Edge Universality of Beta Ensembles

Introduction

Eigenvalues of random matrices were envisioned by Wigner as universal models for highly correlated systems. A manifestation of this general principle is the universality of random matrix statistics, i.e., that the eigenvalue distributions of large matrices are universal in the sense that they depend only on the symmetry class of the matrix ensemble, but not on the distributions of the matrix elements. These universal eigenvalue distributions are different for eigenvalues in the interior of the spectrum and for the extreme eigenvalues near the spectral edges. In this paper, we will focus on the edge universality.

Let $\lambda_{N}$ be the largest eigenvalue of an $N\times N$ random Wigner matrix with normalization chosen such that the bulk spectrum is $$. The probability distributions of$\lambda_{N}$ for the classical Gaussian ensembles are identified by Tracy and Widom to be

where $F_{\beta}(s)$ can be computed in terms of Painlevé equations and $\beta=1,2,4$ corresponds respectively to the classical orthogonal, unitary or symplectic ensemble. The edge universality means that the distributions of $\lambda_{N}$ are given by $F_{\beta}$ for non Gaussian ensembles as well. In fact, this holds not only for the largest eigenvalue, but the joint distributions of any finitely many “edge eigenvalues” are universal as well.

The edge universality for a large class of Wigner matrices was first proved via the moment method by Soshnikov for unitary and orthogonal ensembles. This method requires that the distribution of the matrix elements be symmetric. The symmetry assumption was partially removed in and it was completely removed in . In addition to the symmetry assumption, the moment method also requires that sufficient high moments of the matrix elements be finite. This assumption was greatly relaxed in and it was finally proved by Lee and Yin that essentially the finiteness of the fourth moment is the sufficient and necessary condition for the Tracy-Widom edge universality to hold (an almost optimal necessary condition was established earlier in ).

We now turn to the edge universality for invariant ensembles. These are matrix models with probability density on the space of $N\times N$ matrices $H$ given by $Z^{-1}e^{-N\beta{\rm Tr}V(H)/2}$ where $V$ is a real valued potential and $Z=Z_{N}$ is the normalization. The parameter $\beta$ is determined by the symmetry class of $H$. The probability distribution of the ordered eigenvalues of $H$ on the simplex determined by $\lambda_{1}\leqslant\dots\leqslant\lambda_{N}$ is given by

For classical invariant ensembles, i.e., $\beta=1,2,4$, it is well-known that the correlation functions can be expressed in terms of orthogonal polynomials. Historically, they have been first analyzed in the bulk. The analysis at the edges is not a straightforward generalization of that in the bulk and serious technical hurdles had to be overcome. Nevertheless, the edge universality was proved by Deift-Gioev for general polynomial potentials, by Pastur-Shcherbina and Shcherbina for real analytic, even potentials.

We now compare the notions of edge and bulk universality. The edge universality refers to the distributions of individual eigenvalues. However, according to Wigner’s original vision, the bulk universality concerns differences of neighboring eigenvalues, i.e., gap distributions. The bulk universality is often formulated in terms of local correlation functions. These two notions are equivalent only after a certain averaging in the energy parameter. Strictly speaking, there are three notions of bulk universality: (i) in a weak sense which allows for energy averaging; (ii) correlation function universality at a fix energy; (iii) gap universality at a fixed label $j$. Clearly, universality in the sense (ii) or (iii) implies (i).

The bulk universality in the sense of (ii) for classical invariant ensembles was proved in using methods related to orthogonal polynomials. For Wigner ensembles, universality for Hermitian matrices in the sense (ii) was proved in and for all symmetry classes in the sense (i) in . The gap universality, i.e., (iii), is in fact much harder to obtain; it was proved only recently in both for invariant and Wigner ensembles using new ideas from parabolic regularity theory (the special case of hermitian matrices with the first four moments of the matrix elements matching those of GUE was proved earlier in ). The bulk universality for log-gases for general $\beta$ was proved in the sense (i) and (iii) in . The bulk universality in the sense (ii) for Wigner ensembles with $\beta\not=2$ and for log-gases with $\beta\not\in\{1,2,4\}$ remains open problems.

Returning to the edge universality, we will establish the following two results in this paper: (1) edge universality for $\mathscr{C}^{4}$ potentials and for all $\beta\geqslant 1$; (2) edge universality for generalized Wigner matrices (these are matrices with independent but not necessarily identically distributed entries, see Definition 2.6). An important ingredient of the proof will be an optimal location estimate for the particles up to the edge, for external potentials of class $\mathscr{C}^{4}$. This rigidity will also allow us to remove the analyticity assumption from previous results about bulk universality .

We now outline the technique used in this paper. For the edge universality of invariant ensembles, the basic idea is to consider a local version of the log-gas (1.1). This is the measure on $K$ consecutive particles that is obtained by fixing all other particles which act as boundary conditions. Following the standard language in statistical physics, we will refer to these local measures as local log-gases. Our core result is the “uniqueness” of this local measure in the limit $K\to\infty$ assuming that the boundary conditions are “good”. By uniqueness, we mean that the distributions of the particles far away from the boundaries are independent of choice of the “good” boundary conditions. This idea first appeared in for proving the bulk universality of log-gases. However, the uniqueness of the local Gibbs state in the bulk was defined slightly differently in ; only the gap distributions were required to be independent of the boundary conditions.

It is well-known that the uniqueness of local Gibbs measures in the thermodynamical limit is closely related to the decay of correlation functions. The work of Gustavsson for the special $\beta=2$ and Gaussian case (i.e., the GUE case) indicates that in general the point-point correlation function decays only logarithmically in the bulk, i.e., $\langle\lambda_{i};\lambda_{j}\rangle_{\mu_{\beta}}\sim\log|i-j|$. Gibbs measures with such a slow decay are typically not unique in the usual sense. The key reason why we were able to prove the uniqueness of the gap distributions of local log-gases in the bulk is the observation that the point-gap correlation, $\langle\lambda_{i};\lambda_{j}-\lambda_{j+1}\rangle_{\mu_{\beta}}\sim\partial_{j}\langle\lambda_{i};\lambda_{j}\rangle_{\mu_{\beta}}$ is expected to decay much faster due to the simple reason that $\partial_{j}\log|i-j|\sim 1/|j|$. In real statistical physics system, however, it is very difficult to compute derivatives of correlation functions unless they are expressed almost explicitly by some expansion method. The Dirichlet form inequality , a main tool in , allows us to take advantage of the fact that the observables are functions of the gaps.

In the subsequent work , the correlation functions were expressed in terms of off-diagonal matrix elements of heat kernels describing random walks in random environments. This representation in a lattice setting was given in . In a slightly different formulation it already appeared in the earlier paper of Naddaf and Spencer , which was a probabilistic formulation of the idea of Helffer and Sjöstrand . Using this representation, the decay of the point-gap correlation amounts to the Hölder continuity of the heat kernel for the random walk dynamics . We note that the jump rates in this random walk dynamics are long ranged and contain short distance singularities depending on the stochastically driven environment. The proof of the Hölder continuity in requires extending the De Giorgi-Nash-Moser type method of Caffarelli, Chan and Vasseur to the singular coefficient case and providing a priori estimates such as rigidity and level repulsion.

It should be stressed that, despite these efforts, only gap distributions but not those of individual eigenvalues were identified in . Edge universality, however, is exactly about individual eigenvalues and not about gaps. The surprising fact is that correlation functions of log-gases decay as a power law near the edges! Thus we do not need the Hölder regularity argument from to analyze the edges. Instead, in this paper we rely on the energy method from parabolic PDE’s and on certain new Sobolev type inequalities for nonlocal operators to prove the decay of off-diagonal elements of the heat kernel. For this purpose, we will need rigidity and level repulsion estimates near the edges. We will extend the multi-scale analysis of the loop equation, first appeared in , in two directions. First, this analysis will be performed along the whole spectrum, including the edge, where the change of scaling poses a major difficulty; second, analyticity of the external potential is not required, thanks to a new analysis of the loop equation.

For the edge universality of Wigner ensembles, we will use the idea of the local relaxation flow initiated in and the Green function comparison theorem from . This theorem can be used for both the bulk or the edge universality. In particular, the eigenvalue distributions of two Wigner ensembles near the edges are the same provided that the variances of the matrix elements of the two ensembles are identical. This implied the edge universality for Wigner matrices .

On the other hand, if the variances of the matrix elements are allowed to vary, then the matrix cannot be matched to a Gaussian Wigner matrix. Thus the edge universality for generalized Wigner matrices cannot be proved directly with the Green function comparison theorem. Using the uniqueness of local log-gases, we can identify the distributions of the edge particles in the Dyson Brownian Motion (DBM). This implies the edge universality for general classes of Gaussian divisible ensembles with varying variance. Finally, we will use the Green function comparison theorem to bridge the gap between generalized Wigner matrices and their Gaussian divisible counterparts.

We emphasize that the uniqueness of local log-gases plays a central role both in the edge universality of log-gases and in our analysis of edge points in DBM. For log-gases, it is natural to localize the problem so that the external potential can be replaced by its first order approximation and thus it becomes universal after scaling. However, localization of the measure in general introduces very large errors in strongly correlated systems. The key observation is that there are strong cancellations in the effective potential for “good” boundary conditions. The significance of the local log-gases in the proof of proof of universality for Wigner matrices is subtler, and will be explained in details in Section 5.

Main results

We will have two related results, one concerns the generalized Wigner ensembles, the other one the general beta ensembles.

Consider the probability distribution on $\Xi_{N}$ given by

for some $\alpha>0$, if $|x|$ is large enough. It is known that under these (in fact, even weaker) conditions the measure is normalizable, $Z^{(N)}<\infty$. Moreover, the averaged density of the empirical spectral measure, defined as

converges weakly to a continuous function $\varrho=\varrho_{V}$, the equilibrium density, with compact support. We assume that $\varrho$ is supported on a single interval $[A,B]$, and that $V$ is regular in the sense of . We recall that $V$ is regular if its equilibrium density $\varrho$ is positive on $(A,B)$ and vanishes like a square root at each of the endpoints of $[A,B]$, that is

for some constants $s_{A},\,s_{B}>0$. We remark that this regularity assumption is not a strong constraint; regular potentials $V$ form a dense and open subset in the space of the potentials with a natural topology .

Let the limiting classical location of the $k$-th particle, $\gamma_{k}=\gamma_{k}(N)$, be defined by

We will be interested in the usual $n$-point correlation functions, generalizing $\varrho_{1}^{(N)}$, and defined by

where \mbox{\boldmath\lambda}^{(\sigma)}=(\lambda_{\sigma(1)},\dots,\lambda_{\sigma(N)}), with $\lambda_{\sigma(1)}<\dots<\lambda_{\sigma(N)}$. Our main result is the following.

In the following theorem, we consider two regular potentials, $V$ and $\widetilde{V}$, such that their equilibrium densities $\varrho_{V}$ and $\varrho_{\widetilde{V}}$ are supported on a single interval. Without loss of generality (by applying a simple scaling and shift), we may also assume that the singularities at the left edge match and both occur at $A=0$, with the same constant $s_{A}=1$:

Let $\beta\geqslant 1$ and $V$, $\widetilde{V}$ be $\mathscr{C}^{4}$, regular and satisfy (2.3), (2.4). Assume that the equilibrium density $\varrho_{V}$ and $\varrho_{\widetilde{V}}$ are supported on a single interval and satisfy (2.8).

Remark. Note that one may define $\gamma_{j}$ in (2.6) with respect to the measure $\varrho_{V}$ or $\varrho_{\widetilde{V}}$, it does not make any difference in the above theorem when $\kappa<2/5$: from (2.8) one obtains $\gamma_{j}-\widetilde{\gamma}_{j}=\OO\left(\left(j/N\right)^{4/3}\right)$, which is of smaller order than the scale $N^{-2/3}j^{-1/3}$ detected in (2.9). We also remark that Theorem 2.1 is formulated for points near the lower spectral edge $A$, but a similar statement holds near the upper spectral edge $B$.

The first results on edge universality for invariant ensembles concerned the classical values of $\beta=1,2,4$. The case $\beta=2$ and real analytic $V$ was solved in . The $\beta=1,4$ cases are considerably harder than $\beta=2$. For $\beta=1,4$ universality was first solved for polynomial potentials in , then the real analytic case for $\beta=1$ in , which also give an alternative proof for $\beta=2$. Finally, independently of our work with a completely different method, edge universality for any $\beta>0$ and convex polynomial $V$ was recently proved in .

Choosing $S=\llbracket 1,m\rrbracket$ and $\widetilde{V}(x)=x^{2}$ in the previous theorem allows us to identify the universal distribution from Theorem 2.1 with the Tracy-Widom distribution with parameter $\beta>0$. This distribution can be represented via the stochastic Airy operator. We refer to for its proper definition, the Hilbert space it acts on, and the proof that its smallest eigenvalues describe the asymptotic edge fluctuations of the Gaussian beta ensembles.

Theorem 2.1 can be used to show Gaussian fluctuations for the points in an intermediate distance from the edge. Indeed, such fluctuations were proved by Gustavsson in in the $\beta=2$ Gaussian case (GUE) for all eigenvalues, and this was extended $\beta=1$ and $4$ in . Combining these results with Theorem 2.1 immediately gives the following statement (here $k\sim N^{\vartheta}$ means $\log k/\log N\to\vartheta$).

Let $\beta=1,2$ or $4$ and the potential $V$ be $\mathscr{C}^{4}$, regular such that the equilibrium density $\varrho_{V}$ is supported on a single interval and satisfies (2.8). Consider the measure $\mu_{\beta,V}^{(N)}$. We define

where $c=(3/2)^{1/3}\pi\beta^{1/2}$. Fix $\kappa<2/5$. Then for any sequence $i=i_{N}\to\infty$, with $i\leqslant N^{\kappa}$, we have $X_{i}\to\mathcal{N}(0,1)$ in distribution.

Moreover, for some fixed $m>0$ and $\delta\in(0,2/5)$, let $k_{1}<\dots<k_{m}$ satisfy $k_{1}\sim N^{\delta}$, and $k_{i+1}-k_{i}\sim N^{\vartheta_{i}}$, $0<\vartheta_{i}<\delta$. Then $(X_{k_{1}},\dots,X_{k_{m}})$ converges to a Gaussian vector with covariance matrix $\Lambda_{ij}=1-\delta^{-1}\max\{\vartheta_{k},i\leqslant k<j\}$ if $i<j$, $\Lambda_{ii}=1$.

We note that if Gustavsson’s result on Gaussian fluctuations were known for the general Gaussian beta ensembles, then this corollary would prove a central limit theorem for general beta ensembles near the edge.

An important element in the proof of Theorem 2.1 consists in proving the following rigidity estimate asserting that any particle $\lambda_{k}$ is very close to its limiting classical location. For any $k\in\llbracket 1,N\rrbracket$ we define

where $A\sim B$ means $c\leqslant A/B\leqslant C$. More precisely, by the square-root singularity of $\varrho$ near the left edge,

and similar asymptotics hold near the right edge. The following theorem states that all particles will be close to their classical locations on this scale, up to a factor $N^{\xi}$ with an arbitrary small exponent $\xi>0$. Following , we will call such a precise bound on the locations of particles a rigidity estimate. The rigidity estimate in some weaker forms has already been used as a fundamental input to prove the universality for Wigner matrices. It also played a key role in the proof of the bulk universality for the log-gases in . The following result extends the rigidity estimate from the bulk to the edges, and removes the analyticity assumption.

Let $\beta>0$, $V$ be $\mathscr{C}^{4}$, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (2.3), (2.4). For any $\xi>0$, there are constants $c>0$ and $N_{0}$ such that for any $N\geqslant N_{0}$ and $k\in\llbracket 1,N\rrbracket$ we have

Related bounds on the concentration of the empirical density on a scale far from the optimal one (2.11) were established previously , see also references in .

Thanks to Theorem 2.4, bulk universality holds for beta ensembles, as stated in , without the analyticity assumption.

Let $V$ be $\mathscr{C}^{4}$, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (2.3), (2.4). Then the following two results hold.

Here $\varrho_{sc}(E)=\frac{1}{2\pi}\sqrt{4-E^{2}}$ is the Wigner semicircle law and $\varrho_{{\rm Gauss},n}^{(N)}$ are the correlation functions of the Gaussian $\beta$-ensemble, i.e. with $V(x)=x^{2}$.

where $c_{k}^{\mu}=\varrho^{(\mu)}(\gamma_{k})$ and $c_{m}^{\rm Gauss}=\varrho_{sc}(\gamma_{m}^{\rm Gauss})$ with $\gamma_{m}^{\rm Gauss}$ being the $m$-th quantile of the semicircle law defined by $\int_{-2}^{\gamma_{m}^{\rm Gauss}}\varrho_{sc}(x){\rm d}x=m/N$.

Part $(i)$ was proved in under the assumption that $V$ is analytic, a hypothesis that was only required for proving rigidity in the bulk of the spectrum. Theorem 2.4 proves that $V$ of class $\mathscr{C}^{4}$ is sufficient for rigidity, and the proof of the uniqueness of the Gibbs measure is identical to . The result in these papers were stated in a limiting form, as $N\to\infty$, and for smooth observables $O$, but the proofs hold for any continuously differentiable $O$ and with an effective error bound of order $N^{-\chi}$ with some $\chi>0$ as well. The statement $(ii)$ holds for the same reason, being previously proved for analytic $V$ in . ∎

We finally remark that while the rigidity estimate (2.11) holds for any $\beta>0$, the edge universality in Theorem 2.1 was stated only for $\beta\geqslant 1$. This restriction is mainly due to that the DBM dynamics (8.1) is known to be well-posed only for $\beta\geqslant 1$. We believe that this restriction can be removed, but we will not pursue this issue in this paper.

2 Edge universality of the generalized Wigner matrices

We now define the generalized Wigner ensembles. Let $H=(h_{ij})_{i,j=1}^{N}$ be an $N\times N$ complex Hermitian or real symmetric matrix where the matrix elements $h_{ij}=\bar{h}_{ji}$, $i\leqslant j$, are independent random variables given by a probability measure $\nu_{ij}$ with mean zero and variance $\sigma_{ij}^{2}\geqslant 0$;

The distribution $\nu_{ij}$ and its variance $\sigma_{ij}^{2}$ may depend on $N$, but we omit this fact in the notation. We also assume that the normalized matrix elements satisfy a uniform subexponential decay,

with some fixed constant $\vartheta$, uniformly in $N,i,j$.

The matrix ensemble $H$ defined above is called generalized Wigner matrix if the following assumptions hold on the variances of the matrix elements (2.12)

There exist two positive constants, $C_{1}$ and $C_{2}$, independent of $N$ such that

For Hermitian ensembles, we additionally assume that for each $i,j$ the $2\times 2$ covariance matrix

where $\mu_{G}$ is the standard Gaussian GOE or GUE ensemble, depending on the symmetry class of $H$ (It is well-known that $\mu_{G}$ is also given by (2.2) with potential $V(x)=\frac{1}{2}x^{2}$ and with the choice $\beta=1,2$, respectively).

This theorem immediately implies analogues of Corollaries 2.2 and 2.3 in the case of symmetric or Hermitian generalized Wigner ensembles.

Edge universality for Wigner matrices was first proved in assuming symmetry of the distribution of the matrix elements and finiteness of all their moments. In the consequent works, after partial results in , the symmetry condition was completely eliminated . The moment condition was improved in and the optimal result was obtained in . All these works heavily rely on the fact that the variances of the matrix elements are identical. The main point of Theorem 2.7 is to consider generalized Wigner matrices, i.e., matrices with non-constant variances. In fact, it was shown in that the edge statistics for any generalized Wigner matrix are universal in the sense that they coincide with those of a generalized Gaussian Wigner matrix with the same variances, but it was not shown that the statistics are independent of the variances themselves. Theorem 2.7 provides this missing step and thus it proves the edge universality in the broadest sense.

Local equilibrium measures

Recall that the support of the equilibrium density $\varrho$ was denoted by $[A,B]$. Without loss of generality, by a shift we set $A=0$ and we will study the particles near the lower edge of the support. Fix a small exponent $\delta$ and a parameter $K=K_{N}$ satisfying

Denote by $I=\llbracket 1,K\rrbracket$ the set of the first $K$ indices. We will distinguish the first $K$ particles from the rest by renaming them as

Note that the particles keep their original indices. We recall the notation $\Xi^{(N)}$ for the simplex (2.1). In short we will write

These points are always listed in increasing order and we will refer to the $y$’s as the external points and to the $x$’s as internal points. We will fix the external points (often called boundary conditions) and study the conditional measures on the internal points. Note that for any fixed ${\bf{y}}\in\Xi^{(N-K)}$, all $x_{j}$’s lie in the open configuration interval, denoted by

Define the local equilibrium measure (or local measure in short) on $J^{K}$ with boundary condition ${\bf{y}}$ by

Here $V_{\bf{y}}(x)$ can be viewed as the external potential of a log-gas of the points $\{x_{i}:i\in I\}$. Although this is the natural local measure, it does not have good uniform convexity in the regime $x_{1}\ll 0$. It is more convenient to consider the following modified measure $\sigma$ and its local version $\sigma_{\bf{y}}$. For the proof of the universality of the original measure $\mu$ it will actually be sufficient to consider only the local measure $\sigma_{\bf{y}}$.

We will fix a small parameter $\xi>0$ whose actual value is immaterial; it will be used to provide an multiplicative error bar of size $N^{C\xi}$ in various estimates on the location of the particles. We will not carry $\xi$ in the notation and at the end of the proof it can be chosen sufficiently small, depending on all other exponents along the argument.

We introduce a confined measure by adding an extra quadratic potential $\Theta$ to prevent the $x_{i}$’s from deviating far in the left direction:

The local version of the measure $\sigma$ is defined in the obvious way,

For technical reasons we will also need the following variants of $\sigma$ and $\sigma_{\bf{y}}$ where we added slightly less convexity through $\Theta$:

The measures $\sigma$, $\widehat{\sigma}$ and their local versions depend on the parameters $V,\beta,K$ and $\xi$ but we do not carry this dependence in the notation.

Rigidity estimates proved for the global measure $\mu$ (Theorem 2.4) also hold for the local measures $\sigma_{\bf{y}}$ provided ${\bf{y}}$ lies in the set of “good” boundary conditions that is defined as follows:

The rigidity exponent $\xi$ will always be chosen much smaller than the exponent $\delta$ in (3.1). This guarantees that the typical length of the configuration interval, $|J|\sim\gamma_{K}-\gamma_{1}\geqslant c(K/N)^{2/3}$, be bigger than the largest rigidity precision, $N^{-\frac{2}{3}+\xi}$.

We will need the following two modifications of ${\mathcal{R}}$. The first one requires that $x_{k}$ be good in an expectation sense w.r.t. $\sigma_{\bf{y}}$, and that $x_{1}$ is not too negative. Thus we define the set

Notice that for technical reasons to be clear later on the constraint on $x_{1}$ is w.r.t. the measure $\widehat{\sigma}_{\bf{y}}$. This condition will be important in Sect. 6.5.

Another modification adds the condition of a level repulsion near the boundary, i.e., we define

In the following theorems we establish rigidity and level repulsion estimates for the local log-gas $\sigma_{\bf{y}}$ with good boundary conditions ${\bf{y}}$ up to the spectral edges. These theorems extend similar estimates for the local measure $\mu_{\bf{y}}$ in the bulk of the spectrum established in to the edges for the measure $\sigma_{\bf{y}}$.

Fix $\beta,\xi>0$ and, using the above notations, assume that ${\bf y}\in\mathcal{R}_{K}^{*}(\xi)$. Then there exists constants $C,c>0$ (independent of ${\bf y},K$) such that for large enough $N$ we have, for any $k\in I$, and $u>0$,

As a side comment we remark that the Gaussian decay in (3.7) is an artifact of the additional confinement in the local measure $\sigma_{\bf{y}}$. For the measures $\mu$ or $\mu_{\bf{y}}$, the tail probability of $x_{1}$ has a slower decay $\exp{[-C(\gamma_{1}-x_{1})^{3/2}]}$ in the regime $x_{1}\ll\gamma_{1}$ in accordance with the tail behaviour of the Tracy-Widom law (for the Gaussian beta ensemble, see for a detailed analysis of the edge tail behavior). However, Theorem 3.3 below asserts that $\sigma_{\bf{y}}$ has the correct distribution when $x_{1}-\gamma_{1}\sim N^{-\frac{2}{3}}$.

We also have the following level repulsion estimates. Similar bounds for the measure $\mu_{\bf{y}}$ in the bulk were proved in .

Let $\beta>0$, let $\xi$ be an arbitrary fixed positive constant and assume that $K$ satisfies (3.1). Then there are constants $C,c>0$ such that for ${\bf{y}}\in{\mathcal{R}}={\mathcal{R}}_{K}(\xi)$ and for any $s>0$ we have

Note that these two bounds are complementary. The first one gives optimal level repulsion for arbitrary small $s$, but the constant $K^{2}$ is not optimal. The second bound improves this constant but at the expenses of an exponentially small additive error.

We remark that statements similar to (3.9) hold for any gap $x_{i+1}-x_{i}$, not only for the last one with $i=K$. The proofs are very similar, after conditioning on the points $x_{i+1},x_{i+2},\dots,x_{K}$ being close to their classical locations.

We will prove the rigidity and level repulsion results only for $\sigma_{\bf{y}}$ since these bounds are needed in the proof of the main theorems. The proof of the level repulsion bounds for $\sigma_{\bf{y}}$, however, verbatim applies to $\mu_{\bf{y}}$. For the rigidity bound, from Theorem 3.1 there exists a set $\mathcal{Y}$ of almost full $\mu$-measure such that for any ${\bf{y}}\in{\mathcal{Y}}$ we have

The role of the confinement in the definition of $\sigma$ is to prevent the first particle $x_{1}$ to be very negative, since it would destroy the good convexity bound on the Hessian. The reason we have to introduce $\sigma$ and $\sigma_{\bf{y}}$ is that in a technical step (establishing rigidity for the interpolation between local equilibrium measures with two different boundary conditions, see Section 8) we need a superexponential decaying tail probability of the rigidity estimate. We establish such bound only for the confined measure $\sigma_{\bf{y}}$ and not for $\mu_{\bf{y}}$.

Our main technical result, Theorem 3.3 below, asserts that, for $K$ in a restricted range, the local gap statistics is essentially independent of $V$ and ${\bf{y}}$ for good boundary conditions ${\bf{y}}$ (see (3.4)). For a fixed ${\bf{y}}\in{\mathcal{R}}$, we define the classical locations $\alpha_{j}=\alpha_{j}({\bf{y}})$ of $x_{j}$ by the formula

i.e., $\alpha_{j}$’s are the $j$-th $(K+1)$-quantiles of the density in $J_{\bf{y}}$. Recall that the support of $\varrho$ starts from $A=0$ even though the configuration interval starts from minus infinity.

The core universality result on the local measures is the following theorem. It compares two local measures with potentials $V$ and $\widetilde{V}$ and external configurations ${\bf{y}}$ and $\widetilde{\bf{y}}$. For notational simplicity, we will use tilde to refer to objects related to the measure $\widetilde{\mu}:=\mu_{\widetilde{V}}$.

Let $\beta\geqslant 1$ and $V$, $\widetilde{V}$ be $\mathscr{C}^{4}$ be regular and satisfy (2.3) and (2.4). Assume that the equilibrium density $\varrho_{V}$ and $\varrho_{\widetilde{V}}$ are supported on a single interval and satisfy (2.8). Fix small positive parameters $\xi,\delta>0$ and a parameter $0<\zeta<1$ that satisfy

with a sufficiently large universal constant $C_{0}$, and assume that

We remark that, thanks to the conditions (2.8) and (3.12), the points $\alpha_{j}$ and $\widetilde{\alpha}_{j}$ (defined by (3.10) with $\varrho$ and $\widetilde{\varrho}$) in (3.13) can both be replaced by $\gamma_{j}$. To see this, we claim that for any ${\bf{y}}\in{\mathcal{R}}_{K}$ we have

and these estimates are more accurate than the precision detected by the smooth observable $O$ in (3.13) for any $j\leqslant K^{\zeta}$. To prove (3.14), we recall $\gamma_{K}\sim N^{-2/3}K^{2/3}$ and for ${\bf{y}}\in{\mathcal{R}}$, we have $|y_{K+1}-\gamma_{K+1}|\leqslant N^{-2/3+\xi}K^{-1/3}$. Since the density has a square root singularity near $A=0$ (2.5), by assumption $K\geqslant N^{\delta}\gg N^{\xi}$ we have for ${\bf{y}}\in{\mathcal{R}}$ that

Therefore, for ${\bf{y}}\in{\mathcal{R}}$ we obtain that

As a consequence of the proof of Theorem 3.3, we also have the following correlation decay estimate.

Let $\beta\geqslant 1$, $V$ be $\mathscr{C}^{4}$, regular, and satisfy (2.3), (2.4). Assume that $\varrho_{V}$ satisfies (2.8). Fix small positive parameters $\xi,\delta>0$ and assume (3.11), (3.12). Consider the local measure $\sigma_{\bf{y}}$ with ${\bf{y}}\in{\mathcal{R}}^{\#}_{K,V,\beta}(\xi)\cap{\mathcal{R}}^{*}_{K,V,\beta}(\xi)$. Then there is a constant $C$, independent of $N,K$, such that for any two differentiable functions $f,q$ on $J_{\bf{y}}$ and large enough $N$, we have

We remark that the rigidity estimate (3.7) shows that $N^{2/3}i^{1/3}(x_{i}-i^{2/3})\leqslant N^{C\xi}$ with a very high probability. Therefore, as long as $i\ll j^{1/3}$, (3.16) is stronger than the trivial bound

obtained from the rigidity estimate. We believe that the optimal estimate on the correlation decay is of the following form:

and the same decay rate holds for the global measures $\sigma$ and $\mu$. A heuristic argument that this is the optimal decay rate, at least w.r.t. the GUE measure, will be given in Appendix E. It is based on an extension of the argument in . We note that this decay is quite different from the logarithmic correlation decay in the bulk

which is proven for the GUE measure $\mu$ in and conjectured to hold for other ensembles as well.

Theorem 3.3 is our key result. In Sections 4 and 5 we will show how to use Theorem 3.3 to prove the main Theorems 2.7 and 2.1. The proofs of these two theorems follow the arguments used in . The proofs of the auxiliary Theorem 3.1 will be given in Subsection 6.5, and Theorem 3.2 in Appendix D. The proof of Theorem 3.3 will start from Section 7 and will continue until the end of the paper.

Edge universality of beta ensembles: proof of Theorem 2.1

In this section, we shall use the edge universality Theorem 3.3 to prove global edge universality Theorem 2.1. Recall the definition of the measure $\sigma$ with normalization factor $Z_{\sigma}$. We start with he following lemma on properties of $\sigma$, defined by (3.2).

In particular, this implies that $\mu$ and $\sigma$ have the same local statistics and $\sigma$ also satisfies the following rigidity estimate: for any $\xi>0$ there exists $N_{0}$ and $c>0$ such that for all $N\geqslant N_{0}$, $k\in\llbracket 1,N\rrbracket$, we have

with some positive constant $c^{\prime}>0$.

Clearly, by $\Theta\geqslant 0$, we have the relation $Z^{\sigma}\leqslant Z$ among the normalization constants for $\sigma$ and $\mu$. For a lower bound, from the rigidity estimate (2.11) for $x_{1}$ we have

For any bounded nonnegative observable $O$, we have from the rigidity estimate on $\mu$ that

Using this separately for the positive and negative parts of an arbitrary bounded observable, this proves (4.1). From the rigidity estimate (4.2) we have

(notice that the index of ${\mathcal{R}}$ is $K+1$ instead of $K$ and we use $\xi/3$ instead of $\xi$ for later convenience). Furthermore, for any ${\bf{y}}\in{\mathcal{R}}_{K+1}(\xi/3)$, the level repulsion estimate w.r.t. $\sigma_{\bf{y}}$ in the form proved in (3.9) implies

for $s\geqslant\exp(-K^{\theta})$. Using (4.4), we see that (4.5) also holds with $\sigma_{\bf{y}}$ replaced by $\sigma$. Applying this with $s=N^{-\xi}\ll N^{-7\xi/9}$, we have

The estimates (4.1)–(4.6) also hold for the measure $\widehat{\sigma}$ instead of $\sigma$ with the same proof.

From the rigidity estimate w.r.t. $\sigma$, (4.2), we have for any ${\varepsilon}>0$ that

By the estimate (3.14) on $\gamma_{j}-\alpha_{j}$, (4.7) also holds if $\alpha_{j}$ is replaced by $\gamma_{j}$. From the rigidity estimate w.r.t. $\widehat{\sigma}$, we have

By (4.1) we have and also the parallel version with $\sigma$ replaced by $\widehat{\sigma}$, we have

This guarantees that the second constraint in the definition of ${\mathcal{R}}^{*}$ from (3.5) is satisfied for a set of ${\bf{y}}$’s with a high $\sigma$-probability. The first constraint is easily satisfied for a large set of ${\bf{y}}$’s by the rigidity w.r.t. $\sigma$. Thus we obtain

Combining (4.6) and (4.8), we obtain (4.3). ∎

where we have explicitly indicated the dependence of $\alpha_{j}$ on ${\bf{y}}$. From (3.14) and $j\leqslant K^{\zeta}$ we have

This condition is guaranteed by the condition (3.11) if $\chi>0$ is chosen sufficiently small. Under this condition, we have thus proved that

Recall that the $\sigma$-probability of the complement of the set ${\mathcal{R}}^{\#}_{K}\cap{\mathcal{R}}^{*}_{K}$ is small, see (4.3), and we can choose $\chi<c^{\prime}$ where $c^{\prime}$ is the constant in (4.3). Together with the fact that $O$ is bounded, we can drop the characteristic function $\mathds{1}_{{\bf{y}}\in{\mathcal{R}}^{\#}_{K}\cap{\mathcal{R}}^{*}_{K}}$ at a negligible error and we have

Edge universality of Wigner matrices: proof of Theorem 2.7

We will first prove Theorem 2.7 under the assumption that the matrix elements of the normalized matrix satisfy a uniform subexponential decay (2.13). This will be done in the following two steps. First we show that edge universality holds for Wigner matrices with a small Gaussian component. This argument is based upon the analysis of the Dyson Brownian Motion (DBM). In the second step we remove the small Gaussian component by a moment matching perturbation argument.

We first recall the notion of Dyson’s Brownian motion. It describes the evolution of the eigenvalues of a flow of Wigner matrices, $H=H_{t}$, if each matrix element $h_{ij}$ evolves according to independent (up to symmetry restriction) Ornstein-Uhlenbeck processes. In the Hermitian case, this process for the rescaled matrix elements $v_{ij}:=N^{1/2}h_{ij}$ is given by the stochastic differential equation

where ${\rm B}_{ij}$, $i<j$, are independent complex Brownian motions with variance one and ${\rm B}_{ii}$ are real Brownian motions of the same variance. The real symmetric case is analogous, just $\beta_{ij}$ are real Brownian motions.

Denote the distribution of the eigenvalues \mbox{\boldmath\lambda}=(\lambda_{1},\lambda_{2},\dots,\lambda_{N}) of $H_{t}+2$ at time $t$ by f_{t}(\mbox{\boldmath\lambda})\mu({\rm d}\mbox{\boldmath\lambda}) where the Gaussian measure $\mu$ is given by (2.2) with $V(x)=\frac{1}{2}(x-2)^{2}$. (This simple shift ensures that the convention $A=0$ made at the beginning of Section 3 holds.) The density $f_{t}=f_{t,N}$ satisfies the forward equation

with $\beta=1$ for the real symmetric case and $\beta=2$ in the complex hermitian case. The initial data $f_{0}$ given by the original generalized Wigner matrix. The main result of this section is that edge universality holds for the measure $f_{t}\mu$ if $t$ is at least a small negative power of $N$.

Note that, in this section, we always consider the cases $\beta=1$ or $2$, although the proof of the following theorem could be adapted to general $\beta\geqslant 1$.

Let $\mu$ be the Gaussian beta ensemble, (2.2), with quadratic $V$, and $f_{t}$ be the solution of (5.1) with initial data $f_{0}$ given by the original generalized Wigner matrix. Fix an integer $m>0$ and $\kappa<1/4$. Then there are positive constants ${\mathfrak{b}}$ and $\chi$ such that for any $t\geqslant N^{-{\mathfrak{b}}}$ and for any compactly supported smooth observable $O$ we have

for any $p_{1},\dots,p_{m}\leqslant N^{\kappa}$.

For any $\tau>0$ define an auxiliary potential $W=W^{\tau}$ by

The parameter $\tau>0$ will be chosen as $\tau\sim N^{-{\mathfrak{a}}}$ where ${\mathfrak{a}}$ is some positive exponent with ${\mathfrak{a}}<{\mathfrak{b}}$.

We define the probability measure ${\rm d}\mu^{\tau}:=Z_{\tau}^{-1}e^{-N\beta{\mathcal{H}}^{\tau}}$, where the total Hamiltonian is given by

Here ${\mathcal{H}}$ is the Gaussian Hamiltonian given by (2.2) with $V(x)=x^{2}/2$ and $Z_{\tau}=Z_{\mu^{\tau}}$ is the partition function. The measure $\mu^{\tau}$ will be referred to as the relaxation measure.

Since $H_{t}$ is a generalized Wigner matrix for all $t$, the following rigidity estimate (Theorem 2.2 and Theorem 7.6 ) holds:

where $\gamma_{k}$ is computed w.r.t. the semicircle law and we have used $\delta\xi$ as the small positive exponent needed in the rigidity estimate so that $N^{\delta\xi}\leqslant K^{\xi}$. Together with a trivial tail estimate from (2.13),

for any $\nu>0$ if $N\geqslant N_{0}(\nu)$ is large enough.

Recall the definition of the Dirichlet form w.r.t. a probability measure ${\omega}$

and the definition of the relative entropy of two probability measures $g{\omega}$ and ${\omega}$

The $1/N$ prefactor in the definition of the Dirichlet form as well as in (5.2) originates from the $N^{-1/2}$-rescaling of the matrix elements $h_{ij}=N^{-1/2}v_{ij}$.

By the Bakry-Émery criterion ), the local relaxation measure satisfies the logarithmic Sobolev inequality, i.e.,

for any probability measure $f\mu^{\tau}$.

Now we recall Theorem 2.5 from (the equation (2.37) in has a typo and the correct form should be $S(f_{\tau}\mu|{\omega})\leqslant CN^{m}$). This theorem was first proved in ; a closely related result was obtained earlier in in .

Let $0<\tau\leqslant 1$ be a (possibly $N$-dependent) parameter. Consider the local relaxation measure $\mu^{\tau}$. Set $\psi:=\frac{d\mu^{\tau}}{d\mu}$ and let $g_{t}:=f_{t}/\psi$. Suppose there is a constant $m$ such that

Fix an ${\varepsilon}^{\prime}>0$. Then for any $t\geqslant\tau N^{{\varepsilon}^{\prime}}$ the entropy and the Dirichlet form satisfy the estimates:

where the constants depend on ${\varepsilon}^{\prime}$ and $m$.

We remark that the condition (5.6) is trivially satisfied in our applications for any $\tau\geqslant N^{-2/3+\xi}$ since

and $S(f_{\tau}\mu|\mu)\leqslant S(H_{\tau}|H_{\infty})=N^{2}S((h_{\tau})_{ij}|(h_{\infty})_{ij})\leqslant CN^{m}$, where $H_{\infty}$ is the GOE/GUE matrix. The other two terms in (5.8) satisfy a similar bound by (5.3).

Recall the probability measure $\sigma$ (3.2) and define $q_{t}$ by

From (5.3)–(5.4) and (5.7) (and recalling that we have shifted the eigenvalues in such a way that the left spectral edge $-2$ is now shifted to $0$), we can check that

for any $t\geqslant\tau N^{{\varepsilon}^{\prime}}$.

In this estimate we used that $V^{\prime\prime}$ is bounded from below, see (2.3), and that

holds for any $x\geqslant-N^{-2/3+\xi}$ and ${\bf{y}}\in{\mathcal{R}}_{K}$.

Define $q_{t,{\bf{y}}}$ to be the conditional density of $f_{t}\mu=q_{t}\sigma$ w.r.t. $\sigma_{\bf{y}}$ given ${\bf{y}}$, i.e., it is defined by the relation $q_{t,{\bf{y}}}\sigma_{\bf{y}}=(f_{t}\mu)_{\bf{y}}$. From the bound (5.10) we have the logarithmic Sobolev inequality

Combining it with the entropy inequality, we have

The following Lemma controls the Dirichlet forms $D_{i}^{\sigma_{\bf{y}}}$ for most external configurations ${\bf{y}}$.

Fix $0<{\mathfrak{a}}\leqslant 1$, $\xi,\nu>0$, and $\tau\geqslant N^{-{\mathfrak{a}}}$. Suppose the initial data $f_{0}$ of the DBM is given by a generalized Wigner ensemble. Then, for any ${\varepsilon},{\varepsilon}^{\prime}>0$ and $t\geqslant\tau N^{{\varepsilon}^{\prime}}$ there exists a set ${\mathcal{G}}_{K,t}\subset{\mathcal{R}}_{K}(\xi)$ of good boundary conditions ${\bf{y}}$ with

such that for any ${\bf{y}}\in{\mathcal{G}}_{K,t}$ we have

Furthermore, for any bounded observable $O$, we have

The same bounds hold if $\sigma_{\bf{y}}$ and $q_{t,{\bf{y}}}$ are replaced with $\widehat{\sigma}_{\bf{y}}$ and $\widehat{q}_{t,{\bf{y}}}$ where $\widehat{q}_{t}$ is defined by $\widehat{q}_{t}\widehat{\sigma}=f_{t}\mu$.

Similarly, the rigidity bound (5.3) with respect to $f\mu$ can be translated to the measure $f_{\bf{y}}\mu_{\bf{y}}$ for most ${\bf{y}}$, i.e., there exists a set ${\mathcal{G}}^{2}_{K}\subset{\mathcal{R}}_{K}$ with

such that for any ${\bf{y}}\in{\mathcal{G}}^{2}_{K}$ and for any $k\in I$, we have

In particular, by setting ${\mathcal{G}}_{K}:={\mathcal{G}}^{1}_{K}\cap{\mathcal{G}}^{2}_{K}$ we can conclude (5.16) for any ${\bf{y}}\in{\mathcal{G}}_{K}$. This proves the lemma. ∎

Fix $0<{\mathfrak{a}}<1/6$, $\xi,\nu>0$, and $\tau\geqslant N^{-{\mathfrak{a}}}$. Suppose the initial data $f_{0}$ of the DBM is given by a generalized Wigner ensemble. Then, for any ${\varepsilon}^{\prime}>0$, $t\geqslant\tau N^{{\varepsilon}^{\prime}}$, $k\in I$ and ${\bf{y}}\in{\mathcal{G}}_{K,t}$ (defined in Lemma 5.4), we have

Notice that we need ${\mathfrak{a}}<1/6$ in order that (5.19) has a solution with $K\to\infty$. In our application we will choose ${\mathfrak{a}}$ arbitrarily close to 0, then we can take any $K$ with $K\leqslant N^{1/4-\delta}$ and still find sufficiently small positive exponents $\nu,{\mathfrak{a}},{\varepsilon}^{\prime}$ with ${\mathfrak{a}}+{\varepsilon}^{\prime}\leqslant{\mathfrak{b}}$ so that (5.19) holds. We will not trace the precise interrelation among these exponents. This explains the restriction $\kappa<1/4$ in Theorem 2.7.

The following proof is essentially the same as the one for Lemma 5.5 in .

We claim that the estimate (5.18) follows from

where we have used (5.16), (5.20) and (5.19). To prove (5.20), we run the reversible dynamics

starting from initial data $h_{0}=q_{t,{\bf{y}}}$, where the generator ${\mathcal{L}}_{\bf{y}}$ is the unique reversible generator with the Dirichlet form $D^{\sigma_{\bf{y}}}$, i.e.,

Recall that from the convexity bound (5.10), $\tau_{K}=K^{1/3}/N^{1/3}$ is an upper bound for the time to equilibrium of this dynamics. After differentiation and integration we get,

From the Schwarz inequality with a free parameter $R$, we can bound the last line by

Dropping the trivial subexponential error term and using that the time integral of the Dirichlet form is bounded by the initial entropy, we can bound the last line by

Using the logarithmic Sobolev inequality for $\sigma_{\bf{y}}$ and optimizing the parameter $R$, we can bound the last term by

Combining this bound with (5.14) with the choice ${\varepsilon}={\varepsilon}^{\prime}$, we obtain (5.20). ∎

We note that if we applied (5.15) with the special choice $O({\bf{x}})=x_{k}$ to control (5.20), then the error estimate would have been much worse. We stress that (5.18) is not an obvious fact although we know that it holds for ${\bf{y}}$ with high probability w.r.t. the equilibrium measure $\mu$. The key point of (5.18) is that it holds for any ${\bf{y}}\in{\mathcal{G}}_{K}$, i.e., for a set of ${\bf{y}}$’s with ”high probability” w.r.t $f_{t}\mu$! We also remark that (5.18) holds only in the sense of expectation of $x_{k}$ and have not yet established that

We will finally prove this estimate (Theorem 3.1) but only after we prove the rigidity estimate for $\sigma_{\bf{y}}$.

We can now prove the main result of this section.

We will consider only the case $m=1$ since the general case is only notationally more involved. From the assumption (5.19) the right hand side of (5.15) is smaller than $K^{-1/2}$. Choosing $\chi$ sufficiently small, we thus have

for all ${\bf{y}}\in{\mathcal{G}}_{K}$ and $p\leqslant K^{\zeta}$ with the $f_{t}\mu$-probability of ${\mathcal{G}}_{K}$ satisfying (5.13).

We now apply Theorem 3.3 to the same Gaussian beta ensemble with two different boundary conditions so that

for all ${\bf{y}}\in{\mathcal{G}}_{K}\cap{\mathcal{R}}^{\#}_{K}\cap{\mathcal{R}}^{\ast}_{K}$. Once we prove that

then by averaging (5.22) in ${\bf{y}}$ w.r.t. $f_{t}\mu$ we have

Since $f_{t}\mu$ represents the probability distribution of a generalized Wigner matrix ensemble, from the rigidity estimate (5.3), we have

From the level repulsion estimate (3.9) with $k=K+1$, we have for any ${\bf{y}}\in{\mathcal{R}}_{K+1}$ that

Applying (5.15) with $O({\bf{x}})=\mathds{1}(y_{K+2}-x_{K+1}\leqslant sN^{-2/3}K^{-1/3})$ and using the condition (5.19), we obtain a similar estimate w.r.t. the measure $(f_{t}\mu)_{{\bf{y}}}$, i.e.,

This estimate (5.26) and the bound (5.25) with $K+1$ replaced by $K$ imply (5.24) provided $7\xi^{\prime}/3\ll\xi$ and (5.19) is satisfied.

holds for ${\bf{y}}\in{\mathcal{G}}_{K}$. To prove (5.27), for ${\bf{y}}\in{\mathcal{G}}_{K}$ we have from (5.15) (applied to $\widehat{\sigma}_{\bf{y}}$) that

Under the assumption (5.19), the right hand side of the last equation vanishes as $N\to\infty$. Thus we have

2 Removal of the Gaussian convolution

The last step to complete the proof of edge universality is to approximate arbitrary Wigner matrices by a Gaussian divisible ensemble. We will need the following result.

then there is an ${\varepsilon}>0$ and $\delta>0$ depending on $\vartheta$ in (2.13) such that or any real parameter $s$ (may depend on $N$) we have

for $N\geqslant N_{0}$ sufficiently large, where $N_{0}$ is independent of $s$. Analogous result holds for the smallest eigenvalue $\lambda_{1}$ and also for extensions to the joint distributions of any finite number of eigenvalues $\lambda_{N-i_{1}},\dots,\lambda_{N-i_{k}}$ as long as $|i_{k}|\leqslant N^{\varepsilon}$ (or similar results for the smallest eigenvalues).

Rigidity of the particles

Most of this section is devoted to proving Theorem 2.4 which asserts the rigidity of the particles under the measure $\mu$ at the optimal scale up to the edge (which, for us, means a control throughout the support of the equilibrium measure including the edge). We recall that the same statement holds for the measure $\sigma$ (Lemma 4.1).

Our method to prove rigidity is a multiscale analysis, initiated for the bulk particles in . It is a bootstrap argument where concentration and accuracy bounds are proved in tandem, gradually for smaller and smaller scales. Concentration bound means a control on the fluctuation of a particle around its mean; this is obtained by a local logarithmic Sobolev inequality (for non-convex $V$ we need an extra convexification argument). To estimate the log-Sobolev constant we use rigidity on a larger scale. The next step is to identify the mean, this is achieved by the first loop equation, where the error term involves the improved concentration bound. This leads to a better accuracy and thus better rigidity. This information can be used to improve the concentration bound on a smaller scale, etc. In this paper we prove rigidity up to the edge, which involves new difficulties: the loop equation is less stable since the density vanishes near the edge. Moreover, the loop equation is used to improve the accuracy of one specific particle (the leftmost one, $\lambda_{1}$), whose rigidity cannot originate in the pairwise interaction from surrounding particles.

This extra difficulty (lack of a natural boundary on the left) is also critical in the last subsection, where we prove Theorem 3.1, i.e., the rigidity of the particles under the conditional measure $\sigma_{\bf y}$ with a Gaussian tail. Extra convexity (hence rigidity) on the left of the first particle is the reason for introducing the modification $\sigma_{\bf{y}}$ of $\mu_{\bf y}$ which artificially confines the first particle.

Another extra difficulty consists in improving the accuracy without assuming that $V$ is analytic. This analyticity condition was essential in the works and the previous optimal bulk rigidity estimates . It turns out that the analyticity condition can be replaced by a much weaker smoothness assumption by a more careful analysis of the first loop equation, see (6.18) and (6.38).

In this section we disregard the shift convention which sets $A=0$.

For any fixed $N$, let the classical position $\gamma^{(N)}_{k}$ of the $k$-th particle under $\mu^{(N)}$ be defined by

where $\varrho_{1}^{(N)}$ is the density of $\mu^{(N)}$. Recall that $\gamma_{k}$ from (2.6) denotes the limiting classical location.

In the following definitions, the potential $V$ and $\beta>0$ are fixed.

We say that rigidity at scale $a$ holds if for any ${\varepsilon}>0$, there are constants $c>0$ and $N_{0}$ such that for any $N\geqslant N_{0}$ and $k\in\llbracket 1,N\rrbracket$ we have

We say that concentration at scale $a$ holds if for any ${\varepsilon}>0$, there are constants $c>0$ and $N_{0}$ such that for any $N\geqslant N_{0}$ and $k\in\llbracket 1,N\rrbracket$ we have

We say that accuracy at scale $a$ holds if for any ${\varepsilon}>0$, there is a constant $N_{0}$ such that for any $N\geqslant N_{0}$ and $k\in\llbracket 1,N\rrbracket$ we have

For the proof of Theorem 2.4 the main steps are the concentration and accuracy improvements hereafter, proved in the following subsections.

Let $V$ be $\mathscr{C}^{2}$, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (2.3), (2.4). Then rigidity at scale $a$ implies concentration at scale $a/2$.

Let $V$ be $\mathscr{C}^{4}$, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (2.3), (2.4). Then rigidity at scale $a$ implies accuracy at scale $11a/12$.

Notice that the accuracy improves from scale $a$ only to scale $11a/12$ instead of $3a/4$ as it was achieved in the bulk case (see Proposition 3.13 in ). This weaker control is due to some difficult estimates near the edge that have not been optimized.

It is known that rigidity at scale 1 holds. More precisely, for any ${\varepsilon}>0$ there are positive constants $c_{1}$, $c_{2}$ such that, for all $N\geqslant 1$,

For eigenvalues in the bulk, (6.2) follows from the large deviations for the empirical spectral measure with speed $N^{2}$, see . For the extreme eigenvalues the large deviations principle with speed $N$ is proved in for the GOE case, and extended in Theorem 2.6.6, for the general case (up to a condition on the partition function that follows from Theorem 1 (iii) in ).

We now use Propositions 6.2 and 6.3 to obtain that concentration and accuracy hold at scale $11/12$. We just need to prove that concentration and accuracy at some scale $b>0$ imply rigidity the same scale $b$. Then a simple induction on scales shows that rigidity holds on scale $(11/12)^{m}$ for any integer $m$, i.e., it holds at any positive scale $\xi$.

To show the key part of the induction step, assume that concentration and accuracy hold at scale $b$. Fix any $k\in\llbracket 1,N\rrbracket$. Then for any $\varepsilon>0$ we have

for large enough $N$. As we have accuracy at scale $b$, the same conclusion holds when replacing $\gamma_{k}^{(N)}$ by $\gamma_{k}$. We thus proved, for any $\varepsilon>0$, the existence of some $C>0$ such that for all $N$ and $k$ we have

The first term can be bounded by the concentration hypothesis, the second term is 0 for large enough $N$, thanks to (6.3). ∎

2 Initial estimates for non-analytic potentials

Let $h$ be a continuous and bounded function. Consider the probability distribution on the simplex $\lambda_{1}\leqslant\dots\leqslant\lambda_{N}$ given by

where $\mathcal{H}$ is defined in (1.1). We denote by $m_{N,h}$ the Stieltjes transform for the measure $\mu^{(N,h)}$:

In the following, it will be useful to have the density supported strictly in a compact interval: for given $\kappa>0$, define the following variant of $\mu^{(N,h)}$ conditioned to have all particles in $[A-\kappa,B+\kappa]$:

We will choose $\kappa$ to be small, fixed number. Let $\varrho_{k}^{(N,h,\kappa)}$ denote the correlation functions and $m_{N,h,\kappa}(z)$ the Stieltjes transform, defined in the same way as (2.7) and (6.4), but for the underlying measure $\mu^{(N,h,\kappa)}$. Then Lemma 1 in (strictly speaking this result is given in only for $h\equiv 0$, but the proof works for any fixed $h$) states that under condition (2.4), for some large enough $\kappa$ there exists some $c>0$, depending only on $V$, such that for any $x_{1},\dots,x_{k}\in[A-\kappa,B+\kappa]$, we have

and for $x_{1},\dots,x_{j}\not\in[A-\kappa,B+\kappa]$, $x_{j+1},\dots,x_{k}\in[A-\kappa,B+\kappa]$,

The estimates (6.6) and (6.7) actually also hold for arbitrarily small fixed $\kappa>0$ thanks to the large deviations estimates (6.2), which holds not only for $\mu^{(N)}$ but also for $\mu^{(N,h)}$. From now we fix this small parameter $\kappa>0$. The following Lemma relates estimates on $m_{N,h}-m$ and concentration of linear statistics of the particles.

Let $V$ be $\mathscr{C}^{4}$, regular such that the equilibrium density $\varrho_{V}$ is supported on a single interval $[A,B]$ and satisfies (2.3), (2.4). Let $h_{1},h_{2}$ be $\mathscr{C}^{2}$ functions such that $\|h_{1}\|_{\infty},\|h_{1}^{\prime}\|_{\infty},\|h_{1}^{\prime\prime}\|_{\infty}<\infty$, and the same for $h_{2}$. Let $a\in(0,1/2)$ and ${\varepsilon}>0$. Assume that for any $\Im(z)=\eta\in(N^{-1/2},N^{-a})$ and $s\in(-\beta,\beta)$ we have

Then there is a constant $c>0$ such that, for any $N\geqslant 1$, we have

Let $\kappa>0$ be a small constant and $C>0$ be chosen such that for any $E\in I_{\kappa}:=[A-\kappa,B+\kappa]$, $\eta\in(0,N^{-a})$, and $s\in(-\beta,\beta)$ we have

This inequality was proved in , Theorem 2.3 (ii) Let $\chi$ be a smooth nonnegative cutoff function; $\chi=1$ on $[0,N^{-a}/2]$, $\chi=0$ on $[N^{-a},\infty)$, $\|\chi^{\prime}\|_{\infty}=\OO(N^{a})$. Let $\widetilde{h}_{2}$ be $\mathscr{C}^{2}$, compactly supported on $I_{\kappa}$, such that $h_{2}=\widetilde{h}_{2}$ on $I_{\kappa/2}$, for some $\kappa>0$. From the large deviations estimate (6.2) we have, for any ${\varepsilon}>0$,

for some $c>0$. As a consequence, to prove (6.9), we can assume that $h_{2}$ is supported on $I_{\kappa}$. By the Helffer-Sjöstrand formula (see formula (B.13) in ),

The term involving $\chi^{\prime}$ can be evaluated using (6.8), and is bounded by $N^{-1+2a}$. For the $\chi$ term, we bound $m_{N,(1+s)h_{1}}(z)-m(z)$ by (6.8) if $\eta\geqslant N^{-1/2}$ and by (6.10) if $\eta\in(0,N^{-1/2})$. We obtain

The remainder of the proof is a classical argument: using the above estimate we get

and one concludes by the exponential Markov inequality. ∎

The following lemma provides almost optimal estimates for $m_{N,h}-m$ for $\eta=\Im(z)$ till order 1. For non-analytic $V$, it improves previous estimates by Pastur and Shcherbina by a factor $\sqrt{N}$, and relies on their initial estimates proved in .

Let $V$ be $\mathscr{C}^{4}$, regular such that the equilibrium density $\varrho_{V}$ is supported on a single interval $[A,B]$ and satisfy (2.3), (2.4). Let $h$ be a $\mathscr{C}^{2}$ function with $\|h\|_{\infty},\|h^{\prime}\|_{\infty},\|h^{\prime\prime}\|_{\infty}<\infty$. Then for any ${\varepsilon}>0$ there exists a constant $C=C(V,{\varepsilon},\|h^{\prime}\|_{\infty})$ such that, for any $E\in[A-\kappa,B+\kappa]$, $\eta\in(0,N^{-{\varepsilon}})$, we have