Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation

Introduction

Certain spectral statistics of broad classes of $N\times N$ random matrix ensembles are believed to follow a universal behavior in the limit $N\to\infty$. Wigner has observed that the density of eigenvalues of large symmetric or hermitian matrices $H$ with independent entries (up to the symmetry requirement) converges, as $N\to\infty$, to a universal density, the Wigner semicircle law. Dyson has observed that the local correlation statistics of neighboring eigenvalues inside the bulk of the spectrum follows another universal pattern, the Dyson sine-kernel in the $N\to\infty$ limit . Moreover, any $k$-point correlation function can be obtained as a determinant of the two point correlation functions. The precise form of the universal two point function in the bulk seems to depend only on the symmetry class of the matrix ensemble (a different universal behavior emerges near the spectral edge ).

Dyson has proved this fact for the Gaussian Unitary Ensemble (GUE), where the matrix elements are independent, identically distributed complex Gaussian random variables (subject to the hermitian constraint). A characteristic feature of GUE is that the distribution is invariant under unitary conjugation, $H\to U^{*}HU$ for any unitary matrix $U$. Dyson found an explicit formula for the joint density function of the $N$ eigenvalues. The formula contains a characteristic Vandermonde determinant and therefore it coincides with the Gibbs measure of a particle system interacting via a logarithmic potential analogously to the two dimensional Coulomb gas. Dyson also observed that the computation of two point function can be reduced to asymptotics of Hermite polynomials.

His approach has later been substantially generalized to include a large class of random matrix ensembles, but always with unitary (orthogonal, symplectic, etc.) invariance. For example, a general class of invariant ensembles can be given by the measure $Z^{-1}\exp(-\mbox{Tr\,}V(H)){\rm d}H$ on the space of hermitian matrices, where ${\rm d}H$ stands for the Lebesgue measure for all independent matrix entries, $Z$ is the normalization and $V$ is a real function with certain smoothness and growth properties. For example, the GUE ensemble corresponds to $V(x)=x^{2}$.

The joint density function is explicit in all these cases and the evaluation of the two point function can be reduced to certain asymptotic properties of orthogonal polynomials with respect to the weight function $\exp(-V(x))$ on the real line. The sine kernel can thus be proved for a wide range of potentials $V$. Since the references in this direction are enormous, we can only refer the reader to the book by Deift for the Riemann-Hilbert approach, the paper by Levin and Lubinsky and references therein for approaches based on classical analysis of orthogonal polynomials, or the paper by Pastur and Shcherbina for a probabilistic/statistical physics approach. The book by Anderson et al or the book by Metha also contain extensive lists of literatures.

Since the computation of the explicit formula of the joint density relies on the unitary invariance, there have been very little progress in understanding non-unitary invariant ensembles. The most prominent example is the Wigner ensemble or Wigner matrices, i.e., hermitian random matrices with i.i.d. entries. Wigner matrices are not unitarily invariant unless the single entry distribution is Gaussian, i.e. for the GUE case. The disparity between our understanding of the Wigner ensembles and the unitary invariant ensembles is startling. Up until the very recent work of , there was no proof that the density follows the semicircle law in small spectral windows unless the number of eigenvalues in the window is at least $\sqrt{N}$. This is entirely due to a serious lack of analytic tools for studying eigenvalues once the mapping between eigenvalues and Coulomb gas ceases to apply. At present, there are only two rigorous approaches to eigenvalue distributions: the moment method and Green function method. The moment method is restricted to studying the spectrum near the edges ; the precision of the Green function method seems to be still very far from getting information on level spacing .

Beyond the unitary ensembles, Johansson proved the sine-kernel for a broader category of ensembles, i.e., for matrices of the form $H+sV$ where $H$ is a Wigner matrix, $V$ is an independent GUE matrix and $s$ is a positive constant of order one. (Strictly speaking, in the original work , the range of the parameter $s$ depends on the energy $E$. This restriction was later removed by Ben Arous and Péché , who also extended this approach to Wishart ensembles). Alternatively formulated, if the matrix elements are normalized to have variance one, then the distribution of the matrix elements of the ensemble $H+sV$ is given by $\nu\ast{\mathcal{G}}_{s}$, where $\nu$ is the distribution of the Wigner matrix elements and ${\mathcal{G}}_{s}$ is the centered Gaussian law with variance $s^{2}$. Johasson’s work is based on the analysis of the explicit formula for the joint eigenvalue distribution of the matrix $H+sV$ (see also ).

Dyson has introduced a dynamical version of generating random matrices. He considered a matrix-valued process $H+sV$ where $V$ is a matrix-valued Brownian motion. The distribution of the eigenvalues then evolves according to a process called Dyson’s Brownian motions. For the convenience of analysis, we replace the Brownian motions by an Ornstein-Uhlenbeck process so that the distribution of GUE is the invariant measure of this modified process, which we still call Dyson’s Brownian motion. Dyson’s Brownian motion thus can be viewed as a reversible interacting particle system with a long range (logarithmic) interaction. This process is well adapted for studying the evolution of the empirical measures of the eigenvalues, see . The sine kernel, on the other hand, is a very detailed property which typically cannot be obtained from considerations of interacting particle systems. The Hamiltonian for GUE, however, is strictly convex and thus the Dyson’s Brownian motion satisfies the logarithmic Sobolev inequality (LSI). It was noted in the derivation of the Navier-Stokes equations that the combination of the Guo-Papanicolaou-Varadhan approach and LSI provides very detailed estimates on the dynamics.

The key observation of the present paper is that this method can also be used to estimate the approach to local equilibria so precisely that, after combining it with existing techniques from orthogonal polynomials, the Dyson sine kernel emerges. In pursuing this approach, we face two major obstacles: 1. Good estimate of the initial entropy, 2. Good understanding of the structure of local equilibria. It turns out that the initial entropy can be estimated using the explicitly formula for the transition kernel of the Dyson’s Brownian motion (see and ) provided strong inputs on the local semicircle law and level repulsion are available.

The structure of local equilibria, however, is much harder to analyze. Typically, the local equilibrium measures are finite volume Gibbs measures with short range interaction and the boundary effects can be easily dealt with in the high temperature phase. In the GUE case, the logarithmic potential does not even decay at large distance and the equilibrium measure can depend critically on the boundary conditions. The theory of orthogonal polynomials provides explicit formulae for the correlation functions of this highly correlated Gibbs measure. These formulae can be effectively analyzed if the external potential (or logarithm of the weight function in the terminology of the orthogonal polynomials) is very well understood. Fortunately, we have proved the local semicircle law up to scales of order $1/N$ and the level repulsion, which can be used to control the boundary effects. By invoking the theorem of Levin and Lubinsky and the method of Pastur and Shcherbina we are led to the sine kernel.

It is easy to see that adding a Gaussian component of size much smaller than $N^{-1}$ to the original Wigner matrix would not move the eigenvalues sufficiently to change the local statistics. Our requirement that the Gaussian component is at least of size $N^{-3/4}$ comes from technical estimates to control the initial global entropy and it does not have any intrinsic meaning. The case that the variance is of order $N^{-1}$, however, is an intrinsic barrier which is difficult to cross. Nevertheless, we believe that our method may offer a possible strategy to prove the universality of sine kernel for general Wigner matrices.

After this manuscript had been completed, we found a different approach to prove the Dyson sine kernel , partly based on a contour integral representation for the two-point correlation function . Shortly after our manuscripts were completed, we learned that our main result was also obtained by Tao and Vu in with a different method under no regularity conditions on the initial distribution $\nu$ provided the third moment of $\nu$ vanishes.

Although the results in this paper are weaker than those in and , we believe that the method presented here has certain independent interest. Unlike and , this approach does not use the contour integral representation of the two point correlation function. Hence, it may potentially have a broader applicability to other matrix ensembles for which such representation is not available.

Acknowledgements. We would like to thank the referees for suggesting several improvements of the presentation.

Main theorem and conditions

We assume that the probability measures $\nu$ and $\widetilde{\nu}$ have a small Gaussian component of variance $N^{-3/4+\beta}$ where $\beta>0$ is some fixed positive number. More precisely, we assume there exist probability measures $\nu_{0}$ and $\widetilde{\nu}_{0}$ with zero expectation and variance $\frac{1}{2}$ and $1$, respectively, such that

where $G_{s}(x)=(2\pi s)^{-1}\exp(-x^{2}/2s)$ is the Gaussian law with variance $s^{2}$ and $\nu_{s}$, $\widetilde{\nu}_{s}$ are the rescaling of the laws $\nu_{0}$, $\widetilde{\nu}_{0}$ to ensure that $\nu$ and $\widetilde{\nu}$ have variance $1/2$ and 1; i.e, explicitly

This requirement is equivalent to considering random matrices of the form

where $\widehat{H}$ is a Wigner matrix with single entry distribution $\nu_{0}$ and $\widetilde{\nu}_{0}$, and $V$ is a GUE matrix whose elements are centered Gaussian random variables with variance $1/N$.

Furthermore, we assume that $\nu$ is absolutely continuous with positive density functions $h(x)>0$, i.e. we can write it as ${\rm d}\nu(x)=h(x){\rm d}x=\exp(-g(x)){\rm d}x$ with some real function $g$. We assume the following conditions:

The measure ${\rm d}\nu$ satisfies the logarithmic Sobolev inequality, i.e. there exists a constant $S$ such that

holds for any density function $u>0$ with $\int u\,{\rm d}\nu=1$.

The Fourier transform of the functions $h$ and $h(\Delta g)$ satisfy the decay estimates

with some constants ${\omega},\widetilde{\omega}>0$.

There exists a $\delta_{0}>0$ such that for the distribution of the diagonal elements

Although the conditions are stated directly for the measures $\nu$ and $\widetilde{\nu}$, it is easy to see that it is sufficient to assume that $\nu_{0}$ satisfies (2.4) and (2.5) and $\widetilde{\nu}_{0}$ satisfies (2.6). We remark that (2.4) implies that (2.6) holds for $\nu$ instead of $\widetilde{\nu}$ as well (see ).

The eigenvalues of $H$ are denoted by $\lambda_{1},\lambda_{2},\ldots\lambda_{N}$. The law of the matrix ensemble induces a probability measure on the set of eigenvalues whose density function will be denoted by $p(\lambda_{1},\lambda_{2},\ldots,\lambda_{N})$. The eigenvalues are considered unordered for the moment and thus $p$ is a symmetric function. For any $k=1,2,\ldots,N$, let

be the $k$-point correlation function of the eigenvalues. The $k=1$ point correlation function (density) is denoted by $\varrho(\lambda):=p^{(1)}(\lambda)$. With our normalization convention, the density $\varrho(\lambda)$ is supported in $$and in the$N\to\infty$ limit it converges to the Wigner semicircle law given by the density

The main result of this paper is the following theorem:

Fix arbitrary positive constants $\beta>0$ and $\kappa>0$. Consider the Wigner matrix ensemble with a Gaussian convolution of variance $s^{2}=N^{-3/4+\beta}$ given by (2.3) and assume (2.4)–(2.6). Let $p^{(2)}$ be the two point correlation function of the eigenvalues of this ensemble. Let $|E_{0}|<2-\kappa$ and

with $g,h$ smooth and compactly supported functions such that $h\geq 0$ and $\int h=1$. Then we have

The factor $g$ in the observable (2.8) tests the eigenvalue differences. The factor $h$, that disappears in the right hand side of (2.9), is only a normalization factor. Thus the special form of observable (2.8) directly exhibits the fact that the local statistics is translation invariant.

Our approach has three main ingredients. In the first step, we use the entropy method from hydrodynamical limits to establish a local equilibrium of the eigenvalues in a window of size $N^{-1+\varepsilon}$ (with some small $\varepsilon>0$), i.e. window that typically contains $n=N^{\varepsilon}$ eigenvalues. This local equilibrium is subject to an external potential generated by all other eigenvalues. In the second step we then prove that the density of this equilibrium measure is locally constant by using methods from orthogonal polynomials. Finally, in the third step, we employ a recent result to deduce the sine-kernel. We now describe each step in more details.

We generate the Wigner matrix with a small Gaussian component by running a matrix-valued Ornstein-Uhlenbeck process (3.1) for a short time of order $t\sim N^{-\zeta}$, $\zeta>0$. This generates a stochastic process for the eigenvalues which can be described as Ornstein-Uhlenbeck processes for the individual eigenvalues with a strong interaction (3.10).

This process is the celebrated Dyson’s Brownian motion (DBM) and the equilibrium measure is the GUE distribution of eigenvalues. The transition kernel can be computed explicitly (5.12) and it contains the determinantal structure of the joint probability density of the GUE eigenvalues that is responsible for the sine-kernel. This kernel was analyzed by Johansson assuming that the time $t$ is of order one, which is the same order as the relaxation time to equilibrium for the Dyson’s Brownian motions. The sine-kernel, however, is a local statistics, and local equilibrium can be reached within a much shorter time scale. To implement this idea, we first control the global entropy on time scale $N^{-1}$ by $N^{1+\alpha}$, with $\alpha>1/4$ (Section 5.2).

More precisely, recall that the entropy of $f\mu$ with respect to a probability measure $\mu$ is given by

In our application, the measure $\mu$ is the Gibbs measure for the equilibrium distribution of the (ordered) eigenvalues of the GUE, given by the Hamiltonian

If $f_{t}$ denotes the joint probability density of the eigenvalues at the time $t$ with respect to $\mu$, then the evolution of $f_{t}$ is given by the equation

where the generator $L$ is defined via the Dirichlet form

The evolution of the entropy is given by the equation

The key initial entropy estimate is the inequality that

for any $\alpha>\frac{1}{4}$ and for sufficiently large $N$. The proof of this estimate uses the explicit formula for the transition kernel of (2.11) and several inputs from our previous papers on the local semicircle law and on the level repulsion for general Wigner matrices. We need to strengthen some of these inputs; the new result will be presented in Section 4 with proofs deferred to Appendix A, Appendix B and Appendix C.

It is natural to think of each eigenvalue as a particle and we will use the language of interacting particle systems. We remark that the entropy per particle is typically of order one in the interacting particle systems. But in our setting, due to the factor $N$ in front of the Hamiltonian (2.10), the typical size of entropy per particle is of order $N$. Thus for a system bearing little relation to the equilibrium measure $\mu$, we expect the total entropy to be $O(N^{2})$. So the bound (2.12) already contains nontrivial information. However, we believe that one should be able to improve this bound to $\alpha\sim 0$ and the additional $\alpha>1/4$ power in (2.12) is only for technical reasons. This is the main reason why our final result holds only for a Gaussian convolution with variance larger than $N^{-3/4}$. The additional $N^{\alpha}$ factor originates from Lemma 5.3 where we approximate the Vandermonde determinant appearing in the transition kernel by estimating the fluctuations around the local semicircle law. We will explain the origin of $\alpha>1/4$ in the beginning of Appendix D where the proof of Lemma 5.3 is given.

From the initial entropy estimate, it follows that the time integration of the Dirichlet form is bounded by the initial entropy. For the DBM, due to convexity of the Hamiltonian of the equilibrium measure $\mu$, the Dirichlet form is actually decreasing. Thus for $t=\tau N^{-1}$ with some $\tau\geq 2$ we have

The last estimate says that the Dirichlet form per particle is bounded by $N^{1+\alpha}\tau^{-1}$. So if we take an interval of $n$ particles (with coordinates given by ${\bf{x}}=(x_{1},\ldots,x_{n})$), then on average the total Dirichlet form of these particles is bounded by $nN^{1+\alpha}\tau^{-1}$. We will choose $n=N^{\varepsilon}$ with some very small $\varepsilon>0$. As always in the hydrodynamical limit approach, we consider the probability law of these $n$ particles given that all other particles (denoted by ${\bf{y}}$) are fixed. Denote by $\mu_{\bf{y}}({\rm d}{\bf{x}})$ the equilibrium measure of ${\bf{x}}$ given that the coordinates of the other $N-n$ particles ${\bf{y}}$ are fixed. Let $f_{{\bf{y}},t}$ be the conditional density of $f_{t}$ w.r.t. $\mu_{\bf{y}}({\rm d}{\bf{x}})$ with ${\bf{y}}$ given. The Hamiltonian of the measure $\mu_{\bf{y}}({\rm d}{\bf{x}})$ is given by

If ${\bf{y}}$ are regularly distributed, we have the convexity bound

This implies the logarithmic Sobolev inequality

where in the last estimate some additional $n$-factors were needed to convert the local Dirichlet form estimate per particle on average to an estimate that holds for a typical particle. Thus we obtain

provided we choose $t=N^{-1}\tau=N^{\beta-1}$ with $\beta\geq 10\varepsilon+\alpha$ (Section 6). The last inequality asserts that the two measures $f_{\bf{y}}\mu_{\bf{y}}$ and $\mu_{\bf{y}}$ are almost the same and thus we only need to establish the sine kernel for the measure $\mu_{\bf{y}}$. At this point, we remark that this argument is valid only if ${\bf{y}}$ is regularly distributed in a certain sense which we will call good configurations (Definition 4.1). Precise estimates on the local semicircle law can be used to show that most external configurations are good. Although the rigorous treatment of the good configurations and estimates on the bad configurations occupy a large part of this paper, it is of technical nature and we deferred the proofs of several steps to the appendices.

In Sections 8, 9 and 10, we refine the precision on the local density and prove that the density is essentially constant pointwise. Direct probabilistic arguments to establish the local semicircle law in rely on the law of large numbers and they give information on the density on scales of much larger than $N^{-1}$, i.e. on scales that contain many eigenvalues. The local equilibrium is reached in a window of size $n/N$ and within this window, we can conclude that the local semicircle law holds on scales of size $n^{\gamma}/N$ with an arbitrary small $\gamma>0$. However, this still does not control the density pointwise. To get this information, we need to use orthogonal polynomials.

The density in local equilibrium can be expressed in terms of sum of squares of orthogonal polynomials $p_{1}(x),p_{2}(x),\ldots$ with respect to the weight function $\exp{(-nU_{\bf{y}}(x))}$ generated by the external configuration ${\bf{y}}$ (see Section 8 for precise definitions). To get a pointwise bound from the appropriate bound on average, we need only to control the derivative of the density, that, in particular, can be expressed in terms of derivatives of the orthogonal polynomials $p_{k}$. Using integration by parts and orthogonality properties of $p_{k}$, it is possible to control the $L^{2}$ norm of $p_{k}^{\prime}$ in terms of the $L^{2}$ norm of $p_{k}(x)U_{\bf{y}}^{\prime}(x)$. Although the derivative of the potential is singular, $\|p_{k}U_{\bf{y}}^{\prime}\|_{2}$ can be estimated by a Schwarz inequality at the expense of treating higher $L^{p}$ norms of $p_{k}$ (Lemma 8.1). In this content, we will exploit the fact that we are dealing with polynomials by using the Nikolskii inequality which estimates higher $L^{p}$ norms in terms of lower ones at the expense of a constant depending on the degree. To avoid a very large constant in the Nikolskii inequality, in Section 7 we first cutoff the external potential and thus we reduce the degree of the weight function.

We remark that our approach of using orthogonal polynomials to control the density pointwise was motivated by the work of Pastur and Shcherbina , where they proved sine-kernel for unitary invariant matrix ensembles with a three times differentiable potential function on the real line. In our case, however, the potential is determined by the external points and it is logarithmically divergent near the edges of the window.

Finally, in Section 11, we complete the proof of the sine-kernel by applying the main theorem of . This result establishes the sine-kernel for orthogonal polynomials with respect to an $n$-dependent sequence of weight functions under general conditions. The most serious condition to verify is that the density is essentially constant pointwise – the main result we have achieved in the Step 2 above. We also need to identify the support of the equilibrium measure which will be done in Appendix F.

We remark that, alternatively, it is possible to complete the third step along the lines of the argument of without using . Using explicit formulae from orthogonal polynomials and the pointwise control on the density and on its derivative, it is possible to prove that the local two-point correlation function $p^{(2)}_{n}(x,y)$ is translation invariant as $n\to\infty$. After having established the translation invariance of $p^{(2)}$, it is easy to derive an equation for its Fourier transform and obtain the sine-kernel as the unique solution of this equation. We will not pursue this alternative direction in this paper.

Dyson’s Brownian motion

We can generate our matrix $H$ (2.3) from a stochastic process with initial condition $\widehat{H}$. Consider the following matrix valued stochastic differential equation

where \mbox{\boldmath\beta}_{t} is a hermitian matrix-valued stochastic process whose diagonal matrix elements are standard real Brownian motions and whose off-diagonal matrix elements are standard complex Brownian motions.

For completeness we describe this matrix valued Ornstein-Uhlenbeck process more precisely. The rescaled matrix elements $z_{ij}=N^{1/2}h_{ij}$ evolve according to the complex Ornstein-Uhlenbeck process

For $i\neq j$, $\beta=\beta_{ij}$ is a complex Brownian motion with variance one. The real and imaginary parts of $z=x+iy$ satisfy

with $\beta=\frac{1}{\sqrt{2}}(\beta_{x}+i\beta_{y})$ and where $\beta_{x},\beta_{y}$ are independent standard real Brownian motions. For the diagonal elements $i=j$ in (3.2), $\beta_{ii}$ is a standard real Brownian motion with variance 1.

We note that ${\rm d}\mbox{Tr\,}H^{2}=0$, thus

If the initial condition of (3.1) is distributed according to the law of $\widehat{H}$, then the solution of (3.1) is clearly

where $V$ is a standard GUE matrix (with matrix elements having variance $1/N$) that is independent of $\widehat{H}$. With the choice of $t$ satisfying $(1-e^{-t})=s^{2}=N^{-3/4+\beta}$, i.e. $t=-\log(1-N^{-3/4+\beta})\approx N^{-3/4+\beta}$, we see that $H$ given in (2.3) has the same law as $H_{t}$.

2 Joint probability distribution of the eigenvalues

The measure $\widetilde{\mu}$ has a density with respect to Lebesgue measure given by

where \Delta_{N}(\mbox{\boldmath\lambda})=\prod_{i<j}(\lambda_{i}-\lambda_{j}). This is the joint probability distribution of the eigenvalues of the standard GUE ensemble normalized in such a way that the matrix elements have variance $1/N$ (see, e.g. ). With this normalization convention, the bulk of the one point function (density) is supported in $$and in the$N\to\infty$ limit it converges to the Wigner semicircle law (2.7).

For any finite time $t<\infty$ we will represent the joint probability density of the eigenvalues of $H_{t}$ as f_{t}(\mbox{\boldmath\lambda})\widetilde{u}(\mbox{\boldmath\lambda}), with \lim_{t\to\infty}f_{t}(\mbox{\boldmath\lambda})=1. In particular, we write the joint distribution of the eigenvalues of the initial Wigner matrix $\widehat{H}$ as f_{0}(\mbox{\boldmath\lambda})\widetilde{\mu}({\rm d}\mbox{\boldmath\lambda})=f_{0}(\mbox{\boldmath\lambda})\widetilde{u}(\mbox{\boldmath\lambda}){\rm d}\mbox{\boldmath\lambda}.

3 The generator of Dyson’s Brownian motion

The Ornstein-Uhlenbeck process (3.1) induces a stochastic process for the eigenvalues.

acting on $L^{2}(\widetilde{\mu})$ and let

be the corresponding Dirichlet form, where $\partial_{j}=\partial_{\lambda_{j}}$. Clearly $\widetilde{\mu}$ is an invariant measure for the dynamics generated by $L$.

Let the distribution of the eigenvalues of the Wigner ensemble be given by f_{0}(\mbox{\boldmath\lambda})\widetilde{\mu}({\rm d}\mbox{\boldmath\lambda}). We will evolve this distribution by the dynamics given by $L$:

The corresponding stochastic differential equation for the eigenvalues \mbox{\boldmath\lambda}(t) is now given by (see, e.g. Section 12.1 of )

where $\{B_{i}\;:\;1\leq i\leq N\}$ is a collection of independent Brownian motions and with initial condition \mbox{\boldmath\lambda}(0) that is distributed according to the probability density f_{0}(\mbox{\boldmath\lambda})\widetilde{\mu}({\rm d}\mbox{\boldmath\lambda}).

We remark that \widetilde{u}(\mbox{\boldmath\lambda}) and f_{t}(\mbox{\boldmath\lambda}) are symmetric functions of the variables $\lambda_{j}$ and $\widetilde{u}$ vanishes whenever two points coincide. By the level repulsion we also know that f_{0}(\mbox{\boldmath\lambda})\widetilde{u}(\mbox{\boldmath\lambda}) vanishes whenever $\lambda_{j}=\lambda_{k}$ for some $j\neq k$. We can label the eigenvalues according to their ordering, $\lambda_{1}<\lambda_{2}<\ldots<\lambda_{N}$, i.e. one can consider the configuration space

with $a=\frac{1}{2}(\lambda_{i}+\lambda_{j})$, $b=\frac{1}{2}(\lambda_{i}-\lambda_{j})$. The constant $1$ in front of the drift term is critical for the Bessel process $\frac{1}{2}\partial^{2}_{b}+\frac{1}{b}\partial_{b}$ not to reach the boundary point $b=0$.

The density function of the ordered eigenvalues is thus f_{t}(\mbox{\boldmath\lambda})u(\mbox{\boldmath\lambda}) on $\Xi^{(N)}$. Throughout this paper, with the exception of Section 5.2, we work on the space $\Xi^{(N)}$, i.e., the equilibrium measure \mu({\rm d}\mbox{\boldmath\lambda})=u(\mbox{\boldmath\lambda}){\rm d}\mbox{\boldmath\lambda} with density u(\mbox{\boldmath\lambda}) and the density function f_{t}(\mbox{\boldmath\lambda}) will be considered restricted to $\Xi^{(N)}$.

Good global configurations

Several estimates in this paper will rely on the fact that the number of eigenvalues ${\mathcal{N}}_{I}$ in intervals $I$ with length much larger than $1/N$ is given by the semicircle law . In this section we define the set of good global configurations, i.e. the event that the semicircle law holds on all subintervals in addition to a few other typical properties.

be the empirical density of the eigenvalues. For an interval $I=[a,b]$ we introduce the notation

for the number of eigenvalues in $I$. For the interval $[E-\eta/2,E+\eta/2]$ of length $\eta$ and centered at $E$ we will also use the notation

be the empirical density smoothed out on scale $\eta$. Furthermore, let

be the Stieltjes transform of the empirical eigenvalue distribution and

be the Stieljes transform of the semicircle law. The square root here is defined as the analytic extension (away from the branch cut $$) of the positive square root on large positive numbers. Clearly${\omega}_{y}(x)=\pi^{-1}\mbox{Im}\;m(x+iy)$for$y>0$.

We will need an improved version of Theorem 4.1 from that is also applicable near the spectral edges. The proof of the following theorem is given in Appendix A.

Assume that the Wigner matrix ensemble satisfies conditions (2.4)–(2.6) and assume that $y$ is such that $(\log N)^{4}/N\leq|y|\leq 1$.

where $C_{q}$ is independent of $x$ and $y$.

(ii) Assume that $|x|\leq K$ for some $K>0$. Then there exists $c>0$ such that

for all $\delta>0$ small enough and all $N$ large enough (independently of $\delta$). Consequently, we have

with some $q$-dependent constant $C_{q}$. Moreover,

for all $N$ large enough (independently of $x,y$).

(iii) Assuming $|x|\leq K$ and that $\sqrt{N|y||2-|x||}\geq(\log N)^{2}$ we also have

As a corollary to Theorem 4.1, the semicircle law for the density of states holds locally on very short scales. The next proposition can be proved, starting from Theorem 4.1, exactly as Eq. (4.3) was shown in .

Assuming (2.4)–(2.6), for any sufficiently small $\delta$ and for any $\eta^{*}$ with

(with a sufficiently large constant $C$) we have

We also need an estimate directly on the number of eigenvalues in a certain interval, but this will be needed only away from the spectral edge. The following two results estimate the deviation of the normalized empirical counting function $\frac{1}{N}{\mathcal{N}}[-\infty,E]=\frac{1}{N}\#\{\lambda_{j}\leq E\}$ and its expectation

from the distribution function of the semicircle law, defined as

Assume that the Wigner matrix ensemble satisfies conditions (2.4)–(2.6). Let $\kappa>0$ be fixed. For any $0<\delta<1$ and $|E|\leq 2-\kappa$, we have

with $\kappa$-dependent constants. Moreover, there exists a constant $C>0$ such that

The proof of this proposition will be given in Appendix B.

Next we define the good global configurations; the idea is that good global configurations are configurations for which the semicircle law holds up to scales of the order $(\log N)^{4}/N$ (and so that some more technical conditions are also satisfied). By Proposition 4.1 and Proposition 4.2, we will see that set of these configurations have, asymptotically, a full measure. As a consequence, we will be able to neglect all configurations that are not good.

with respect to any Wigner ensemble. This gives rise to the following definition.

Let $\eta^{*}_{m}=2^{m}n^{\gamma}N^{-1}$ with some small constant $\gamma>0$, $m=0,1,2,\ldots\log N$, and let $K$ be a fixed big constant. The event

will be called the set of good global configurations.

The probability of good global configurations satisfies

with respect to any Wigner ensemble satisfying the conditions (2.4) and (2.5)

Proof. The probability of $\Omega^{(m)}$ was estimated in (4.17). The probability of the second event in (4.18) can be estimated by (4.13) from Proposition 4.2 and from ${\mathfrak{N}}_{sc}(0)=1/2$. The third event is treated by the large deviation estimate on ${\mathcal{N}}_{I}$ for any interval $I$ with length $|I|\geq(\log N)^{2}/N$ (see Theorem 4.6 from ; note that there is a small error in the statement of this theorem, since the conditions $y\geq(\log N)/N$ and $|I|\geq(\log N)/N$ should actually be replaced by the stronger assumptions $y\geq(\log N)^{2}/N$ and $|I|\geq(\log N)^{2}/N$ which are used in its proof):

The fourth event is a large deviation of the largest eigenvalue, see, e.g. Lemma 7.4. in . $\Box$

In case of good configurations, the location of the eigenvalues are close to their equilibrium localition given by the semicircle law. The following lemma contains the precise statement and it will be proven in Appendix C.

Let $\lambda_{1}<\lambda_{2}<\ldots<\lambda_{N}$ denote the eigenvalues in increasing order and let $\kappa>0$. Then on the set $\Omega$ and if $N\geq N_{0}(\kappa)$, it holds that

for any $N\kappa^{3/2}\leq a\leq N(1-\kappa^{3/2})$ (recall the definition of ${\mathfrak{N}}_{sc}$ from (4.12)), and

for any $N\kappa^{3/2}\leq a<b\leq N(1-\kappa^{3/2})$ and $|b-a|\leq CNn^{-\gamma/6}$.

On the set $\Omega$ and with the choice $n$ given in (4.15), we have

with respect to any Wigner ensemble satisfying the conditions (2.4) and (2.5)

Proof. First we partition the interval $[-2+\kappa,2-\kappa]$ into subintervals

that have already been used in the proof of Lemma 4.3. On the set $\Omega$ we have the bound

The second sum is bounded by $Cn^{3\gamma}$. In the first sum, we use the level repulsion estimate by decomposing $I_{r-1}\cup I_{r}\cup I_{r+1}=\bigcup_{m}J_{m}$ into intervals of length $2^{k+2}N^{-1}$ that overlap at least by $2^{k+1}N^{-1}$, more precisely

where $m=1,2,\ldots,3n^{\gamma}\cdot 2^{-k-1}$. Then

Using the level repulsion estimate given in Theorem 3.4 of (here the condition (2.5) is used) and the fact that $J_{m}\subset I_{r-1}\cup I_{r}\cup I_{r+1}\subset[-2+\kappa,2-\kappa]$ since $|r|\leq r_{1}$, we have

Recalling the choice of $n$ completes the proof of Lemma 4.4. $\Box$

Global entropy

Recall the definition of the entropy of $f\mu$ with respect to $\mu$

and let $f_{t}$ solve (3.9). Then the evolution of the entropy is given by the equation

For dynamics with energy ${\mathcal{H}}$ and the convexity condition

for some constant $\Lambda$, the following Bakry-Emery inequality holds:

(notice the additional $N$ factor due to the $N^{-1}$ in front of the second order term in the generator $L$, see (3.7)). This implies the logarithmic Sobolev inequality that for any probability density $g$, with respect to $\mu$,

In this case, the Dirichlet form is a decreasing function in time and we thus have for any $t>s$ that

as a matrix inequality away from the singularities (see remark below how to treat the singular set). Thus we have

This tells us that $S(f_{t})$ in (3.9) is exponential decaying as long as $t\gg 1$. But for any time $t\sim 1$ fixed, the entropy is still the same order as the initial one. Note that $t\sim 1$ is the case considered in Johasson’s work .

Recall that the invariant measure \exp(-{\mathcal{H}}){\rm d}\mbox{\boldmath\lambda} and the dynamics $L=\frac{1}{2N}[\Delta-(\nabla{\mathcal{H}})\nabla]$ are restricted to $\Xi=\Xi^{(N)}$. With $h=\sqrt{f}$ we have

Computing $\partial_{t}D(\sqrt{f_{t}})$, we have

2 Bound on the entropy

Let $s=N^{-1}$. For any $\alpha>\frac{1}{4}$ we have

Given the density f_{0}(\mbox{\boldmath\lambda})\widetilde{\mu}({\rm d}\mbox{\boldmath\lambda}) of the eigenvalues of the Wigner matrix as an initial distribution, the eigenvalue density f_{s}(\mbox{\boldmath\lambda}) for the matrix evolved under the Dyson’s Brownian motion is given by

where $c=c(s)=e^{-s/2}$ for brevity. The derivation of (5.12) follows very similarly to Johansson’s presentation of the Harish-Chandra/Itzykson-Zuber formula (see Proposition 1.1 of ) with the difference that in our case the matrix elements move by the Ornstein-Uhlenbeck process (3.1) instead of the Brownian motion.

In particular, formula (5.12) implies that $f_{s}$ is an analytic function for any $s>0$ since

with an explicit analytic function h_{s}(\mbox{\boldmath\lambda}). Since the determinant is analytic in $\lambda$, we see that f_{s}(\mbox{\boldmath\lambda}) is meromorphic in each variables and the only possible poles of f_{s}(\mbox{\boldmath\lambda}) come from the factors $(\lambda_{i}-\lambda_{j})^{-1}$ in \Delta_{N}(\mbox{\boldmath\lambda}) near the coalescence points. But f_{s}(\mbox{\boldmath\lambda}) is a non-negative function, so it cannot have a singularity of order $-1$, thus these singular factors cancel out from a factor $(\lambda_{i}-\lambda_{j})$ from the integral. Alternatively, using the Laplace expansion the determinant, one can explicitly see that each 2 by 2 subdeterminant from the $i$-th and $j$-th columns carry a factor $\pm(\lambda_{i}-\lambda_{j})$.

Then, by Jensen inequality from (5.11) and from the fact that f_{0}(\mbox{\boldmath\nu})\widetilde{u}(\mbox{\boldmath\nu}) is a probability density, we have

Expanding this last expression we find, after an exact cancellation of the term $(N/2)\log(2\pi)$,

Since $s=N^{-1}$, we have $\log{c}=-1/2N$ and $\log(1-c^{2})=-\log N+O(N^{-1})$. Hence

For the determinant term, we use that each entry is at most one, thus

The last term in (5.13) can be estimated using Stirling’s formula and Riemann integration

thus the $\frac{1}{2}N^{2}\log{N}$ terms cancel. For the $N^{2}$ terms we need the following approximation

With respect to any Wigner ensemble whose single-site distribution satisfies (2.4)–(2.6) and for any $\alpha>1/4$ we have

where the constant in the error term depends on $\alpha$ and on the constants in (2.4)–(2.6).

Note that (2.6), (2.5) hold for both the initial Wigner ensemble with density $f_{0}$ and for the evolved one with density $f_{t}$. These conditions ensure that Theorem 3.5 of is applicable.

Proof of Lemma 5.2. The quadratic term can be computed explicitly using (3.4):

The second (determinant) term will be approximated in the following lemma whose proof is postponed to Appendix D.

With respect to any Wigner ensemble whose single-site distribution satisfies (2.4)–(2.6) and for any $\alpha>1/4$ we have

Finally, explicit calculation then shows that

Hence, continuing the estimate (5.13), we have the bound

Local equilibrium

Choose $t=\tau N^{-1}$ with some $\tau\geq 2$. Thus from (5.4) with $s=N^{-1}$, we have

by using (5.10). Recall that the eigenvalues are ordered, $\lambda_{1}<\lambda_{2}<\ldots<\lambda_{N}$. Let $L\leq N-n$ ($n$ was defined in (4.15)) and define

its complement. For convenience, we will relabel the elements of $\Pi_{L}$ as ${\bf{x}}=\{x_{1},x_{2},\ldots x_{n}\}$ in increasing order. The elements of $\Pi_{L}^{c}$ will be denoted by

again in increasing order ($\Xi$ was defined in (3.11)). We set

to be the index set of the $y$’s. We will refer to the $y$’s as external points and to the $x_{j}$’s as internal points. Note that the indices are chosen such that for any $j$ we have $y_{k}<x_{j}$ for $k<0$ and $y_{k}>x_{j}$ for $k>0$. In particular, for any fixed $L$, we can split any ${\bf{y}}\in\Xi^{(N-n)}$ as ${\bf{y}}=({\bf{y}}_{-},{\bf{y}}_{+})$ where

The set $\Xi^{(N-n)}$ with a splitting mark after the $L$-th coordinate will be denoted by $\Xi^{(N-n)}_{L}$ and we use the ${\bf{y}}\in\Xi^{(N-n)}\Longleftrightarrow({\bf{y}}_{-},{\bf{y}}_{+})\in\Xi^{(N-n)}_{L}$ one-to-one correspondance.

For a fixed $L$ we will often consider the expectation of functions $O({\bf{y}})$ on $\Xi^{(N-n)}$ with respect to $\mu$ or $f\mu$; this will always mean the marginal probability:

For a fixed $L\leq N-n$ and ${\bf{y}}\in\Xi^{(N-n)}$ let

be the conditional density of ${\bf{x}}$ given ${\bf{y}}$ with respect to the conditional equilibrium measure

Here $f_{\bf{y}}^{L}$ also depends on time $t$, but we will omit this dependence in the notation. Note that for any fixed ${\bf{y}}\in\Xi^{(N-n)}$, any value $x_{j}$ lies in the interval $I_{\bf{y}}:=[y_{-1},y_{1}]$, i.e. the functions $u_{\bf{y}}({\bf{x}})$ and $f_{\bf{y}}({\bf{x}})$ are supported on the set

Now we localize the good set $\Omega$ introduced in Definition 4.1. For any fixed $L$ and ${\bf{y}}=({\bf{y}}_{-},{\bf{y}}_{+})\in\Xi^{(N-n)}_{L}$ we define

Note that ${\bf{y}}\in\Omega_{1}$ also implies, for large $N$, that there exists an ${\bf{x}}\in I_{\bf{y}}^{n}$ such that $({\bf{y}}_{-},{\bf{x}},{\bf{y}}_{+})\in\Omega$. This ensures that those properties of \mbox{\boldmath\lambda}\in\Omega that are determined only by ${\bf{y}}$’s, will be inherited to the ${\bf{y}}$’s. E.g. ${\bf{y}}\in\Omega_{1}$ will guarantee that the local density of ${\bf{y}}$’s is close to the semicircle law on each interval away from $I_{\bf{y}}$. More precisely, note that for any interval $I=[E-\eta_{m}^{*}/2,E+\eta_{m}^{*}/2]$ of length $\eta_{m}^{*}=2^{m}n^{\gamma}N^{-1}$ and center $E$, $|E|\leq 2-\kappa$, that is disjoint from $I_{\bf{y}}$, we have, by (4.16),

Moreover, for any interval $I$ with $|I|\geq n^{\gamma}N^{-1}$ we have, by (4.18),

For any $L$ with $N\kappa^{3/2}\leq L\leq N(1-\kappa^{3/2})$, let $E_{L}={\mathfrak{N}}_{sc}^{-1}(LN^{-1})$, i.e.

Using (4.21) and (4.22) from Lemma 4.3 on the set $\Omega$ (see (4.18)), we for any ${\bf{y}}\in\Omega_{1}(L)$ we have

with some large constant $K$. On the set $\Omega$ we have $|I_{\bf{y}}|\leq Kn/N$ (see (6.14)), thus $\Pi_{L}^{c}(\Omega)\subset\Omega_{2}(L)$, i.e.

2 Localization of the Dirichlet form

For any $L\leq N-n$ and any ${\bf{y}}\in\Xi^{(N-n)}_{L}$, we define the Dirichlet form

for functions $f=f({\bf{x}})$ defined on $\Xi^{(n)}_{\bf{y}}$. Hence from (6.1) we have the inequality

and therefore, when we sum over all $L\in\{N\kappa^{3/2},\dots,N(1-\kappa^{3/2})\}$ as on the l.h.s. of (6.17), every local Dirichlet form is summed over at most $n$ times, so we get the total Dirichlet form with a multiplicity at most $n$.

then the above inequality guarantees that for the cardinality of ${\mathcal{G}}_{1}$,

3 Local entropy bound

Suppose that $L\in{\mathcal{G}}_{1}$ and fix it. For any ${\bf{y}}\in\Xi^{(N-n)}_{L}$ denote by

for any ${\bf{x}}\in I_{\bf{y}}^{n}$ as a matrix inequality. On the set ${\bf{y}}\in\Omega_{2}(L)$ we have

We can apply the logarithmic Sobolev inequality (5.3) to the local measure $\mu_{\bf{y}}$, taking into account Remark 5.1. Thus we have

for $\mu=\mu_{\bf{y}}$ and $f=f_{\bf{y}}$, we have also have

We will choose $t=N^{-1}\tau$ with $\tau=N^{\beta}$ such that

4 Good external configurations

The set of good $L$-indices is defined by

Lemma 4.4 together with (6.19) imply that

Notice that for any fixed $L$ we can write

and similar formulae hold when $\lambda_{L}$ is replaced with $\lambda_{L+n+1}$ and $y_{-1}$ with $y_{1}$.

We also want to ensure that the density on scale $\eta:=\eta_{0}^{*}=n^{\gamma}N^{-1}$ is close to the semicircle law. Let

be the characteristic function of the interval $[E-\eta/2,E+\eta/2]$. Consider $\Omega^{(0)}$ defined in (4.16), then $\Omega\subset\Omega^{(0)}$ and (4.19) imply that

Fix $L\in{\mathcal{G}}$, consider ${\bf{y}}\in\Xi^{(N-n)}_{L}$ and define

so that if $E\in I^{*}_{\bf{y}}$ then $[E-\eta/2,E+\eta/2]\subset I_{\bf{y}}$. Moreover, on the set $\Omega$ we know that $I_{\bf{y}}\subset[-2+\kappa/2,2-\kappa/2]$ (see (6.14)). Therefore

This gives rise to the following definition:

Let $L\in{\mathcal{G}}$. The set of good external points is given by

It follows from (6.8), (6.16), (6.21), (6.28) and (6.30) that

5 Bounds in equilibrium

In this section we translate the bounds in the second and third lines of (6.31) into similar bounds with respect to equilibrium using that the control on the local Dirichlet form also controls the local entropy for the good indices:

Let $A>0$ be arbitrary and ${\bf{y}}\in{\mathcal{Y}}_{L}$. If $\tau\geq n^{4A+8}N^{\alpha}$, i.e. $\beta\geq(4A+8)\varepsilon+\alpha$, then for $p=1,2$ we have

by the entropy inequality (6.25). If $L\in{\mathcal{G}}$ and ${\bf{y}}\in\Omega_{2}(L)$, then we have by (6.26) that

For a given ${\bf{y}}\in{\mathcal{Y}}_{L}$, we set the observable

with $\|\mathcal{O}\|_{\infty}\leq Cn^{Ap+1}\leq cn^{2A+1}$. Then, for $\tau\geq n^{4A+8}N^{\alpha}$ we obtain from (6.31) and (6.35) that

Combining the last two estimates proves (6.33).

The proof of (6.34) is analogous, here we use that the corresponding observable has an $L^{\infty}$ bound

This completes the proof of Lemma 6.1. $\Box$

Cutoff Estimates

In this section, we cutoff the interaction with the far away particles. We fix a good index $L\in{\mathcal{G}}$ and a good external point configuration ${\bf{y}}\in{\mathcal{Y}}_{L}$. Consider the measure $\mu_{\bf{y}}=e^{-{\mathcal{H}}_{\bf{y}}}/Z_{\bf{y}}$ with

The measure $\mu_{\bf{y}}$ is supported on the interval $I_{\bf{y}}=(y_{-1},y_{1})$.

where $B$ is a large positive number with $B\varepsilon<1/2$. We define the measure

Let $L\in{\mathcal{G}}$ and ${\bf{y}}\in{\mathcal{Y}}_{L}$. For $B\geq 20$, we have

This lemma will imply that one can cutoff all $y_{k}$’s in the potential with $|k|\geq n^{B}$.

then, by (6.15) and ${\bf{y}}\in{\mathcal{Y}}_{L}$, we have

In Lemma 7.2 we will give an upper bound on $\|V^{\prime}_{2}\|_{\infty}$, and then we have, for $B\geq 20$, that

For $B\geq 20$ and for any $L\in{\mathcal{G}}_{1}$, ${\bf{y}}\in{\mathcal{Y}}_{L}$ we have

Proof. Recall that ${\bf{y}}\in{\mathcal{Y}}_{L}\subset\Omega_{1}$ implies that the density of the $y$’s is close the semicircle law in the sense of (6.9). Let

Since ${\bf{y}}\in\Omega_{1}$, we know that $|y_{-1}|,|y_{1}|\leq 2-\kappa/2$ (see (6.14)), thus $\varrho_{sc}(y_{-1})\geq c>0$. Taking the imaginary part of (4.3) for $|z|\leq 2$ and renaming the variables, we have the identity

Furthermore, with $\bar{y}=\frac{1}{2}(y_{-1}+y_{1})$ we have

since $\bar{y}$ is away from the spectral edge thus $\varrho_{sc}$ is continuously differentiable on the interval of integration $[\bar{y}-d,\bar{y}+d]$. Thus

therefore to prove (7.7) it is sufficient to show that

We will consider only $k\geq n^{B}$ and compare the sum with the integral on the regime $y\geq\bar{y}+d$, the sum for $k\leq-n^{B}$ is similar.

Since ${\bf{y}}\in{\mathcal{Y}}_{L}\subset\Omega_{1}$, i.e. $\max|y_{k}|\leq K$, there will be no $y_{k}$ above the last interval $I_{\log N}$. We subdivide each $I_{m}$ into $n^{B/2}$ equal disjoint subintervals of length $2^{m}dn^{-B/2}$

For ${\bf{y}}\in{\mathcal{Y}}_{L}\subset\Omega_{1}$, the estimate (4.22) holds for $y_{1}$ and $y_{n^{B}}$, i.e.

(using $B\varepsilon<1/2$, $n^{B}\leq N^{1/2}$), i.e.

by using the definition of $d$ from (7.8), the fact that $\varrho_{sc}(y_{\pm 1})$ is separated away from zero and that $|I_{\bf{y}}|\leq CnN^{-1}$ from (6.14).

To see the last estimate, we notice that in the first summand we have $\bar{y}+d\leq y_{j}\leq y_{n^{B}}\leq\bar{y}+d+Cn^{4B/5}N^{-1}$ by (7.11), i.e. all these $y_{j}$’s lie in an interval of length $Cn^{4B/5}N^{-1}$, so their number is bounded by $Cn^{4B/5}$ by (6.10). Thus the first term in the right hand side of (7.12) is bounded by $Cn^{4B/5}N^{-1}d^{-1}\leq Cn^{1-B/5}$; the estimate of the second term is similar.

Finally, the second term on the left hand side of (7.14) is a Riemann sum of the integral in (7.9) with an error

Combining (7.12), (7.13), (7.14) and (7.15), we have proved (7.9) which completes the proof of Lemma 7.2. $\Box$

Derivative Estimate of Orthogonal Polynomials

In the next few sections, we will prove the boundedness and small distance regularity of the density. Our proof follows the approach of (cf: Lemma 3.3 and 3.4 in ), but the estimates are done in a different way due to the singularity of the potential. For the rest of this paper, it is convenient to rescale the local equilibrium measure to the interval $$ as we now explain.

Suppose $L\in{\mathcal{G}}$ and ${\bf{y}}\in{\mathcal{Y}}_{L}$. We change variables by introducing the transformation

then $T(I_{\bf{y}})=$. Let $\widetilde{\mu}_{\widetilde{\bf{y}}}$ be the measure $\mu^{(1)}_{\bf{y}}$ (see (7.5)) rescaled to the interval $$, i.e.,

Let $p_{j}(\lambda)$, $j=0,1,\ldots$ denote the real orthonormal polynomials on $$corresponding to the weight function$e^{-nU_{\widetilde{\bf{y}}}(\lambda)}$, i.e.$\mbox{deg}\;p_{j}=j$ and

to be orthonormal functions with respect to the Lebesgue measure on $$. Everything depends on${\bf{y}}$, but${\bf{y}}$ is fixed in this section and we will omit this dependence from the notation.

following the standard identities in orthogonal polynomials. For the rest of the paper we drop the tilde and all variables will denote the rescaled ones, i.e. all $x$ variables will be on the interval $$. All integrals in this section are understood on$$.

The basic ingredients of the approach can be described as follows: Suppose that the following two properties hold for the normalized function $\psi=\psi_{j}$, $j=n-1,n$, and for some fixed $\kappa>0$

for some positive $\sigma,\delta,\bar{\varepsilon}$ with $\sigma<1$. We will take take $\delta=1/4$, same as in . Let

Note that $|\psi(x_{0})|=O(n^{\frac{1}{2}-\varepsilon^{\prime}})$ with some $\varepsilon^{\prime}>0$ provided that $\sigma+\bar{\varepsilon}<\delta$. Suppose we can also prove that

with some small power $\varepsilon^{\prime\prime}$, then it will follow that $|\varrho^{\prime}(x)|\leq o(n)$ and this proves the regularity of the density over a distance of order $1/n$. Together with the fact that the density is well approximated with the semicircle law on scales bigger than $1/n$ this will show that the density is close to the semicirle law pointwise. In the regularity of the density on larger scales followed from the smoothness of the potential (Theorem 2.2 of ). In our case this follows from (6.34) which is a consequence of the fact that the semicircle law is precise on scales slightly larger than $1/N$ that corresponds to scales bigger than $1/n$ after rescaling.

In proving (8.9), (8.10) and (8.12), one basic assumption in requires the potential to be in $C^{2+\nu}$ for some $\nu>0$. The potential for our probability measure (8.2), parametrized by the boundary conditions ${\bf{y}}$, is singular near the boundary points $\{\pm 1\}$. In order to control these singularities, besides using some special properties of orthogonal polynomials, we rely on via (6.33) to provide essential estimates such as level repulsions. It turns out that we can only establish (8.9) and (8.10) for $\psi_{j},j\leq n-1$ following this idea. The case of $j=n$ has to be treated completely differently. We now start to prove (8.9) for $\psi_{j},j=n-1,n-2$.

Suppose that $L\in{\mathcal{G}}$, ${\bf{y}}\in{\mathcal{Y}}_{L}$ and, after rescaling that sets $y_{-1}=-1$, $y_{1}=1$, let the ${\bf{y}}$-configuration satisfy

(note that the boundary terms $k=\pm 1$ are not included in the summations). Furthermore, assume that the density $\varrho_{n}$ satisfies

for some $A\geq 60B$. Then for the orthonormal functions $\psi_{j}$ from (8.4) we have

Notice that the assumptions (8.13) and (8.14) follow from (6.31) and (6.33).

In this section and in the subsequent Sections 9 and 10 we work with orthogonal polynomials on $$with respect to the potential$U_{\widetilde{\bf{y}}}(x)$(see (8.2)). For brevity, we set$V(x)=U_{\widetilde{\bf{y}}}(x)$in these three sections and we make the convention that the summation over the index$k$that labels the elements of the external configuration${\bf{y}}$will always run over integers with for$1\leq|k|

Proof. For simplicity, let $p(x)=p_{j}(x)$ and $\psi(x)=\psi_{j}(x)$. Then

Note that $e^{-nV(x)}$ is zero at the boundary $x=\pm 1$ so the boundary term vanishes in the integration by parts. Since $p(x)$ is an orthogonal polynomial, it is orthogonal to all polynomials of lower degree, thus the first integral vanishes. By Schwarz inequality, the second integral is bounded by

From (8.13), and the normalization of $\psi$ we have

To control the term $I_{1}$, we separate the integration regimes $|x\pm 1|\leq n^{-A}$ and $-1+n^{-A}\leq x\leq 1-n^{-A}$ for some big constant $A$. In the inside regime, we can use $|\psi(x)|^{2}=|\psi_{j}(x)|^{2}\leq n\varrho_{n}(x)$ since $j\leq n-1$. From (8.14) we obtain

To estimate the singular part of the integral in $I_{1}$ near the boundary points, we can focus in estimating

Notice that $g(x)$ is a polynomial of degree $\mbox{deg}\;g\leq 2n^{2B}+n$. From the Nikolskii inequality (see, e.g., Theorem A.4.4 of )

with some universal constant $C$. Here $\|g\|_{p}$ is defined as \big{(}\int_{-1}^{1}|g(x)|^{p}{\rm d}x\big{)}^{1/p} for any $0<p<\infty$. Notice that Nikolskii inequality holds between $L^{p}$ spaces even with exponents $p<1$. By the Hölder inequality,

Thus from (8.22) we have $\|g\|_{4}\leq Cn^{15B}$ and by Hölder inequality we have

provided $A\geq 60B$. Together with (8.21), this proves $I_{1}\leq Cn^{1+4\gamma}$. Combining this with (8.20) we obtain a bound $Cn^{2+6\gamma}$ for (8.18) which proves (8.15).

Using this estimate and (8.17) we obtain that

by using (8.18). This completes the proof. $\Box$

Bound on smeared-out orthogonal polynomials

Let $\kappa,\delta_{0}>0$ be arbitrary positive numbers. Let $L\in{\mathcal{G}}$, ${\bf{y}}\in{\mathcal{Y}}_{L}$, suppose that the ${\bf{y}}$-configuration satisfies (8.13), (8.14) and the density $\varrho_{n}(x)\geq\delta_{0}>0$ for all $|x|\leq 1-\kappa$. Let $\psi=\psi_{n-1}$ or $\psi_{n-2}$ be an orthogonal function. Then we have

with a constant $C$ depending on $\kappa$ and $\delta_{0}$.

denote the Stieltjes transform of the density and denote by

the truncated correlation function, where $\widetilde{p}^{(2)}_{n}$ was defined in (8.3) and computed from (8.8). We will again drop the tilde in this proof.

This identity follows from expressing $\varrho_{n}$ by an integral over $n-1$ variables of the equilibrium measure and then integrating by parts (see also (2.81) of ). Hence, by using (8.6), we have

where, to estimate the last integral, we have used the Christoffel-Darboux formula

We define a new measure $\mu^{-}_{{\bf{y}}}$ on $^{n-1}$ as

where we already omitted the tildes and recall that $V(x)=U_{\bf{y}}(x)$. Note that this measure differs from (8.1) written in $n-1$ variables in that we kept the prefactor $n$ in front of the potential. Define

where $\psi_{j}$’s are defined in (8.4). This latter formula follows from the recursive relation of the correlation functions for GUE-like ensembles, therefore

be the Stieltjes transform of $\varrho_{n}^{-}$; then we have the analogue of (9.6)

Assume that $u=\mbox{Re}\,z$ satisfies $|u-x_{0}|\leq n^{-1/4}$. By adding $n(m_{n}(z)-m_{n}^{-}(z))V^{\prime}(u)$ to the both sides of (9.7), we obtain

We divide the integral into $|x-x_{0}|\leq\nu/2$ and $|x-x_{0}|\geq\nu/2$. In the first integration regime, since $|x_{0}|\leq 1-\nu$, we have

Since $|x|\leq 1-\nu/2$, $|u|\leq 1-\nu/2$, we have $|y_{k}-u|\geq 2\nu^{-1}$ for any $k$. Thus, by (8.13), the prefactor in (9.8) is bounded, uniformly in $|x|\leq 1-\nu/2$, by

where the constant $C$ depends on $\nu$ and we recall that $y_{-1}=-1$, $y_{1}=1$ in the rescaled variables.

In the second integration regime we use $|x-u|\geq|x-x_{0}|-|x_{0}-u|\geq\nu/4$ and obtain

where we have used (8.15) and Hölder inequality to estimate the first term in the last line and using (8.13) for the second term.

using that ${\text{I}m}\,m_{n}^{-}(z)>0$. Since $\varrho_{n}(x)\geq\delta_{0}>0$ by assumption, ${\text{I}m}\,m_{n}(z)$ is bounded from below. Thus, choosing $\eta=n^{-1/4}$, we obtain