gaussian distribution.

for any Hermitian matrix $H$, or by linearity for $H=E_{jk}$, $1\leq j,k\leq N$ where $E_{jk}$, $1\leq j,k\leq N$ is the canonical basis of the complex space of $N\times N$ matrices.

Now, we consider the normalized sum $\frac{1}{N^{2}}\sum_{ij}$ of the previous identities to obtain

Now, it is well known (see CD; HT and Lemma 3.1) that

Thus, in the case where $X_{N}=X_{N}^{G}$ we obtain:

The Stieltjes transform $g_{N}$ satisfies the following inequality:

We now explain how to obtain the corresponding (30) in the Wigner case. Since the computations are the same as in CD This paper treats the case of several independent non-Deformed Wigner matrices. and KKP, The authors considered a non-Deformed Wigner matrix in the symmetric real setting. we just give some hints of the proof.

The integration by parts formula for the Gaussian distribution is replaced by the following tool:

We apply this lemma with the function $\phi(\xi)$ given, as before, by $\phi(\xi)=G_{ij}$ and $\xi$ is now one of the variables $\Re e((X_{N})_{kl})$, $\Im m((X_{N})_{kl})$. Note that, since the above random variables are symmetric, only the odd derivatives in (31) give a nonnull term. Moreover, as we are concerned by estimation of order $\frac{1}{N^{2}}$ of $g_{N}$, we only need to consider (31) up to the third derivative (see CD). The computation of the first derivative will provide the same term as in the Gaussian case.

We refer to CD or KKP for a detailed study of the third derivative. Using some bounds on $G_{N}$ (see Lemma 3.2), we can prove that the only term arising from the third derivative in the master equation, giving a contribution of order $\frac{1}{N}$, is

In conclusion, the first master equation in the Wigner case reads as follows:

where $\kappa_{4}$ is the fourth cumulant of the distribution $\mu$.

1.2 Estimation of |gN−gσ||g_{N}-g_{\sigma}|

To estimate $|g_{N}-g_{\sigma}|$ from (21) and (33), we follow the method initiated in HT and S. We do not develop it here since it follows exactly the lines of Section 3.4 in CD but we briefly recall the main arguments and results which will be useful later on. We define the open connected set

One can prove that for any $z$ in $\mathcal{O}^{\prime}_{N}$:

writing (21) at the point $\Lambda_{N}(z)$, we easily get that

on the nonempty open subset $\mathcal{O}^{\prime\prime}_{N}=\{z\in\mathcal{O}^{\prime}_{N},\Im m(z)>\sqrt{2}\sigma\}$ and then on $\mathcal{O}^{\prime}_{N}$ by the principle of uniqueness of continuation.

this allows us to get an estimation of $|g_{N}(z)-g_{\sigma}(z)|$ on $\mathcal{O}^{\prime}_{N}$ and then to deduce:

From now on and until the end of Section 4.1, we denote by $\gamma_{1},\ldots,\gamma_{r}$ the nonnull eigenvalues of $A_{N}$ ($\gamma_{i}=\theta_{j}$ for some $j\not=j_{0}$) in order to simplify the writing. Let $U_{N}:=U$ be a unitary matrix such that $A_{N}=U^{*}\Delta U$ where $\Delta$ is the diagonal matrix with entries $\Delta_{ii}=\gamma_{i},\ i\leq r;\Delta_{ii}=0,\ i>r$. We set

Our aim is to express $h_{N}(z)$ in terms of the Stieltjes transform $g_{N}(z)$ for $N$ large, using the integration by parts formula. Note that since we want an estimation of order $O(N^{-2})$ in the master inequality (4.1), we only need an estimation of $h_{N}(z)$ of order $O(N^{-1})$. As in the previous subsection, we first write the equation in the Gaussian case and then study the additional term (third derivative) in the Wigner case.

(a) Gaussian case. Apply (29) to $\Phi(X_{N})=G_{jl}$ and $H=E_{il}$ to get

Expressing $GX_{N}$ in terms of $GA_{N}$, we obtain

Now, we consider the sum $\sum_{i,j}U^{*}_{ik}U_{kj}I_{ji}$, $k=1,\ldots,r$ fixed and we denote $\alpha_{k}=\sum_{i,j}U^{*}_{ik}U_{kj}G_{ji}=(UGU^{*})_{kk}$. Then, we have the following equality, using that $U$ is unitary:

(b) The general Wigner case. We shall prove that (42) still holds. We now rely on Lemma 4.2 to obtain the analogue of (41):

The term $A_{i,j,l}$ is a fixed linear combination of the third derivative of $\Phi:=G_{jl}$ with respect to $\mathcal{R}e(X_{N})_{il}$ (i.e., in the direction $e_{il}=E_{il}+E_{li})$ and $\Im m(X_{N})_{il}$ [i.e., in the direction $f_{il}:=\sqrt{-1}(E_{il}-E_{li})$]. We do not need to write the exact form of this term since we just want to show that this term will give a contribution of order $O(N^{-1})$ in the equation for $h_{N}(z)$. Let us write the derivative in the direction $e_{il}$:

which is the sum of eight terms of the form

where if $i_{2q+1}=i$ (resp., $l$), then $i_{2q+2}=l$ (resp., $i$), $q=0,1,2$.

$F(N)$ is the sum of eight terms corresponding to (45). Let us write, for example, the term corresponding to $i_{1}=i$, $i_{3}=i$, $i_{5}=i$:

We give the majoration for the term corresponding to $i_{1}=l$, $i_{3}=l$, $i_{5}=l$:

As in the Gaussian case, we now consider the sum $\sum_{i,j}U^{*}_{ik}U_{kj}J_{ji}$. From Lemma 4.3 and the bound (using the Cauchy–Schwarz inequality)

we still get (42) and thus (43). More precisely, we proved:

We now study the last term in the master inequality of Theorem 4.1. For the non-Deformed Wigner matrices, it is shown in KKP that

Moreover, Proposition 3.2 in CD, in the more general setting of several independent Wigner matrices, gives an estimate of $|R_{N}(z)-g_{\sigma}^{4}(z)|$. The above convergence holds true in the Deformed case. We just give some hints of the proof of the estimate of $|R_{N}(z)-g_{\sigma}^{4}(z)|$ since the computations are almost the same as in the non-Deformed case. Let us set

For the last term, we apply an integration by parts formula (Lemma 4.2) to obtain (see KKP; CD)

It remains to see that the additional term due to $A_{N}$ is of order $O(N^{-1})$:

We thus obtain (again with the help of a variance estimate)

Then using (39) and since $d_{N}(z)$ is bounded we deduce that

We can now give our final master inequality for $g_{N}(z)$ following our previous estimates:

where ${E_{\sigma}(z)=\sum_{k=1}^{r}\frac{\gamma_{k}}{z-\sigma^{2}g_{\sigma}(z)-\gamma_{k}}+\frac{\kappa_{4}}{2}g_{\sigma}^{4}(z)},$ $\kappa_{4}$ is the fourth cumulant of the distribution $\mu$.

Note that $E_{\sigma}(z)$ can be written in terms of the distinct eigenvalues $\theta_{j}$ of $A_{N}$ as

where $s$ is a centered semicircular random variable with variance $\sigma^{2}$.

2 Estimation of |gσ​(z)−gN​(z)+1N​Lσ​(z)||g_{\sigma}(z)-g_{N}(z)+\frac{1}{N}L_{\sigma}(z)|

is roughly the same as the one described in Section 3.6 in CD. Nevertheless we choose to develop it here for the reader’s convenience. We have for any $z$ in $\mathcal{O}^{\prime}_{n}$, by using (34) and (38),

We get from Theorem 4.2, (35), (39), (4.2), (23),

Finally, using also (36), we get for any $z$ in $\mathcal{O}^{\prime}_{n}$,

Now, for $z\notin\mathcal{O}^{\prime}_{n}$, such that $\Im m(z)>0$,

Thus, for any $z$ such that $\Im m(z)>0$,

Let us denote for a while $g_{N}=g_{N}^{A_{N}}$ and $L_{\sigma}=L_{\sigma}^{A_{N}}$. Note that we get exactly the same estimation (50) dealing with $-A_{N}$ instead of $A_{N}$. Hence since $g_{\sigma}(z)=-g_{\sigma}(-z)$, $g_{N}^{-A_{N}}(z)=-g_{N}^{A_{N}}(-z)$ (using the symmetry assumption on $\mu$) and $L_{\sigma}^{-A_{N}}(z)=L_{\sigma}^{A_{N}}(-z)$, it readily follows that (50) is also valid for any $z$ such that $\Im m(z)<0$. In conclusion:

3 The spectrum of MNM_{N}

The following step now consists of deducing Proposition 4.6 from Proposition 4.4 (from which we will easily deduce the appropriate inclusion of the spectrum of $M_{N}$). Since this transition is based on the inverse Stieltjes transform, we start with establishing the fundamental Proposition 4.5 below concerning the nature of $L_{\sigma}$. To this aim, it will be relevant to rewrite $L_{\sigma}$ as

We recall that $J_{+\sigma}$ (resp., $J_{-\sigma}$) denotes the number of $j$’s such that $\theta_{j}>\sigma$ (resp., $\theta_{j}<-\sigma$). As in the Introduction, we define

which is $>2\sigma$ (resp., $<-2\sigma$) when $\theta_{j}>\sigma$ (resp., $<-\sigma$).

$L_{\sigma}$ is the Stieltjes transform of a distribution $\Lambda_{\sigma}$ with compact support

The proof relies on the following characterization already used in S.

$l(z)\rightarrow 0$ as $|z|\rightarrow\infty,$

The following properties of the Stieltjes transform $g_{\sigma}$ will be useful for showing that $L_{\sigma}$ fulfills the previous conditions.

The complement of the support of $\mu_{\sigma}$ is characterized as follows:

Now, we are going to show that $L_{\sigma}$ satisfies (c1) and (c2) of Theorem 4.3. We have obviously that

Using also (26)–(28), we get readily that for $|z|>\alpha$,

Then, it is clear that $|L_{\sigma}(z)|\rightarrow 0$ when $|z|\rightarrow+\infty$ and (c1) is satisfied.

Now we follow the approach of S (Lemma 5.5) to prove (c2). Denote by $\mathcal{E}$ the convex envelope of $K_{\sigma}(\theta_{1},\ldots,\theta_{J})$ and define the interval

Hence (c2) is satisfied with $C=\max(C_{0},C_{1},C_{2})$ and $n=7$ and Proposition 4.5 follows from Theorem 4.3.

We are now in position to deduce the following proposition from the estimate (51).

For any smooth function $\varphi$ with compact support,

Consequently, for $\varphi$ smooth, constant outside a compact set and such that $\operatorname{supp}(\varphi)\cap K_{\sigma}(\theta_{1},\ldots,\theta_{J})=\varnothing$,

We refer the reader to the Appendix of CD where it is proved using the ideas of HT that

Dealing with $h(z)=N^{2}r_{N}(z)$, we deduce that

Following the proof of Lemma 5.6 in S, one can show that $\Lambda_{\sigma}(1)=0$. Then, the rest of the proof of (54) sticks to the proof of Lemma 6.3 in HT (using Lemma 3.1).

Such a method can be carried out in the case of Wigner real symmetric matrices; then the approximate master equation is the following [compare with (4.1)]:

where $s$ is a centered semicircular variable with variance $\sigma^{2}$. Hence by similar arguments as in the complex case, one gets the master equation

Let $(M_{N})_{N}$ be any real or complex Deformed model satisfying (i) and (ii) in Section 2. Let $J_{+\sigma}$ (resp., $J_{-\sigma}$) be the number of j’s such that $\theta_{j}>\sigma$ (resp., $\theta_{j}<-\sigma$). Then for any $\varepsilon>0$, almost surely, there is no eigenvalue of $M_{N}$ in

As soon as $\epsilon>0$ is small enough, the union (4.4) is made of nonempty disjoint intervals.

4 The almost sure convergence result

As announced in the Introduction, Theorem 2.1 is the analogue of the main statement of BS3 established for general spiked population models (1). The previous Theorem 4.4 is the main step of the proof since now, we can adapt the arguments needed for the conclusion of BS3 viewing the Deformed Wigner model (2) as the additive analogue of the spiked population model (1).

Let us consider one of the positive eigenvalues $\theta_{j}$ of the $A_{N}$’s. We recall that this implies that $\lambda_{k_{1}+\cdots+k_{j-1}+i}(A_{N})=\theta_{j}$ for all $1\leq i\leq k_{j}$. We want to show that if $\theta_{j}>\sigma$ (i.e., with our notation, if $j\in\{1,\ldots,J_{+\sigma}\}$), the corresponding eigenvalues of $M_{N}$ almost surely jump above the right endpoint $2\sigma$ of the semicircle support as

whereas the rest of the asymptotic spectrum of $M_{N}$ lies below $2\sigma$ with

Analogous results hold for the negative eigenvalues $\theta_{j}$ [see points (c) and (d) of Theorem 2.1]. To describe the phenomenon, one can say that, when $N$ is large enough, the (first extremal) eigenvalues of $M_{N}$ can be viewed as a “smoothed” deformation of the (first extremal) eigenvalues of $A_{N}$. According to the analysis made in the previous section [Lemma 4.4(b)], we already know that the limits $\rho_{\theta_{j}}$ are related to the $\theta_{j}$’s through the Stieltjes transform $g_{\sigma}$. More precisely, one has

Our main purpose now is to establish the asymptotic link between the spectra of the matrices $M_{N}=X_{N}+A_{N}$ and $A_{N}$.

Intuitively, this link seems rather natural when $\sigma$ is close to zero. Indeed, when $N$ goes to infinity, since the spectrum of $X_{N}$ is concentrated in $[-2\sigma,2\sigma]$ [recall (11)], the spectrum of $M_{N}$ should be close to the one of $A_{N}$ as soon as $\sigma$ will be close to zero (in other words, the spectrum of $M_{N}$ is, viewed as a deformation of the one of $A_{N}$, continuous in $\sigma$ in a neighborhood of zero). Thus given an interval $[a,b]\subset{}{{}^{c}\!K}_{\sigma}(\theta_{1},\ldots,\theta_{J})$, the result of Theorem 4.4 saying that $[a,b]$ does not contain eigenvalues of $M_{N}$ should be improved: it should correspond to $[a,b]$ some interval $I$ close to $[a,b]$, lying outside the spectrum of $A_{N}$ and such that the number of eigenvalues of $M_{N}$ in one side of $[a,b]$ is equal to the one of $A_{N}$ in the corresponding side of $I$. Following BS2, we will say that there is exact separation of eigenvalues of the matrices $A_{N}$ and $M_{N}$.

In the following section, we justify that the exact separation phenomenon occurs regardless of the size of $\sigma$. The proof of Theorem 2.1 will then follow from some suitable choices of $[a,b]$ (see Section 4.4.2).

According to the previous discussion, we need to refine the analysis made on $g_{\sigma}$ in order to identify and understand the link between intervals in ${}^{c}K_{\sigma}(\theta_{1},\ldots,\theta_{J})$ and the complement of the spectrum of the $A_{N}$’s. We also need to understand the dependence on $\sigma$. This is the aim of the following important Lemma 4.5.

As before, we denote (recall Lemma 4.4) by $z_{\sigma}$ the inverse function of $g_{\sigma}$ which is given by

Using Lemma 4.4, one readily sees that the set ${}^{c}K_{\sigma}(\theta_{1},\ldots,\theta_{J})$ can be characterized as follows:

Obviously, one has $g=g_{\sigma}(x)$ if $x\in{}{{}^{c}\!K}_{\sigma}(\theta_{1},\ldots,\theta_{J})$.

Let $[a,b]$ be a compact set contained in ${}^{c}K_{\sigma}(\theta_{1},\ldots,\theta_{J})$. Then:

${[\frac{1}{g_{\sigma}(a)},\frac{1}{g_{\sigma}(b)}]}\subset(\operatorname{Spect}(A_{N}))^{c}$.

For all $0<\hat{\sigma}<\sigma$, the interval $[z_{\hat{\sigma}}(g_{\sigma}(a)),z_{\hat{\sigma}}(g_{\sigma}(b))]$ is contained in ${}^{c}K_{\hat{\sigma}}(\theta_{1},\ldots,\theta_{J})$ and $z_{\hat{\sigma}}(g_{\sigma}(b))-z_{\hat{\sigma}}(g_{\sigma}(a))\geq b-a$.

The function $1/{g_{\sigma}}$ being increasing, (i) readily follows from (56).

Noticing that $\mathcal{G}_{{\sigma}}\subset\mathcal{G}_{\hat{\sigma}}$ for all $\hat{\sigma}<\sigma$ implies (recall also that $g_{\sigma}$ decreases on $[a,b]$) that $[g_{\sigma}(b),g_{\sigma}(a)]\subset\mathcal{G}_{\hat{\sigma}}$. Relation (56) combined with the fact that the function $z_{\hat{\sigma}}$ is decreasing on $[g_{\sigma}(b),g_{\sigma}(a)]$ leads to

and the first part of (ii) is stated. Now, we have

The exact separation result can now be stated. Let $[a,b]$ be an interval contained in ${}^{c}K_{\sigma}(\theta_{1},\ldots,\theta_{J})$. By Theorem 4.4, $[a,b]$ is outside the spectrum of $M_{N}$. Moreover, from Lemma 4.5(i), there corresponds an interval $I=[a^{\prime},b^{\prime}]$ outside the spectrum of $A_{N}$, that is, there is $i_{N}\in\{0,\ldots,N\}$ such that

$a$ and $a^{\prime}$ (resp., $b$ and $b^{\prime}$) are linked as follows:

We claim that $[a,b]$ splits the eigenvalues of $M_{N}$ exactly as $I$ splits the spectrum of $A_{N}$. In other words:

With $i_{N}$ satisfying $(\ref{sep1})$, one has

This result is the analogue of the main statement of BS2 (cf. Theorem 1.2 of BS2) established in the spiked population setting (and in fact for quite general sample covariance matrices). Its proof is quite technical and is inspired by the work BS2. It mainly relies on results on eigenvalues of the rescaled Wigner matrix $X_{N}$ combined with the following classical result (due to Weyl).

Let B and C be two $N\times N$ Hermitian matrices. For any pair of integers $j,k$ such that $1\leq j,k\leq N$ and $j+k\leq N+1$, we have

For any pair of integers $j,k$ such that $1\leq j,k\leq N$ and $j+k\geq N+1$, we have

Note that this lemma is the additive analogue of Lemma 1.1 of BS2 needed for the investigations of the spiked population model.

In particular, Lemma 4.6 gives that $\lambda_{i_{N}+1}(M_{N})\leq\lambda_{i_{N}+1}(A_{N})+\lambda_{1}(X_{N})$ and $\lambda_{i_{N}}(M_{N})\geq\lambda_{i_{N}}(A_{N})+\lambda_{N}(X_{N}).$ Besides, as both $\lambda_{1}(X_{N})$ and $-\lambda_{N}(X_{N})$ tend toward $2\sigma$ as $N\to\infty$ [this is (11)], the statement of Theorem 4.5 can be quite easily derived when $\sigma$ is close enough to zero. To handle the general case, the key idea is that one can reduce to the previous situation by introducing some parameters. More precisely, given $L>0$ and $k\geq 0$, we will introduce the Wigner matrix

be the Deformed Wigner matrix of parameter

The proof will be organized as follows. On the one hand, as $\sigma_{k,L}\to 0$ when $k\to\infty$ (for any fixed $L>0$), we will readily prove that exact separation occurs for the matrices $A_{N}$ and $M_{N}^{K,L}$ as soon as $K$ is large enough. On the other hand, we will show that exact separation also occurs for the eigenvalues of $M_{N}=M_{N}^{0,L}$ and $M_{N}^{K,L}$ choosing $L$ large enough. This latter point will be established by induction on $k$; the underlying idea is that when the parameter $L$ is large, the matrices $M_{N}^{k,L}$ and $M_{N}^{k+1,L}$ are close to each other and hence split their spectrum in a similar way.

[Proof of Theorem 4.5] With our choice of $[a,b]$ and the very definition of the spectrum of the $A_{N}$’s, one can consider $\epsilon^{\prime}>0$ small enough such that, for all large $N$,

Given $L>0$ and $k\geq 0$ (their size will be determined later), we define

where we recall that $z_{\sigma_{k,L}}(g)={1}/{g}+\sigma_{k,L}^{2}g.$ Note that for all $L>0$, one has $a_{0,L}=a$ and $b_{0,L}=b$.

We first choose the size of $L$ as follows. We take $L_{0}$ large enough such that for all $L\geq L_{0}$,

From the very definition of the $a_{k,L}$’s and $b_{k,L}$’s, one can easily see that $b_{k,L}-a_{k,L}\geq b-a$ [using the last point of (ii) in Lemma 4.5] and that this choice of $L_{0}$ ensures that, for all $L\geq L_{0}$ and for all $k\geq 0$,

Now, we fix $L$ such that $L\geq L_{0}$ and we write $a_{k}=a_{k,L}$, $b_{k}=b_{k,L}$ and $\sigma_{k}=\sigma_{k,L}$.

We first show that there exists $K$ large enough such that, for all $k\geq K$, there is exact separation of the eigenvalues of the matrices $A_{N}$ and $M_{N}^{k,L}$, that is,

Furthermore, according to (11), the two first extremal eigenvalues of $X_{N}$ are such that almost surely and for all $N$ large enough,

Thus for all $k$, almost surely, at least for $N$ large enough ($N$ does not depend on $k$),

As $\sigma_{k}\to 0$ when $k\to+\infty$, there is $K$ large enough such that for all $k\geq K$,

and then, almost surely, for all $N$ large enough

Since $\lambda_{N+1}(M_{N}^{k,L})=-\lambda_{0}(M_{N}^{k,L})=-\infty$, (63) [resp., (64)] is obviously satisfied if $i_{N}=N$ (resp., $i_{N}=0$). Thus, we have established that for any $i_{N}\in\{0,\ldots,N\}$ satisfying (58), (62) holds for all $k\geq K$. In particular,

Now, we shall show that with probability $1$: for $N$ large, $[a_{K},b_{K}]$ and $[a,b]$ split the eigenvalues of, respectively, $M_{N}^{K,L}$ and $M_{N}$ having equal amount of eigenvalues to the left sides of the intervals. To this aim, we will proceed by induction on $k$ and establish that, for all $k\geq 0$, $[a_{k},b_{k}]$ and $[a,b]$ split the eigenvalues of $M_{N}^{k,L}$ and $M_{N}$ (recall that $M_{N}=M_{N}^{0,L}$) in exactly the same way. To begin, let us consider for all $k\geq 0$, the set

This can be done by induction calling, one more time, on Lemma 4.6. By (66), this is true for $k=0$. Now, let us assume that (67) holds true. One has

But, for $N$ large enough, $0<-\lambda_{N}(X_{N})\leq 3\sigma$ a.s., so by the condition (60) on $L$,

By (61), one readily observes that $\hat{a}_{k}-a_{k+1}=a_{k}-a_{k+1}+({b-a})/{4}>0$ and similarly that $\hat{b}_{k}-b_{k+1}<0$. This implies that

As a consequence, (67) holds for all $k\geq 0$ and in particular for $k=K$. Comparing this with (65), we deduce that $j_{N}=i_{N}$ a.s. and

Now, we are in position to prove the main Theorem 2.1.

4.2 Proof of Theorem 2.1

Our reasoning is close to the last Section 4 of BS3. It is enough to establish parts (a) and (b) since the assertions (c) and (d) can then be deduced by taking $-M_{N}$ instead of $M_{N}$.

The proof of (a) is mainly based on successive applications of Theorem 4.5. Fix an integer $1\leq j\leq J_{+\sigma}$, and let us consider for $\epsilon>0$, the interval $[a,b]=[\rho_{\theta_{j}}+\epsilon,\rho_{\theta_{j-1}}-\epsilon]$ which is included in the union (4.4) (at least for $\epsilon$ small enough). We define $K_{j(-1)}=k_{1}+\cdots+k_{j(-1)}$. We also take $\theta_{0}:=+\infty$ and recall the conventions that $\lambda_{0}(M_{N})=\lambda_{0}(A_{N})=+\infty$ and $K_{0}=0$. Since $1/g_{\sigma}(\rho_{\theta_{k}})=\theta_{k}$ for $k=j-1$ and $j$ and since the function $1/g_{\sigma}$ is continuous and increasing on $[a,b]$, the compact interval $[a,b]$ satisfies (58) with $i_{N}=K_{j-1}$. Hence by Theorem 4.5, one has

Similar arguments imply that for all $j\in\{1,\ldots,J_{+\sigma}-1\}$,

As a result, we deduce that for all $1\leq j\leq J_{+\sigma}-1$,

So, letting $\epsilon$ go to zero, we obtain (a) for each integer $j$ of $\{1,\ldots,J_{+\sigma}-1\}$.

Let us now quickly consider the case where $j=J_{+\sigma}$. Note first that, from the preceding discussion, we still have (for $\epsilon$ small enough)

Then, using the fact that $1/g_{\sigma}$ increases continuously on $]2\sigma,+\infty[$ with $1/\penalty g_{\sigma}(]2\sigma,+\infty[)=\,]\sigma,+\infty[$, one can show that once $\epsilon>0$ is small enough, the compact set $[a,b]=[2\sigma+\epsilon,\rho_{\theta_{J_{+\sigma}}}-\epsilon]$ satisfies the assumptions of Theorem 4.5 with $i_{N}=K_{J_{+\sigma}}$. This leads to

Letting $\epsilon\to 0$, we deduce that (4.4.2) holds for $j=J_{+\sigma}$ and the assertion (a) is established. For point (b), the preceding analysis gives that $\limsup_{N}\lambda_{K_{J_{+\sigma}}+1}(M_{N})\leq 2\sigma\mbox{ a.s.}$ and it remains to prove that

This inequality follows from the fact that the spectral measure of $M_{N}$ converges a.s. toward the semicircle law $\mu_{sc}$ which is compactly supported in $[-2\sigma,2\sigma]$. This completes the proof of Theorem 2.1.

Fluctuations

The (complex or real) Wigner matricial models under consideration are the same as previously [i.e., defined by (i) in Section 2] but now we assume that the perturbation $A_{N}$ is diagonal: $A_{N}=\operatorname{diag}(\theta,0,\ldots,0)$ with unique nonnull eigenvalue $\theta>\sigma$. According to the previous section, the a.s. convergence of $\lambda_{1}(M_{N})$ toward $\rho_{\theta}$ is universal in the sense that it does not depend on $\mu$.

In the first part of this section, we will show that the fluctuations of $\lambda_{1}(M_{N})$ around this universal limit are not universal any more. Indeed, we are going to prove that $\sqrt{N}(1-{\sigma^{2}}/{\theta^{2}})^{-1}(\lambda_{1}(M_{N})-\rho_{\theta})$ converges in distribution toward the convolution of $\mu$ and a Gaussian distribution. Hence, the limiting distribution clearly varies with $\mu$ and in particular cannot be Gaussian unless $\mu$ is Gaussian.

In the second part of this section, we will sharpen the analysis of the particular Deformed GOE model and explain how this gives Theorem 2.4.

Let $B=(b_{ij})$ be an $N\times N$ Hermitian matrix and $Y_{N}$ be a vector of size $N$ which contains i.i.d. standardized entries with bounded fourth moment. Then there is a constant $K>0$ such that

there exists a constant $a>0$ (not depending on $N$) such that $\|B\|\leq a$,

$\frac{1}{N}\operatorname{Tr}B^{2}$ converges in probability to a number $a_{2}$,

$\frac{1}{N}\sum_{i=1}^{N}b_{ii}^{2}$ converges in probability to a number $a_{1}^{2}$.

Then the random variable $({1}/{\sqrt{N}})(Y_{N}^{*}BY_{N}-\operatorname{Tr}B)$ converges in distribution to a Gaussian variable with mean zero and variance

where $t=4$ when $y_{1}$ is real and is $2$ when $y_{1}$ is complex.