Fluctuations at the edges of the spectrum of the full rank deformed GUE

Introduction and results

Enormous progress has been accomplished in the very recent years in the study of asymptotic spectral properties of large random matrices. A Hermitian Wigner random matrix is a $N\times N$ matrix $W_{N}=\frac{1}{\sqrt{N}}(W_{ij})_{i,j=1}^{N}$,with i.i.d. entries off the diagonal $W_{ij},i<j$ (modulo the symmetry assumption) and independent diagonal real entries. The entries are standardized to be centered and of variance $\sigma^{2}$. The asymptotic local properties of the spectrum of Wigner random matrices are now quite well understood thanks to the fantastic work of Erdös-Schlein-Yau (see , and references therein) and Tao-Vu . In particular, it is known (assuming that the matrix elements admit enough moments) that the fluctuations of eigenvalues in the bulk or at the edges of the spectrum are universal. In particular, they coincide with those identified for a Gaussian (GUE) matrix with variance $\sigma^{2}$. In other words, the limiting asymptotic spectral properties of a Wigner matrix in the large $N$ limit do not depend on the detail of the distribution of the matrix elements $W_{ij}$, $1\leq i,j\leq N.$ In this article, we are interested in deformed random matrix ensembles. A deformation of a standard random matrix can be more or less understood as the modification of the distribution of some of the entries of a Wigner matrix. The set of possible deformations is non exhaustive (one can force some of the entries to be zero such as for sparse matrices) but we here restrict to some additive deformations. More precisely, we consider a matrix $A_{N}$ of size $N$, which is deterministic. Our study could be extended to the case where it is random but we do not wish to pursue this direction here. We consider the deformed matrices

where $W_{N}$ is a standard Wigner matrix. The question is to understand the asymptotic properties of the eigenvalues and eigenvectors of the deformed matrix, knowing that of $A_{N}$ and $W_{N}$. Such ensembles have first been introduced by , and when $W_{N}$ is a GUE.

a.s. Such eigenvalues $\lambda_{i}$ outside the support of the semi-circle distribution are called outliers. Interestingly, and then , have proved that the fluctuations of spikes are not universal in general. More precisely

where the distribution $\mu$ may depend explicitly on the distribution of the matrix elements $W_{ij}$. It can be shown that eigenvectors of the matrix $A_{N}$ play a fundamental role in the universality/non universality of the deformation matrix $A_{N}$. On the contrary, when there is no spike, the limiting distribution of extreme eigenvalues is the same as in the non deformed case. In particular, extreme eigenvalues stick to the bulk of the spectrum. The scale of their fluctuations is $N^{-2/3}$ and the limiting distribution of the largest (and smallest) eigenvalues is the Tracy-Widom distribution, provided the matrix elements $W_{ij}$ admit enough moments. A complete study of such deformed ensembles has been achieved in and and we refer the reader to these articles for a complete state of the art in finite rank deformations of Wigner matrices.

Let us diagonalize $A_{N}$ through $A_{N}=V\text{diag}(y_{1},\ldots,y_{N})V^{*}$. Roughly speaking the deformed model is now understood in the sense that $A_{N}$ is a ”small” perturbation of the matrix $W_{N}+VA_{0}V^{*}$ where $A_{0}$ would be a diagonal matrix made up with quantiles of the probability $\nu.$ The asymptotic global behavior of the spectrum is well-known in this case. Indeed, let $\mu_{N}$ be the empirical eigenvalue distribution of $W_{N}+A_{N}.$ Its Stieltjes transform is

According to , $m_{N}$ converges as $N\to\infty$ to the Stieltjes transform $m_{\tau}$ of a probability distribution $\tau$, called the free convolution of $\nu$ and the semi-circle distribution. This probability distribution $\tau$ is uniquely characterized by a fixed point equation satisfied by $m_{\tau}$, as we review in Section 2; it has a density $p$. We emphasize that the support of the probability distribution $\tau$ may have distinct connected components, depending on $\nu.$

The question of the asymptotic behavior of extreme eigenvalues naturally arises in this setting also. This question has been much less investigated actually. So far, only the case where $W_{N}$ is a GUE has been investigated. In , the author considers the case where $\mu_{A_{N}}$ concentrate quite fast to the measure $\nu$. In particular, there are no spikes. When $W_{N}$ is a GUE, she investigates the local edge regime which deals with the behavior of the eigenvalues near any extremity point $u_{0}$ of a connected component of $\text{supp}(\tau)$. More precisely let some $\epsilon>0$ be given and assume that either

makes a technical assumption on the uniform convergence of the Stieltjes transform of $\mu_{A_{N}}$ to $m_{\nu}$:

where $K$ is some compact subset of the complex plane at a positive distance of the support of $\nu.$ This is a rather strong assumption on the rate of convergence of $\mu_{A_{N}}$ to $\nu$. proves that the joint distribution of the largest (or smallest) eigenvalues converging to $u_{0}$ have universal asymptotic behavior, characterized by the famous Tracy-Widom distribution. We note that also investigates the asymptotic spacing distribution of eigenvalues in the bulk of the spectrum. The same behavior as for non deformed ensemble is obtained (and described by the sine kernel). The extension to a non Gaussian matrix $W$ has recently been obtained by in the case where $A_{N}$ is diagonal. In and , the authors consider the case where $\mu_{A_{N}}=\nu$ is a finite combination of Dirac delta masses. They identify different possible limiting statistics at the edges of the support of $\tau$, after suitable normalization of the eigenvalues. If $u_{0}$ is a point such that $p(u)=0,\>u_{0}-\epsilon\leq u\leq u_{0}$, $p(u)>0,u_{0}<u\leq u_{0}+\epsilon$ for some $\epsilon>0$, the asymptotic distribution of eigenvalues close to $u_{0}$ is the Tracy-Widom distribution. The authors also consider the case where $u_{0}$ is a point where two connected components of $\text{supp}(\tau)$ merge so that $p(u)>0,\forall u\in(u_{0}-\epsilon,u_{0}+\epsilon)\setminus\{u_{0}\}$ and $p(u_{0})=0$. In this case, the limiting eigenvalue statistics are described by the so-called Pearcey kernel (whose definition is reviewed hereafter).

In both cases, a strong assumption is made on the rate of convergence of $\mu_{A_{N}}$ to $\nu$. We here remove this assumption. We identify all the possible limiting eigenvalue statistics at the edges of the spectrum of the deformed GUE, namely at a spike, at the edge of a connected component of the support or at a point where two connected components merge. We emphasize that we do not make any assumptions on the rate of convergence of $\mu_{A_{N}}$ to $\nu$. To state our results, we use a deterministic equivalent of the empirical eigenvalue distribution of $M_{N}$. This equivalent is the free convolution of the semi-circle distribution and $\mu_{A_{N}}$. The choice of the deformed GUE is motivated by the fact that all eigenvalues statistics can be explicitly computed for this ensemble of deformed random matrices. We expect that one can extend these results to full rank deformations of an arbitrary Wigner matrix, as in the fixed rank case (with universal or non universal results). We intend to consider this general case in a forthcoming paper. The techniques needed are completely different.

2 Model and results

We consider the following deformed GUE ensemble

$A_{N}$ is a deterministic Hermitian matrix whose eigenvalues $y_{i}=y_{i}(N)$, $1\leq i\leq N$, are such that the spectral measure $\mu_{A_{N}}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{y_{i}}$ converges weakly to some probability measure $\nu$ with compact support. We assume that

where ${\rm supp}(\nu)$ denotes the support of $\nu$.

We also assume that there exists a fixed integer $r\geq 0$ (independent from $N$) and an integer $0\leq J\leq r$ such that the following holds. There are $J$ fixed real numbers $\theta_{1}>\ldots>\theta_{J}$ independent of $N$ which are outside the support of $\nu$ and such that each $\theta_{j}$ is an eigenvalue of $A_{N}$ with a fixed multiplicity $k_{j}$ (with $\sum_{j=1}^{J}k_{j}=r$). The $\theta_{j}$’s are called the spikes or the spiked eigenvalues of $A_{N}$ and we set

The remaining $N-r$ eigenvalues of $A_{N}$, denoted by $\beta_{j}(N)$, $j=1,\ldots,N-r$, satisfy

Denote by $\mu_{sc}$ the semicircle distribution whose density is given by

According to , the spectral distribution of $M_{N}$ weakly converges almost surely to the so-called free convolution $\mu_{sc}\boxplus\nu$ which has a continuous density $p$ (see ). We recall some important facts about the free convolution with a semi-circular distribution in Section 2.

We are now in position to state our results. Let first consider a real number $d$ which is a right edge of $\text{supp}(\mu_{sc}\boxplus\nu)$ that is which satisfies (1). Assume moreover that for any $\theta_{j}$ such that $\int\frac{d\nu(s)}{(\theta_{i}-s)^{2}}=1$, we have $d\neq\theta_{j}+m_{\nu}(\theta_{j})$. We show in Proposition 3.1 that for $\eta$ small enough, for all large $N$, there exists a unique right edge $d_{N}$ of $\text{supp}(\mu_{sc}\boxplus\mu_{A_{N}})$ in $]d-\eta;d+\eta[$. We derive the asymptotic distribution of eigenvalues in the vicinity of $d_{N}$. Before exposing our results, we need a few notations. Let $Ai(u)$ be the Airy function defined by

where the contour is from $\infty e^{5i\pi/6}$ to $\infty e^{i\pi/6}$. The Airy kernel (see e.g. ) is then given by

Let $\mathbf{A}_{x}$ be the operator acting on $L^{2}((x,\infty))$ with kernel $\mathbf{A}(u,v)$. The GUE Tracy-Widom distribution for the largest eigenvalue is ()

We refer to for the more complicated definition of the GUE distribution for the $k$ largest eigenvalues ($k>1$).

We first prove the following result. Let $k$ be a given fixed integer. Let $\lambda_{max}\geq\lambda_{max-1}\geq\cdots\lambda_{max-k+1}$ denote the $k$ largest of those eigenvalues of $M_{N}$ converging to $d.$

There exists $\alpha>0$ depending on $d_{N}$ only such that the vector

converges in distribution as $N\to\infty$ to the so-called Tracy-Widom GUE distribution for the $k$ largest eigenvalues.

Condition 4 is necessary to obtain Tracy-Widom asymptotics at the edges of the spectrum. If condition 4 fails e.g. at the top edge of the spectrum, meaning that the density of $\nu$ vanishes too fast at the edge, the limiting eigenvalue statistics at the edge can be proved to be Gaussian.

We now turn to the behavior of outliers. Let $\theta_{i}$ be a spiked eigenvalue with multiplicity $k_{i}$, such that $\int\frac{1}{(\theta_{i}-x)^{2}}d\nu(x)<1$. In , the authors prove that the spectrum of $M_{N}$ exhibits $k_{i}$ eigenvalues in a neighborhood of

Note that such a result is obtained when the support of $\nu$ has a finite number of connected components. However this assumption can be easily relaxed (see Remark 2.2). In Proposition 3.4, we prove that for $\epsilon>0$ small enough, for all large $N$, $\text{supp}(\mu_{sc}\boxplus\mu_{A_{N}})$ has a unique connected component $[L_{i}(N);D_{i}(N)]$ inside $]\rho_{\theta_{i}}-\epsilon;\rho_{\theta_{i}}+\epsilon[$. Define

It can be shown that for all large $N$, $\rho_{N}(\theta_{i})\in[L_{i}(N);D_{i}(N)]$ and $\rho_{N}(\theta_{i})=\frac{L_{i}(N)+D_{i}(N)}{2}+o(\frac{1}{\sqrt{N}}).$

To define the limiting correlation function at an outlier, we consider for $k=1,2,\ldots,$ the distribution $G_{k}(\cdot)$ given by

In other words, $G_{k}$ is the distribution of the largest eigenvalue of $k\times k$ GUE. It has been shown (see or e.g.) that

where $\mathbf{H}^{(k)}_{x}$ is the operator acting on $L^{2}((x,\infty))$ defined by the Christoffel Darboux kernel of some rescaled Hermite polynomials satisfying the orthogonality relationship $\int_{-\infty}^{\infty}p_{m}(x)p_{n}(x)e^{-\frac{1}{2}x^{2}}dx=\delta_{mn}$ . We refer the reader to , Section 1.2.2 for a more complete statement of this fact.

Let us denote by $\lambda_{max}$ the largest of the $k_{i}$ outliers around $\rho_{N}(\theta_{i})$.

There exists $c>0$ depending on $\theta_{i}$ and $\nu$ only such that

We actually prove that the $k_{i}$ outliers around $\rho_{N}(\theta_{i})$ fluctuate as the eigenvalues of a $k_{i}\times k_{i}$ GUE.

The contour $\Gamma_{0}$ is formed by two curves lying respectively to the right and left of : one goes from $\infty e^{i\frac{\pi}{4}}$ to $\infty e^{-i\frac{\pi}{4}}$ and the other from $-\infty e^{i\pm\frac{\pi}{4}}$ to $-\infty e^{-i\frac{\pi}{4}}.$ See Figure 1 below.

The article is organized as follows. In Section 2, we review the fundamental properties of the free convolution that we later need in the proof. Section 3 gives fine estimates on the comparison of the support of the spectral distribution of $M_{N}$ on the one hand and that of $\mu_{sc}\boxplus\nu$ on the other hand. These are the fundamental tools for the asymptotic analysis of eigenvalue statistics in Section 4. Therein the basic tool is a saddle point analysis of the correlation functions of the deformed GUE.

Free convolution by a semicircular distribution

on which $m_{\tau}$ is univalent. Let $K_{\tau}$ be its inverse function, defined on $m_{\tau}(D_{\alpha,\beta})$, and

Given two probability measures $\tau$ and $\nu$, there exists a unique probability measure $\lambda$ such that

on a domain where these functions are defined. The probability measure $\lambda$ is called the free convolution of $\tau$ and $\nu$ and denoted by $\tau\boxplus\nu$.

This phenomenon was first observed by D. Voiculescu under a genericity assumption in , and then proved in generality in Theorem 3.1. Later, a new proof of this result was given in , using a fixed point theorem for analytic self-maps of the upper half-plane. In , P. Biane provides a deep study of the free convolution by a semicircular distribution, based on this subordination property.

The previous results of allows to conclude that $\mu_{sc}\boxplus\nu$ is absolutely continuous with respect to the Lebesgue measure and to obtain the following description of the support.

$\Psi_{\nu}$ is a homeomorphism and, at the point $\Psi_{\nu}(t)$, the measure $\mu_{sc}\boxplus\nu$ has a density given by

The support of the measure $\mu_{sc}\boxplus\nu$ is the image of the closure of the open set $U_{\nu}$ by the homeomorphism $\Psi_{\nu}$. $\Psi_{\nu}$ is strictly increasing on $U_{\nu}$.

The following result will be useful later on.

If $t_{0}$ is a point in the complement of the support of $\nu$ where two components of the set $U_{\nu}$ merge into one, then

In , when $\nu$ is a compactly supported probability measure, the authors establish the following results.

Each connected component of $\overline{U_{\nu}}$ contains at least one connected component of ${\rm supp}(\nu)$.

We also need the following additional basic results.

If $t\notin{\rm supp}(\nu)$ is such that there exists $\delta>0$ such that

If $t^{\prime}\notin{\rm supp}(\nu)$ is such that there exists $\delta>0$ such that

Note that $f^{{}^{\prime\prime}}(s)=6\int\frac{d\nu(x)}{(s-x)^{4}}>0$ so that $f^{{}^{\prime}}$ is strictly increasing on $]t-\epsilon;t+\epsilon[$. Therefore if $-f^{{}^{\prime}}(t)=2\int\frac{d\nu(x)}{(t-x)^{3}}\leq 0$ then $f^{{}^{\prime}}>0$ on $]t;t+\epsilon[$ and $\int\frac{d\nu(x)}{(s-x)^{2}}>1$ for $s\in]t;t+\epsilon[$ which leads to a contradiction with (19). Similarly, one can prove (iv).$\Box$

In the rest of the article, since we deal with a measure $\nu$ satisfying (4), we have ${\rm supp}(\nu)\subset U_{\nu}$.

In , the authors prove that a precise localization of the spectrum of $M_{N}$ can be described thanks to the support of the free convolution $\mu_{sc}\boxplus\mu_{A_{N}}$. In this section, we recall some of their results that we need afterwards.

An outlier in the spectrum of $M_{N}$ is an eigenvalue of $M_{N}$ lying outside the support of $\mu_{sc}\boxplus\nu.$ As we now explain, it is possible to describe outliers thanks to the support of $\mu_{sc}\boxplus\mu_{A_{N}}$.

Throughout the rest of the article, we denote $U_{\nu}$, $H_{\nu}$, $\Psi_{\nu}$, $v_{\nu}$ and $p_{\nu}$ by $U$, $H$, $\Psi$, $v$ and $p$ respectively. We also denote $U_{\mu_{A_{N}}}$, $H_{\mu_{A_{N}}}$, $\Psi_{\mu_{A_{N}}}$, $v_{\mu_{A_{N}}}$, $p_{\mu_{A_{N}}}$ by $U_{N}$, $H_{N}$, $\Psi_{N}$, $v_{N}$ and $p_{N}$ respectively. Last, we define the probabily measure $\hat{\nu}_{N}$ by

It is easy to see that $\hat{\nu}_{N}$ weakly converges to $\nu$. We define

Furthermore, for any $\theta_{j}\in\Theta_{\nu}$, we set

Note that $\rho_{\theta_{j}}$ lies outside of the support of $\mu_{sc}\boxplus\nu$ according to (17). Define also

In , the authors obtain moreover the following inclusion of the support of $\mu_{sc}\boxplus\mu_{A_{N}}$.

In , the authors proved this theorem when the $\text{supp}(\nu)$ has a finite number of connected components. Nevertheless, it is still true in our more general setting as we prove in the following lines. We will use the following lemma in which proof does not care about the number of connected components of the supports.

Proof of Theorem 2.3: First, one can readily observe that if $x$ satisfies ${\rm dist}(x,{\rm supp}(\nu))\geq 1$ then $-m_{\nu}^{\prime}(x)\leq 1$. This implies that the open set $U$ is included in the compact set $\{x,\,{\rm dist}(x,{\rm supp}(\nu))\leq 1\}.$ Then we can choose $K$ large enough such that $\{x,\text{dist}(x,\overline{U}\cup\Theta_{\nu})\leq 1\}\subset[-K;K]$ and, since $\lim_{y\rightarrow\pm\infty}\Psi(y)=\pm\infty$ and $(\text{supp}(\mu_{A_{N}}\boxplus\mu_{sc}))_{N}$ are uniformly bounded,

Let $\epsilon>0$. Since $\Psi$ is uniformly continuous on $[-K;K]$, there exists $0<\alpha<1$ such that

Since according to Lemma 2.4, $\overline{U_{N}}\subset\{x,\text{dist}(x,\overline{U}\cup\Theta_{\nu})<\alpha/2\}$ for all large $N$, we have

Thus, using Lemma 2.2 for $\Psi_{N}$, we get that

Now, using the assumptions ($H_{3}$) on the spectrum of $A_{N}$, it is easy to see that $\Psi_{N}$ converges uniformly towards $\Psi$ on the compact set ${\cal A}_{\alpha/2,K}$. Moreover, since $\Psi$ is continuous on the compact set ${\cal A}_{\alpha,K-1}$, we have

Therefore since for all large $N$, $\sup_{{\cal A}_{\alpha/2,K}}|\Psi_{N}(x)-\Psi(x)|<m$ and using also Lemma 2.2 for $\Psi$, we get that for all large $N$, for all $x\in{\cal A}_{\alpha,K-1}$, $\Psi_{N}(x-\frac{\alpha}{2})<\Psi(x)<\Psi_{N}(x+\frac{\alpha}{2})$ and therefore

Then, the result readily follows from (24) and (25). $\Box$

In , the authors proved this theorem when the support of $\nu$ has a finite number of connected components; nevertheless it is still true in our more general setting since it follows from Theorems 2.3 and 2.2 and an exact separation phenomenon (see Theorem 7.1 in ) which proof does not care about the number of connected components of the support of $\nu$.

As we show in the next Section 4, the support of $\mu_{sc}\boxplus\mu_{A_{N}}$ plays a fundamental role in the study of the fluctuations of eigenvalues at the edges of the spectrum. Due to assumptions $(H_{2})$ and $(H_{3}),$ we are able to show that the supports of $\mu_{sc}\boxplus\nu$ and $\mu_{sc}\boxplus\mu_{A_{N}}$ exhibit very similar features at edges which are distant from outliers as we explain in Subsection 3.1 below. In Subsection 3.2, we prove that $\mu_{sc}\boxplus\mu_{A_{N}}$ has a connected component in the vicinity of each outlier. Subsections 3.3 and 3.4 are devoted to the proof of the propositions stated in Subsection 3.1 and Subsection 3.2.

The two following results will be fundamental for considering asymptotics of the correlation kernel at the edges of the support of $\mu_{sc}\boxplus\nu$.

Assume that for a sufficiently small $\epsilon>0$,

where for $t$ in a small neighborhood of $t_{0}$,

Similarly we have the following result involving the left edges of the support of $\mu_{sc}\boxplus\nu$.

Assume that for a sufficiently small $\epsilon>0$,

where for $t$ in a small neighborhood of $t_{0}$,

It is clear that, under the assumption (3) of Shcherbina (), Theorem 1.1 and (28) imply her result.

The following proposition will be fundamental to study the asymptotics of the correlation kernel in a neighborhood of any point of the support of $\mu_{sc}\boxplus\nu$ where the density vanishes.

2 In the vicinity of outliers

It turns out that the support of $\mu_{sc}\boxplus\mu_{A_{N}}$ exhibits a small connected component in the vicinity of each outlier.

Let $\theta_{i}$ be such that $\int\frac{d\nu(x)}{(\theta_{i}-x)^{2}}<1$ and $\rho_{\theta_{i}}=H(\theta_{i})$. Then, for $\epsilon>0$ small enough, for all large $N$, $\text{supp}(\mu_{sc}\boxplus\mu_{A_{N}})$ has a unique connected component $[L_{i}(N);D_{i}(N)]$ inside $]\rho_{\theta_{i}}-\epsilon;\rho_{\theta_{i}}+\epsilon[$. Moreover, setting $\rho_{N}(\theta_{i})=\frac{1}{N}\sum_{y_{j}\neq\theta_{i}}\frac{1}{\theta_{i}-y_{j}}+\theta_{i}$, we have

Thus, $\rho_{N}(\theta_{i})=\frac{L_{i}(N)+D_{i}(N)}{2}+o(\frac{1}{\sqrt{N}}).$

3 Some technical lemmas

In the proof of the previous propositions, we will use the following lemmas.

Let $t_{0}$ $\notin{\rm supp}(\nu)\cup\Theta$ be such that $\int\frac{1}{(t_{0}-x)^{2}}d\nu(x)=1$ and $\int\frac{1}{(t_{0}-x)^{3}}d\nu(x)\neq 0$. Then for small enough $\epsilon>0$, for all large $N$, there exists one and only one $t_{0}(N)\in]t_{0}-\epsilon;t_{0}+\epsilon[$ such that $\int\frac{1}{(t_{0}(N)-x)^{2}}d\mu_{A_{N}}(x)=1$. $t_{0}(N)$ satisfies

Proof: One can readily see that $t\notin\{\theta_{i},i=1,\ldots,J,\beta_{j},j=1,\ldots,N-r\}$ is in $U_{N}$ if and only if $P_{N}(t)>0$ where $P_{N}(t)$ is the polynomial defined by

Condition ($H_{3}$) on the spectrum of $A_{N}$ allows us to choose $\epsilon>0$ small enough such that for $N$ large enough $[t_{0}-2\epsilon;t_{0}+2\epsilon]$ is in the complement of the support of $\nu$ and the support of $\mu_{A_{N}}$. $P_{N}(t)=0$ for $t\in]t_{0}-\epsilon;t_{0}+\epsilon[$ if and only if

Since we have $\int\frac{1}{(t_{0}-x)^{3}}d\nu(x)\neq 0$, it readily follows that for $\epsilon>0$ small enough and for all $z$ such that $|z-t_{0}|\leq\epsilon$, $\int\frac{(z-x+t_{0}-x)}{(z-x)^{2}(t_{0}-x)^{2}}d\nu(x)\neq 0$. Therefore, there exists $C_{1}(\epsilon)>0$ and $C_{2}(\epsilon)>0$ such that for any $z$ such that $|z-t_{0}|\leq\epsilon$, $0<C_{1}(\epsilon)<|\int\frac{(z-x+t_{0}-x)}{(z-x)^{2}(t_{0}-x)^{2}}d\nu(x)|<C_{2}(\epsilon).$ Define on $\{z;|z-t_{0}|\leq\epsilon\}$,

Using Lemma 3.1, by Rouché theorem, for large $N$, the function

has exactly one zero $z_{0}$ in $\{z;|z-t_{0}|<\epsilon\}$. Since $\bar{z_{0}}$ is obviously a zero too, we can conclude that $z_{0}$ is real. Hence, for $\epsilon$ small enough, for all large $N$, $P_{N}$ has exactly one zero $t_{0}(N)$ in $]t_{0}-\epsilon;t_{0}+\epsilon[$ and

where $0<K_{1}(\epsilon)<|h(t_{0}(N))|<K_{2}(\epsilon).$

Proof: Since we assume that for any $x$ in $\text{supp}(\nu)$, $\int\frac{d\nu(s)}{(x-s)^{2}}>1$ (i.e $\text{supp}(\nu)\subset U$) and that $x_{0}\neq\theta_{j},\forall 1\leq j\leq J$, it readily follows that $x_{0}\notin\text{supp}(\nu)\cup\Theta.$ Let $[a;b]$ be such that $x_{0}\in]a;b[$, $[a;b]\subset]d_{1};d_{2}[.$ Since $\int\frac{d\nu(s)}{(x_{0}-s)^{2}}=1$ and there exists $\tau>0$ such that, $\forall x\in]x_{0}-\tau;x_{0}+\tau[\setminus\{x_{0}\},~{}~{}\int\frac{d\nu(s)}{(x-s)^{2}}>1$, the strict convexity of $z\mapsto\int\frac{d{\nu}(s)}{(z-s)^{2}}$ on $[a;b]$ implies that

For each i such that $\int\frac{1}{(\theta_{i}-x)^{2}}d\nu(x)<1$, for $\epsilon>0$ small enough, for all large $N$, $U_{N}\bigcap]\theta_{i}-\epsilon;\theta_{i}+\epsilon[=]t_{1}^{i}(N),t_{2}^{i}(N)[$ where $t_{1}^{i}(N)$ and $t_{2}^{i}(N)$ satisfy

with $\phi_{N}(t)=\frac{1}{1-\frac{N-r}{N}\int\frac{1}{(t-x)^{2}}d\hat{\nu}_{N}(x)-\frac{1}{N}\sum_{j\neq i}\frac{k_{j}}{(t-\theta_{j})^{2}}}$ and $1\leq\phi_{N}(t)\leq K(\epsilon)$ for any $t\in]\theta_{i}-\epsilon;\theta_{i}+\epsilon[$ .

$P_{N}(t)=0$ for $t$ such that $|t-\theta_{i}|<2\epsilon$ if and only if

Now, since $P_{N}(\theta_{i})>0$, it is clear that $U_{N}\bigcap]\theta_{i}-\epsilon;\theta_{i}+\epsilon[=]t_{1}^{i}(N),t_{2}^{i}(N)[$ . The proof of Lemma 3.5 is complete.$\Box$

4 Proof of Propositions 3.1, 3.2, 3.3 and 3.4

with $h(t)=\frac{1}{\int\frac{(t-x+t_{0}-x)}{(t-x)^{2}(t_{0}-x)^{2}}d\nu(x)}$ and $0<K_{1}(\tau)<|h(t)|<K_{2}(\tau),\forall t\in]t_{0}-\tau;t_{0}+\tau[.$ Moreover, for all large $N$,

Since according to Theorem 2.1, $\Psi_{N}$ is strictly increasing on $U_{N}$, we have

It readily follows from (33), (34), (35), (36), (15) and (16) that for any $\eta>0$ small enough, for all large $N$,

Now, for $t$ in a small neighborhood of $t_{0}$ and $N$ large enough let us define

The proof of Proposition 3.1 is complete $~{}~{}~{}~{}~{}~{}~{}\Box$

The proof of Proposition 3.2 is similar and left to the reader.

Using Lemma 3.2, we can deduce that for all large $N$,

Moreover since $\lim_{N\rightarrow+\infty}x_{0}(N)=x_{0}$, we have for all large $N$, $x_{0}(N)\in]\Psi^{-1}(u_{0}-\eta/2);\Psi^{-1}(u_{0}+\eta/2)[$ so that $u_{0}(N)=\Psi_{N}(x_{0}(N))\in]u_{0}-\eta;u_{0}+\eta[$ for all large $N$ by using oncemore Lemma 3.2. The proof is complete $\Box$

According to Lemma 3.5, for $\epsilon>0$ small enough, for all large $N$, $U_{N}\bigcap]\theta_{i}-\epsilon;\theta_{i}+\epsilon[=]t_{1}^{i}(N),t_{2}^{i}(N)[$ where $t_{1}^{i}(N)$ and $t_{2}^{i}(N)$ satisfy

with $\rho_{N}(\theta_{i}):=\frac{1}{N}\sum_{y_{j}\neq\theta_{i}}\frac{1}{\theta_{i}-y_{j}}+\theta_{i}$ and $\tau_{i}=2\sqrt{k_{i}}\sqrt{1-\int\frac{1}{(\theta_{i}-x)^{2}}d\nu(x)}.$ In the same way

Proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3

where $K_{N}$ is the so-called correlation kernel of the deformed GUE, which has been explicited by . We here state his result.

The correlation of the deformed GUE $M_{N}$ is given by the double complex integral:

where $\Gamma$ encircles the poles $y_{1},\ldots,y_{N}$ and $\gamma$ is a line parallel to the $y-$axis not crossing $\Gamma.$

At this point, it is worth mentioning that correlation functions and thus local eigenvalue statistics are invariant through conjugation of the correlation kernel. Indeed, one has that

for any non vanishing function $h$. This fact will be used many times in this article.

Before starting the asymptotic analysis, we list some important facts and notations that are needed hereafter.

Let $u_{0}$ be given. Assume that both $u$ and $v$ satisfy $|u-u_{0}|\leq N^{-{\delta}}$ for some $\delta>0$. Let us set

Note that $F_{u_{0}}$ is the first order approximation (as $N\to\infty$) of the true exponential term arising in both $z$ and $w$ integrals in the correlation kernel $K_{N}$. Indeed the true exponential term arising in both integrals is given by

We neglect for a while the fake singularity introduced by the logarithm (as $e^{F_{u_{0},N}}$ is holomorphic). By definition, critical points satisfy

and one can note that $F_{u_{0},N}^{\prime\prime}=1-\frac{1}{N}\sum_{i=1}^{N}\frac{1}{(z-y_{i})^{2}}$ does not depend on $u_{0}$. It is also convenient for the following to define the curve of critical points of both $F_{u}$ and $F_{u,N}$. Let us define

One can check that a critical point of $F_{u}$ with non null imaginary part lies on

For any $u\in\Psi(U)$, we denote by $z_{c}^{\pm}(u)$ these two critical points:

A critical point of $F_{u,N}$ with non zero imaginary part lies on

For any $u\in\Psi_{N}(U_{N})$, denote by $z_{c,N}^{\pm}(u)$ these two critical points of $F_{u,N}$:

We note that $F_{u,N}$ necessarily admits $N-1$ other critical points, which are real interlaced with the $y_{i}$’s. We disregard these critical points. Then one has that

Actually in all the cases we study, it turns out that the critical points, that we here denote by $\mathbf{z_{c}},$ lie on the real axis. We may therefore need to modify $F_{u,N}$ so that there is no singularity in the logarithm. It may happen in particular that $\exists\>1\leq i\leq N$, $y_{i}<\mathbf{z_{c}}<y_{i+1}$. However by the assumptions we have made, in all cases there exists $\epsilon>0$ such that $[\mathbf{z_{c}}-\epsilon,\mathbf{z_{c}}+\epsilon]$ contains no eigenvalue $y_{j},j=1,\ldots,N$. In that case we set

The contour $\Gamma$ will be split into two parts: $\Gamma_{1}$ lying to the left of $\mathbf{z_{c}}+\epsilon$ and $\Gamma_{2}$ to its right (encircling all the eigenvalues $y_{i}>\mathbf{z_{c}}+\epsilon$). The contour $\gamma$ will be chosen so that it lies to the left of $\mathbf{z_{c}}+\epsilon$. All these contours cross the real axis at a point where $F_{u,N}$ has no singularity. Note that with this new definition of $F_{u,N}$, it is still true that

Thus all the subsequent derivatives and the curve $\mathcal{C}_{N}$ are unchanged with this new definition. The asymptotic exponential term at $\mathbf{z_{c}}$ is then given by

2 Asymptotics of the correlation kernel at the edges of the support

We start from a right extremity point $d$ of a connected component of $\text{supp}(\nu\boxplus\mu_{sc})$ so that $p(x)=0,\forall x\in[d,d+\epsilon]$ for some small $\epsilon>0.$ We assume moreover that for any $\theta_{j}$ such that $\int\frac{d\nu(s)}{(\theta_{i}-s)^{2}}=1$, we have $d\neq\theta_{j}+m_{\nu}(\theta_{j})$. According to Proposition 3.1, such a point $d$ satisfies $d=H(\mathbf{z_{0}})$ where $\mathbf{z_{0}}$ is a real solution of

Since $\mathbf{z_{0}}\notin\text{supp}(\nu)\cup\Theta$, $(H_{3})$ implies that for all large $N$, one also has that $\inf_{k=1,\ldots,N}\text{dist}(\mathbf{z_{0}},y_{k})>0$. By Proposition 3.1, there exists a unique extremity point $d_{N}$ which is the right endpoint of a connected component of $\text{supp}(\mu_{N}\boxplus\mu_{sc})$ and such that $|d-d_{N}|\leq\epsilon$ for any $\epsilon.$ Then there exists a point $\mathbf{z_{N}}$ such that

Let $F_{d_{N},N}$ be defined as in (38) with $\mathbf{z_{c}}=\mathbf{z_{N}}.$ By definition, one has that $\mathbf{z_{N}}$ is the real degenerate critical point associated to $d_{N}:$

We assume that there exists a real number $M_{0}>0$ such that $x,y\geq-M_{0}.$ If $u_{0}$ is not the top edge of the support $\text{supp}(\mu_{A_{N}}\boxplus\mu_{\sigma})$, then $x$ and $y$ shall be bounded from above by $\epsilon_{0}N^{2/3}$ with $\epsilon_{0}$ small enough so that $u_{0}+\frac{\alpha x}{N^{2/3}}$ is smaller than the left edge of the next connected component of $\text{supp}(\mu_{A_{N}}\boxplus\mu_{\sigma})$. The associated rescaled correlation kernel is then

We now consider the asymptotics of the correlation kernel and prove that the rescaled kernel $\frac{\alpha}{N^{2/3}}K_{N}(u,v)$ uniformly converges to the Airy kernel when $-M_{0}\leq x,y\leq\epsilon_{0}N^{2/3}.$

Theorem 1.1 is an easy consequence of the following Proposition. Set

$\alpha$ is well defined using Lemma 2.3, (ii).

There exist constants $q,C,c>0$ such that for any $x,y\in[-M_{0},\epsilon_{0}N^{2/3}],$

where $\mathbf{A}$ denotes the Airy kernel.

By Cauchy’s theory and using the fact proved in Lemma 2.3 that

one deduces that $F_{u_{0},N}^{(3)}(\mathbf{z_{N}})\geq a_{i}/2$ and that there exist $a>0,M>0$ and a small $\delta$-neighborhood of $\mathbf{z_{N}}$ such that

We now rewrite the correlation kernel. To this aim, we split $\Gamma$ into two contours lying respectively to the left and to the right of $\mathbf{z_{N}}.$ This is possible as we assume that $\Delta:=\inf_{k=1,\ldots,N}\text{dist}(\mathbf{z_{0}},y_{k})>0$ and $|z_{N}-z_{0}|<\Delta/2$ for $N$ large enough. Denote by $\Gamma_{1}$ the part of the contour $\Gamma$ lying to the left of $\mathbf{z_{N}}$ and set $\Gamma_{2}:=\Gamma\setminus\Gamma_{1}.$ In the correlation kernel given by Proposition 4.1, along $\Gamma_{1}$, we first rewrite the singularity

which is valid provided the contour $\gamma$ remains to the right of $\Gamma_{1}.$ This then yields the following expression for the correlation kernel (up to a conjugation factor):

Let us first consider the leading term in the exponential defining $H$ and $G$ that is $F_{u_{0},N}$. By the choice of $u_{0}$, the two first derivatives of the exponential term vanish at the real point $\mathbf{z_{N}}$ so that standard saddle point analysis suggest that the ascent and descent contours shall be given by lines with direction $(2)i\pi/3$ through the critical point $\mathbf{z_{N}}$. This is true in a compact neighborhood of $\mathbf{z_{N}}$, as we see below. We ignore for a while the constraint that the contours do not cross each other. We first check that $\Gamma_{1}$ and $\gamma$ shall follow the directions $2i\pi/3$ or $i\pi/3$. To consider the constraint that they do not cross each other, we later modify these contours in a $N^{-1/3}$ neighborhood of $\mathbf{z_{N}}$. Using (41), there exists $\delta_{0}>0$ and $a=a(\delta_{0})$ such that for any $|s|\leq\delta_{0}$

One can then complete the $w$-contour by a line parallel to the imaginary axis. Indeed one can choose $\delta_{0}$ small enough so that $\mathbf{z_{N}}+\delta_{0}e^{i\pi/3}$ lies in the domain where $1>\frac{1}{N}\sum\frac{1}{|z-y_{i}|^{2}}.$ Thus there exists a constant $a^{\prime}>0$ such that

Because $d_{N}$ may not be the right edge of the support, we need to complete the $z-$contour $\Gamma_{2}$ to the right of $\mathbf{z_{N}}$ too. In this case, define

as long as $Z+x$ lies above $\mathcal{C}_{N}.$ This finishes the definition of the contours, apart from the constraint that the two contours cannot cross each other.

We now slightly modify the contours in a $N^{-\frac{1}{3}}$ neighborhood of $\mathbf{z_{N}}$ so that $\gamma$ does not cross $\Gamma_{1}$. Let $\epsilon>0$ (small) be fixed. The $w$ and $z$ contours do not go through $\mathbf{z_{N}}$ but instead follows an arc of circle of ray $\epsilon N^{-\frac{1}{3}}$ centered at $\mathbf{z_{N}}$ in order to avoid crossing each other (see Figure 2).

We now fix $q=\mathbf{z_{N}}+\frac{\epsilon}{2}N^{-\frac{1}{3}}$ where $\epsilon$ has been defined as above. By the estimates on the decay of $F_{u_{0},N}$ given in (55), we deduce the following.

Assume first that $|x|,|y|\leq M_{0}.$ Using (55), we first deduce that there exists $A>0$ such that

Let us now set $\gamma_{0}:=\{te^{i\pm\pi/3},\epsilon\leq t\leq\delta_{0}N^{1/3}\}\cup C_{\epsilon}$ where $C_{\epsilon}$ is the arc of circle centered at joining $\epsilon e^{-i\pi/3}$ and $\epsilon e^{i\pi/3}$. This contour is oriented from bottom to top. We now make the change of variables $w=\mathbf{z_{N}}+sN^{-\frac{1}{3}}$ where $s\in\gamma_{0}.$ We then obtain that

The last line is obtained by using the fact that

for some constant $a>0$. More detail can be found in Section 3 and we do not develop the computations here.

Similarly we define $\Gamma_{0}:=\{te^{i2\pm\pi/3},\epsilon\leq t\leq\delta_{0}N^{1/3}\}\cup C^{\prime}_{\epsilon}$ where $C^{\prime}_{\epsilon}$ is the arc of circle centered at joining $\epsilon e^{-2i\pi/3}$ and $\epsilon e^{2i\pi/3}$. This contour is again oriented from bottom to top.

where $t$ describes the contour $\Gamma_{0}$ formed with the two half lines in the complex plane with angle $e^{\pm 2i\pi/3}$ with respect to the real axis. The contour is also oriented from bottom to top. We recall that $\alpha$ has been chosen as

We then deduce that for $|x|,|y|\leq M_{0}$, one has that

We can now conclude to the asymptotic behavior of the rescaled correlation kernel $K_{N}^{(l)}(u,v)e^{q(y-x)N^{\frac{1}{3}}}$ when $x$ and/or $y$ are allowed to grow unboundedly positive. Indeed for this part of the kernel we do not need to bound $x$ and $y$ from above by $\epsilon_{0}N^{2/3}$. As by construction the two contours $\Gamma_{1}$ and $\gamma$ lie respectively to the left (resp. right) strictly of $q$, one can deduce (copying the arguments developed in Section 3 ) that there exist constants $C,c>0$ such that

Note that (65) also holds true (modifying the constants $C,c$ if needed) when $|x|,|y|\leq M_{0}$.

for some constant $C>0$. As $|y|,|x|\leq\epsilon_{0}N^{2/3}$, we choose $\epsilon_{0}>0$ small enough so that there exists a constant $C^{\prime}>0$ so that

Combining (67), (68) and (65) then yields Proposition 4.2. $\square$

3 Proof of Theorem 1.2

Consider a spike $\theta_{i_{1}}$ of multiplicity $k_{i_{1}}$ such that $\int\frac{1}{(\theta_{i_{1}}-y)^{2}}d\nu(y)<1$. Then $\theta_{i_{1}}$ makes $k_{i_{1}}$ outliers separate from the bulk at $\rho(\theta_{i_{1}})$ asymptotically, with $\rho(z):=z+\int\frac{1}{z-y}d\nu(y)$. We recall that $\theta_{i_{1}}$ is such that $\text{dist}(\rho(\theta_{i_{1}}),\text{supp}(\mu_{sc}\boxplus\nu))>0.$ Thus there exist (possibly) $\mathbf{z_{N}}$ and $\mathbf{w_{N}}$ such that $H_{N}(\mathbf{z_{N}})$ and $H_{N}(\mathbf{w_{N}})$ are respectively the right and left endpoints of the connected component of $\text{supp}(\mu_{sc}\boxplus\mu_{A_{N}})$ which is respectively on the left hand side and right hand side of $\rho(\theta_{i_{1}})$ and we have $\mathbf{z_{N}}<\theta_{i_{1}}<\mathbf{w_{N}}.$ If there is no connected component of $\text{supp}(\mu_{sc}\boxplus\mu_{A_{N}})$ to the right respectively the left of $\rho(\theta_{i_{1}})$, we then set $\mathbf{w_{N}}=+\infty,$ respectively $z_{N}=-\infty.$