The Eigenvector Moment Flow and local Quantum Unique Ergodicity

Introduction

Wigner envisioned that the laws of the eigenvalues of large random matrices are new paradigms for universal statistics of large correlated quantum systems. Although this vision has not been proved for any truly interacting quantum system, it is generally considered to be valid for a wide range of models. For example, the quantum chaos conjecture by Bohigas-Giannoni-Schmit asserts that the eigenvalue statistics of the Laplace operator on a domain or manifold are given by the random matrix statistics, provided that the corresponding classical dynamics are chaotic. Similarly, one expects that the eigenvalue statistics of random Schrödinger operators (Anderson tight binding models) are given by the random matrix statistics in the delocalization regime. Unfortunately, both conjectures are far beyond the reach of the current mathematical technology.

In Wigner’s original theory, the eigenvector behaviour plays no role. As suggested by the Anderson model, random matrix statistics coincide with delocalization of eigenvectors. A strong notion of delocalization, at least in terms of “flatness of the eigenfunctions”, is the quantum ergodicity. For the Laplacian on a negative curved compact Riemannian manifold, Shnirel’man , Colin de Verdière and Zelditch proved that quantum ergodicity holds. More precisely, let $(\psi_{k})_{k\geqslant 1}$ denote an orthonormal basis of eigenfunctions of the Laplace-Beltrami operator, associated with increasing eigenvalues, on a negative curved manifold $\mathcal{M}$ (or more generally, assume only that the geodesic flow of $\mathcal{M}$ is ergodic) with volume measure $\mu$. Then, for any open set $A\subset\mathcal{M}$, one has

where $N(\lambda)=|\{j:\lambda_{j}\leqslant\lambda\}|$. Quantum ergodicity was also proved for $d$-regular graphs under certain assumptions on the injectivity radius and spectral gap of the adjacency matrices . Random graphs are considered a good paradigm for many ideas related to quantum chaos .

An even stronger notion of delocalization is the quantum unique ergodicity conjecture (QUE) proposed by Rudnick-Sarnak , i.e., for any negatively curved compact Riemannian manifold $\mathcal{M}$, the eigenstates become equidistributed with respect to the volume measure $\mu$: for any open $A\subset\mathcal{M}$ we have

Some numerical evidence exists for both eigenvalue statistics and the QUE, but a proper understanding of the semiclassical limit of chaotic systems is still missing. One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces . For results in the case of general compact Riemannian manifolds whose geodesic flow is Anosov, see .

A major class of matrices for which one expects that Wigner’s vision holds is the Wigner matrices, i.e., random matrices with matrix elements distributed by identical mean-zero random variables. For this class of matrices, the Wigner-Dyson-Mehta conjecture states that the local statistics are independent of the laws of the matrix elements and depend only on the symmetry class. This conjecture was recently solved for an even more general class: the generalized Wigner matrices for which the distributions of matrix entries can vary and have different variances. (See and for a review. For earlier results on this conjecture for Wigner matrices, see for the bulk of the spectrum and for the edge). One key ingredient of the method initiated in proceeds by interpolation between Wigner and Gaussian ensembles through Dyson Brownian motion, a matrix process that induces an autonomous evolution of eigenvalues. The fundamental conjecture for Dyson Brownian motion, the Dyson conjecture, states that the time to local equilibrium is of order $t\gtrsim 1/N$, where $N$ is the size of the matrix. This conjecture was resolved in (see for the earlier results) and is the underlying reason for the universality.

Concerning the eigenvectors distribution, complete delocalization was proved in for generalized Wigner matrices in the following sense : with very high probability

where $C$ is a fixed constant and the maximum ranges over all coordinates $\alpha$ of the ${\rm L}^{2}$-normalized eigenvectors, $u_{1},\dots,u_{N}$ (a stronger estimate was obtained for Wigner matrices in , see also for a delocalization bound for the Laplacian on deterministic regular graphs). Although this bound prevents concentration of eigenstates onto a set of size less than $N(\log N)^{-C\log\log N}$, it does not imply the “complete flatness” of type (1.1). In fact, if the eigenvectors are distributed by the Haar measure on the orthogonal group, the weak convergence

holds, where $\mathscr{N}$ is a standard Gaussian random variable and the eigenvector components are asymptotically independent. Since the eigenvectors of GOE are distributed by the Haar measure on the orthogonal group, this asymptotic normality (1.2) holds for GOE (and a similar statement holds for GUE). For Wigner ensembles, by comparing with GOE, this property was proved for eigenvectors in the bulk by Knowles-Yin and Tao-Vu under the condition that the first four moments of the matrix elements of the Wigner ensembles match those of the standard normal distribution. For eigenvectors near the edges, the matching condition can be reduced to only the first two moments .

In this paper, we develop a completely new method to show that this asymptotic normality (1.2) and independence of eigenvector components hold for generalized Wigner matrices without any moment matching condition. In particular, even the second moments are allowed to vary as long as the matrix stays inside the generalized Wigner class. From the law of large numbers of independent random variables, this implies the local quantum unique ergodicity, to be specified below, with high probability. In fact, we will prove a stronger form of asymptotic normality in the sense that any projection of the eigenvector is asymptotically normal, see Theorem 1.2. This can be viewed as the eigenvector universality for the generalized Wigner ensembles.

The key idea in this new approach is to analyze the “Dyson eigenvector flow”. More precisely, the Dyson Brownian motion is induced by the dynamics in which matrix elements undergo independent Brownian motions. The same dynamics on matrix elements yield a flow on the eigenvectors. This eigenvector flow, which we will call the Dyson eigenvector flow, was computed in the context of Brownian motion on ellipsoids , real Wishart processes , and for GOE/GUE in (see also ). This flow is a diffusion process on a compact Lie group ($O(N)$ or $U(N)$) endowed with a Riemannian metric. This diffusion process roughly speaking can be described as follows. We first randomly choose two eigenvectors, $u_{i}$ and $u_{j}$. Then we randomly rotate these two vectors on the circle spanned by them with a rate $(\lambda_{i}-\lambda_{j})^{-2}$ depending on the eigenvalues. Thus the eigenvector flow depends on the eigenvalue dynamics. If we freeze the eigenvalue flow, the eigenvector flow is a diffusion with time dependent singular coefficients depending on the eigenvalues.

Due to its complicated structure, the Dyson eigenvector flow has never been analyzed. Our key observation is that the dynamics of the moments of the eigenvector entries can be viewed as a multi-particle random walk in a random environment. The number of particles of this flow is one half of the degree of polynomials in the eigenvector entries, and the (dynamic) random environment is given by jump rates depending on the eigenvalues. We shall call this flow the eigenvector moment flow. If there is only one particle, this flow is the random walk with the random jump rate $(\lambda_{i}-\lambda_{j})^{-2}$ between two integer locations $i$ and $j$. This one dimensional random walk process was analyzed locally in for the purpose of the single gap universality between eigenvalues. An important result of is the Hölder regularity of the solutions. In higher dimensions, the jump rates depend on the locations of nearby particles and the flow is not a simple tensor product of the one dimensional process. Fortunately, we find that this flow is reversible with respect to an explicit equilibrium measure. The Hölder regularity argument in can be extended to any dimension to prove that the solutions of the moment flow are locally Hölder continuous. From this result and the local semicircle law (more precisely, the isotropic local semicircle law proved in and ), one can obtain that the bulk eigenvectors generated by a Dyson eigenvector flow satisfy local quantum unique ergodicity, and the law of the entries of the eigenvectors are Gaussian.

Instead of showing the Hölder regularity, we will directly prove that the solution to the eigenvector moment flow converges to a constant. This proof is based on a maximum principle for parabolic differential equations and the local isotropic law previously mentioned. It yields the convergence of the eigenvector moment flow to a constant for $t\gtrsim N^{-1/4}$ with explicit error bound. This immediately implies that all eigenvectors (in the bulk and at the edge) generated by a Dyson eigenvector flow satisfy local quantum unique ergodicity, and the law of the entries of the eigenvectors are Gaussian.

The time to equilibrium $t\gtrsim N^{-1/4}$ mentioned above is not optimal and the correct scaling of relaxation to equilibrium is $t\sim N^{-1}$ in the bulk, similar to Dyson’s conjecture for relaxation of bulk eigenvalues to local equilibrium. In other words, we expect that Dyson’s conjecture can be extended to the eigenvector flow bulk as well. We will give a positive answer to this question in Theorem 7.1. A key tool in proving this theorem is a finite speed of propagation estimate for the eigenvector moment flow. An estimate of this type was first proved in [17, Section 9.6], but it requires a difficult level repulsion estimate. In Section 6, we will prove an optimal finite speed of propagation estimate without using any level repulsion estimate.

In order to prove that the eigenvectors of the original matrix ensemble satisfy quantum ergodicity, it remains to approximate the Wigner matrices by Gaussian convoluted ones, i.e., matrices that are a small time solution to the Dyson Brownian motion. We invoke the Green function comparison theorem in a version similar to the one stated in . For bulk eigenvectors, we can remove this small Gaussian component by a continuity principle instead of the Green function comparison theorem: we will show that the Dyson Brownian motion preserves the detailed behavior of eigenvalues and eigenvectors up to time $N^{-1/2}$ directly by using the Itô formula. This approach is much more direct and there is no need to construct moment matching matrices.

The eigenvector moment flow developed in this paper can be applied to other random matrix models. For example, the local quantum unique ergodicity holds for covariance matrices (for the associated flow and results, see Appendix C) and a certain class of Erdős-Rényi graphs. To avoid other technical issues, in this paper we only consider generalized Wigner matrices. Before stating the results and giving more details about the proof, we recall the definition of the considered ensemble.

Normalization: for any $j\in\llbracket 1,N\rrbracket$, $\sum_{i=1}^{N}\sigma_{ij}^{2}=1$.

In the following, $(u_{i})_{i=1}^{N}$ denotes an orthonormal eigenbasis for $H_{N}$, a matrix from the (real or complex) generalized Wigner ensemble. The eigenvector $u_{i}$ is associated with the eigenvalue $\lambda_{i}$, where $\lambda_{1}\leqslant\dots\leqslant\lambda_{N}$.

in the sense of convergence of moments, where all $\mathscr{N}_{j},\mathscr{N}^{(1)}_{j},\mathscr{N}^{(2)}_{j}$, are independent standard Gaussian random variables. This convergence holds uniformly in $I$ and $|{\bf q}|=1$. More precisely, for any polynomial $P$ in $m$ variables, there exists ${\varepsilon}={\varepsilon}(P)>0$ such that for large enough $N$ we have

respectively for the real and complex generalized Wigner ensembles.

The restriction on the eigenvector in the immediate regime $\llbracket N^{1/4},N^{1-\delta}\rrbracket$ (and similarly for its reflection) was due to that near the edges the level repulsion estimate, Definition 5.1, (or the gap universality) was only written in the region $\llbracket 1,N^{1/4}\rrbracket$ (see the discussion after Definition 5.1 for references regarding this matter). There is no doubt that these results can be extended to the the immediate regime with only minor modifications in the proofs. Here we state our theorem based on existing written results.

The normal convergence (1.4) was proved in under the assumption that the entries of $H_{N}$ have moments matching the standard Gaussian distribution up to order four, and if their distribution is symmetric (in particular the fifth moment vanishes).

This convergence of moments implies in particular joint weak convergence. Choosing ${\bf q}$ to be an element of the canonical basis, Theorem 1.2 implies in particular that any entry of an eigenvector is asymptotically normally distributed, modulo the (arbitrary) phase choice. Because the above convergence holds for any $|{\bf q}|=1$, asymptotic joint normality of the eigenvector entries also holds. Since eigenvectors are defined only up to a phase, we define the equivalence relation $u\sim v$ if $u=\pm v$ in the symmetric case and $u=\lambda v$ for some $|\lambda|=1$ in the Hermitian case.

for the symmetric (resp. Hermitian) generalized Wigner ensembles. Here $\omega$ is independent of $H_{N}$ and uniform on the binary set $\{0,\pi\}$ (resp. $(0,2\pi)$).

By characterizing the joint distribution of the entries of the eigenvectors, Theorem 1.2 and Corollary 1.3 imply that for any eigenvector a probabilistic equivalent of $(\ref{eqn:QUE})$ holds. For $a_{N}:\llbracket 1,N\rrbracket\to$ we denote $|a_{N}|=|\{1\leqslant\alpha\leqslant N:a_{N}(\alpha)\neq 0\}|$ the size of the support of $a_{N}$, and $\langle u_{k},a_{N}u_{k}\rangle=\sum|u_{k}(\alpha)|^{2}a_{N}(\alpha)$.

Under the condition that the first four moments of the matrix elements of the Wigner ensembles match those of the standard normal distribution, (1.7) can also be proved from the results in ; the four moment matching were reduced to two moments for eigenvectors near the edges .

The quantum ergodicity for a class of sparse regular graphs was proved by Anantharaman-Le Masson , partly based on pseudo-differential calculus on graphs from . The main result in is for deterministic graphs, but for the purpose of this paper we only state its application to random graphs (see for details and more general statements). If $u_{1},\dots,u_{N}$ are the (${\rm L}^{2}$-normalized) eigenvectors of the discrete Laplacian of a uniformly chosen $(q+1)$-regular graph with $N$ vertices, then for any fixed $\delta>0$ we have, for any $q\geqslant 1$ fixed,

where $a_{N}$ may be random (for instance, it may depend on the graph). The results in were focused on very sparse deterministic regular graphs and are very different from our setting for generalized Wigner matrices.

Notice that our result (1.7) allows the test function to have a very small support and it is valid for any $k$. This means that eigenvectors are flat even in “microscopic scales”. However, the equation (1.7) does not imply that all eigenvectors are completely flat simultaneously with high probability, i.e., we have not proved the following statement:

for $a_{N}$ with support of order $N$. This strong form of QUE, however, holds for the Gaussian ensembles.

In the following section, we will define the Dyson vector flow and, for the sake of completeness, prove the well-posedness of the eigenvector stochastic evolution. In Section 3 we will introduce the eigenvector moment flow and prove the existence of an explicit reversible measure. In Section 4, we will prove Theorem 1.2 under the additional assumption that $H_{N}$ is the sum of a generalized Wigner matrix and a Gaussian matrix with small variance. The proof in this section relies on a maximum principle for the eigenvector moment flow. We will prove Theorem 1.2 by using a Green function comparison theorem in Section 5. In Section 6, we will prove that the speed of propagation for the eigenvector moment flow is finite with very high probability. This estimate will enable us to prove in Section 7 that the relaxation to equilibrium for the eigenvector moment flow in the bulk is of order $t\gtrsim N^{-1}$. The appendices contain a continuity estimate for the Dyson Brownian motion up to time $N^{-1/2}$, and some basic results concerning the generator of the Dyson vector flow as well as analogue results for covariance matrices.

Dyson Vector Flow

In this section, we first state the stochastic differential equation for the eigenvectors under the Dyson Brownian motion. This evolution is given by (2.3) and (2.5). We then give a concise form of the generator for this Dyson vector flow. We will follow the usual slight ambiguity of terminology by naming both the matrix flow and the eigenvalue flow a Dyson Brownian motion. In case we wish to distinguish them, we will use matrix Dyson Brownian motion for the matrix flow.

Hereafter is our choice of normalization for the Dyson Brownian motion.

Let $B^{(s)}$ be a $N\times N$ matrix such that $B^{(s)}_{ij}(i<j)$ and $B^{(s)}_{ii}/\sqrt{2}$ are independent standard Brownian motions, and $B^{(s)}_{ij}=B^{(s)}_{ji}$. The $N\times N$ symmetric Dyson Brownian motion $H^{(s)}$ with initial value $H^{(s)}_{0}$ is defined as

Let $B^{(h)}$ be a $N\times N$ matrix such that $\Re(B^{(h)}_{ij}),\Im(B^{(h)}_{ij})(i<j)$ and $B^{(h)}_{ii}/\sqrt{2}$ are independent standard Brownian motions, and $B^{(h)}_{ji}=(B^{(h)}_{ij})^{*}$. The $N\times N$ Hermitian Dyson Brownian motion $H^{(h)}$ with initial value $H^{(t)}_{0}$ is

We refer to the following stochastic differential equations as the Dyson Brownian motion for (2.2) and (2.4) and the Dyson vector flow for (2.3) and (2.5).

Let $\bm{\lambda}_{0}\in\Sigma_{N}=\{\lambda_{1}<\dots<\lambda_{N}\}$, \mbox{\boldmathu}_{0}\in\OO(N), and $B^{(s)}$ be as in Definition 2.1. The symmetric Dyson Brownian motion/vector flow with initial condition $(\lambda_{1},\dots,\lambda_{N})=\bm{\lambda}_{0}$, (u_{1},\dots,u_{N})=\mbox{\boldmathu}_{0}, is

Let $\bm{\lambda}_{0}\in\Sigma_{N}$, \mbox{\boldmathu}_{0}\in\UU(N), and $B^{(h)}$ be as in Definition 2.1. The Hermitian Dyson Brownian motion/vector flow with initial condition $(\lambda_{1},\dots,\lambda_{N})=\bm{\lambda}_{0}$, (u_{1},\dots,u_{N})=\mbox{\boldmathu}_{0}, is

The theorem below contains the following results. (a) The above stochastic differential equations admit a unique strong solution, this relies on classical techniques and an argument originally by McKean . (b) The matrix Dyson Brownian motion induces the standard Dyson Brownian motion (for the eigenvalues) and Dyson eigenvector flow. This statement was already proved in . (c) For calculation purpose, one can condition on the trajectory of the eigenvalues to study the eigenvectors evolution. For the sake of completeness, this theorem is proved in the appendix.

With a slight abuse of notation, we will write $\bm{\lambda}_{t}$ either for $(\lambda_{1}(t),\dots,\lambda_{N}(t))$ or for the $N\times N$ diagonal matrix with entries $\lambda_{1}(t),\dots,\lambda_{N}(t)$.

The following statements about the Dyson Brownian motion and eigenvalue/vector flow hold.

Existence and strong uniqueness hold for the system of stochastic differential equations (2.2), (2.3). Let (\bm{\lambda}_{t},\mbox{\boldmathu}_{t})_{t\geqslant 0} be the solution. Almost surely, for any $t\geqslant 0$ we have $\bm{\lambda}_{t}\in\Sigma_{N}$ and \mbox{\boldmathu}_{t}\in\OO(N).

Let $(H_{t})_{t\geqslant 0}$ be a symmetric Dyson Brownian motion with initial condition H_{0}=\mbox{\boldmathu}_{0}\bm{\lambda}_{0}\mbox{\boldmathu}_{0}^{*}, $\bm{\lambda}_{0}\in\Sigma_{N}$. Then the processes $(H_{t})_{t\geqslant 0}$ and (\mbox{\boldmathu}_{t}\bm{\lambda}_{t}\mbox{\boldmathu}_{t}^{*})_{t\geqslant 0} have the same distribution.

Existence and strong uniqueness hold for (2.2). For any $T>0$, let $\nu_{T}^{H_{0}}$ be the distribution of $(\bm{\lambda}_{t})_{0\leqslant t\leqslant T}$ with initial value the spectrum of a matrix $H_{0}$. For $0\leqslant T\leqslant T_{0}$ and any given continuous trajectory $\bm{\lambda}=(\bm{\lambda}_{t})_{0\leqslant t\leqslant T_{0}}\subset\Sigma_{N}$, existence and strong uniqueness holds for (2.3) on $[0,T]$. Let $\mu^{H_{0},\bm{\lambda}}_{T}$ be the distribution of (\mbox{\boldmathu}_{t})_{0\leqslant t\leqslant T} with the initial matrix $H_{0}$ and the path $\bm{\lambda}$ given.

The analogous statements hold in the Hermitian setting.

Eigenvector Moment Flow

With this $\sqrt{N}$ normalization, the typical size of $z_{k}$ is of order $1$. We assume that the eigenvalue trajectory $(\lambda_{k}(t),0\leqslant t\leqslant T_{0})_{k=1}^{N}$ in the simplex $\Sigma^{(N)}$ is given. Furthermore, $u$ is the unique strong solution of the stochastic differential equation (2.3) (resp. (2.5)) with the given eigenvalue trajectory. Let ${\rm P}^{(s)}(t)={\rm P}^{(s)}(z_{1},\dots,z_{N})(t)$ and ${\rm P}^{(h)}={\rm P}^{(h)}(z_{1},\dots,z_{N})(t)$ be smooth functions. Then a simple calculation yields

For $m\in\llbracket 1,N\rrbracket$, denote by $j_{1},\dots,j_{m}$ positive integers and let $i_{1},\dots,i_{m}$ in $\llbracket 1,N\rrbracket$ be $m$ distinct indices. The test functions we will consider are:

For any $m$ fixed, linear combinations of such polynomial functions are stable under the action of the generator. More precisely, the following formulas hold.

In the Hermitian setting, we note that the polynomials ${\rm P}^{(h)}$ are invariant under the permutation $z_{i}\to\overline{z}_{i}$. Thus the action of the generator ${\rm L}^{(h)}$ (2.9) on such functions ${\rm P}^{(h)}$ simplifies to

We now normalize the polynomials by defining

Thanks to the scalings (3.3) and (3.4), on the right hand sides of the above four equations, the sums of the coefficients vanish. This allows us to interpret them as multi-particle random walks (in random environments) in the next subsection.

2 Multi-particle random walk.

if the configuration of $\bm{\eta}$ is the same as the one given by the $i,j$’s. Here $\bm{\lambda}$ denotes the whole path of eigenvalues for $0\leqslant t\leqslant 1$. The dependence in the initial matrix $H_{0}$ will often be omitted so that we write $f^{(s)}_{\bm{\lambda},t}=f^{H_{0},(s)}_{\bm{\lambda},t}$, $f^{(h)}_{\bm{\lambda},t}=f^{H_{0},(h)}_{\bm{\lambda},t}$. The following theorem summarizes the results from the previous subsection. It also defines the eigenvector moment flow, through the generators (3.7) and (3.8). They are multi-particles random walks (with $n=\mathcal{N}(\bm{\eta})$ particles) in random environments with jump rates depending on the eigenvalues.

Suppose that $u$ is the solution to the symmetric Dyson vector flow (2.3) and $f^{(s)}_{\bm{\lambda},t}(\bm{\eta})$ is given by (3.5) where $\bm{\eta}$ denote the configuration $\{(i_{1},j_{1}),\dots,(i_{m},j_{m})\}$. Then $f^{(s)}_{\bm{\lambda},t}$ satisfies the equation

Suppose that $u$ is the solution to the Hermitian Dyson vector flow (2.5), and $f^{(h)}_{\bm{\lambda},t}$ is given by (3.5). Then it satisfies the equation

An important property of the eigenvector moment flow is reversibility with respect to a simple explicit equilibrium measure. In the Hermitian case, this is simply the uniform measure on the configuration space.

Recall that a measure $\pi$ on the configuration space is said to be reversible with respect to a generator ${\rm L}$ if $\sum_{\bm{\eta}}\pi(\bm{\eta})g(\bm{\eta}){\rm L}f(\bm{\eta})=\sum_{\bm{\eta}}\pi(\bm{\eta})f(\bm{\eta}){\rm L}g(\bm{\eta})$ for any functions $f$ and $g$. We then define the Dirichlet form by

For the eigenvector moment flow, the following properties hold.

Define a measure on the configuration space by assigning the weight

Then $\pi^{(s)}$ is a reversible measure for $\mathscr{B}^{(s)}$ and the Dirichlet form is given by

The uniform measure ($\pi^{(h)}(\bm{\eta})=1$ for all $\bm{\eta}$) is reversible with respect to $\mathscr{B}^{(h)}$. The associated Dirichlet form is

We first consider $(i)$, concerning the symmetric eigenvector moment flow. The measure $\pi^{(s)}$ is reversible for $\mathscr{B}^{(s)}$ for any choice of the coefficients satisfying $c_{ij}=c_{ji}$ if and only if, for any $i<j$,

A sufficient condition is clearly that both of the following equations hold:

Consider the left hand side of the first one of these two equations. Let $\bm{\xi}=\bm{\eta}^{ij}$. If $\xi_{j}>0$ then $\bm{\eta}=\bm{\xi}^{ji}$, $\eta_{i}=\xi_{i}+1$ and $\eta_{j}=\xi_{j}-1$. For the right hand side of the second equation, we make the change of variables $\bm{\xi}=\bm{\eta}^{ji}$. Finally, rename all the variables on the right hand sides by $\xi$. Thus the above equations are equivalent to

Clearly, both equations hold provided that

If the measure is of type $\pi^{(s)}(\bm{\eta})=\prod_{x}\phi(\eta_{x})$ and we note $\xi_{i}=a,\xi_{j}=b$, this equation is equivalent to

and the second equation yields the same condition with the roles of $a$ and $b$ switched. This holds for all $a$ and $b$ if $\phi(k+1)=((2k+1)/(2k+2))\phi(k)$, which gives (3.9) provided we normalize $\phi(0)=1$. In the case $(ii)$, the same reasoning yields that $\phi$ is constant.

Finally, the Dirichlet form calculation is standard: for example, for $(i)$, $\sum_{\bm{\eta}}\pi^{(s)}(\bm{\eta})\mathscr{B}^{(s)}(f^{2})(\bm{\eta})=0$ by reversibility. Noting $\mathscr{B}^{(s)}(f^{2})(\bm{\eta})=2f(\bm{\eta})\mathscr{B}^{(s)}f(\bm{\eta})-\sum\pi^{(s)}(\bm{\eta})2\eta_{i}(1+2\eta_{j})(f(\bm{\eta})-f(\bm{\eta}^{ij}))^{2}$ allows to conclude. ∎

Maximum principle

From now on we only consider the symmetric ensemble. The Hermitian case can be treated with the same arguments and only notational changes. Given a typical path $\bm{\lambda}$, we will prove in this section that the solution to the eigenvector moment flow (3.7) converges uniformly to $1$ for $t=N^{-1/4+{\varepsilon}}$. It is clear that the maximum (resp. minimum) of $f$ over $\bm{\eta}$ decreases (resp. increases). We can quantify this decrease (resp. increase) in terms of the maximum and minimum themselves (see (4.17)). This yields an explicit convergence speed to 1 by a Gronwall argument.

In the statement below, we will also need $m(z)$, the Stieltjes transform of the semicircular distribution, i.e.

where the square root is chosen so that $m$ is holomorphic in the upper half plane and $m(z)\to 0$ as $z\to\infty$. The following isotropic local semicircle law (Theorem 4.2 in ) gives very useful bounds on $\langle{\bf q},u_{k}\rangle$ for any eigenvector $u_{k}$ via estimates on the associated Green function.

Let $H$ be an element from the generalized Wigner ensemble and $G(z)=(H-z)^{-1}$. Suppose that (1.3) holds. Then for any (small) $\xi>0$ and (large) $D>0$ we have, for large enough $N$,

An important consequence of this theorem, to be used in multiple occasions, is the following isotropic delocalization of eigenvectors: under the same assumptions as Theorem 4.1, for any $\xi>0$ and $D>0$, we have

Under the same assumptions the Stieltjes transform was shown to satisfy the estimate

2 Rescaling.

Recall the definition (2.1) of the evolution matrix $H^{(s)}_{t}$. The variance $\sigma_{ij}^{2}(t)$ of the matrix element $h_{ij}(t)$ is given by $\sigma_{ij}^{2}(t)=\sigma_{ij}^{2}+t/N$ if $i\neq j$, $\sigma_{ij}^{2}(t)=\sigma_{ij}^{2}+2t/N$ if $i=j$. Denote by $\alpha(t)=\left(1+\frac{N+1}{N}t\right)^{-1/2}.$ Then $\alpha(t)H^{(s)}_{t}$ is a generalized Wigner ensemble. In particular, the previously mentionned rigidity estimates hold along our dynamics if we rescale $H^{(s)}_{t}$ into $\alpha(t)H^{(s)}_{t}$. Consider the simple time change of our dynamics $u(t)=\int_{0}^{t}\alpha(s)^{-2}{\rm d}s$. Then $\widetilde{f}_{t}(\bm{\eta}):=f_{u(t)}(\bm{\eta})$ satisfies

In the rest of the paper it will always be understood that the above time rescaling $t\to u(t)$ and matrix scaling $H^{(s)}_{t}\to\alpha(t)H^{(s)}_{t}$ are performed so that all rigidity estimates hold as presented in the previous subsection, for all time.

3 Maximum Principle and regularity.

Note that the two conditions (4.5) and (4.6) follow from (4.4), i.e., $A_{1}\subset A_{2}\cap A_{3}$ , by standard arguments. More precisely, (4.6) can be proved by the argument in the proof of Corollary 3.2 in . The condition (4.5) is exactly the content of the rigidity of eigenvalues, i.e., Theorem 2.2 in . Its proof in Section 5 of used only the estimate (4.4).

The following lemma shows that these conditions hold with high probability.

For any $\omega>\xi>0,D>0$ and $N$ large enough, we have

where the probability denotes the joint law of the random variable $H_{0}$ and the paths of \bm{\lambda},\mbox{\boldmathu}.

For any fixed time, by (2.6) (4.2) and (4.3), the condition (4.4) holds with probability $1-N^{-C}$ for any $C$. As $C$ can be arbitrary, the same condition hold for any time and $z$ in a discrete set of size $N^{C/2}$, say. For any two matrices $H$ and $H^{\prime}$ with Green functions $G(z)$ and $G^{\prime}(z)$, we have

Applying this inequality to $H^{(s)}_{t}$ and $H^{(s)}_{s}$, we have with very high probability that

From the previous lemma, one easily sees that for any $\omega>\xi$, $\nu$ and $D>0$, we have, for large enough $N$,

The constant $C$ depends on ${\varepsilon},\omega,\delta$ and $n$ but not on ${\bf q}$.

We have the following asymptotic normality for eigenvectors of a Gaussian divisible Wigner ensemble with a small Gaussian component.

Let $\delta$ be an arbitrarily small constant and $t=N^{-1/4+\delta}$. Let $H_{t}$ be the solution to (2.1) and $(u_{1}(t),\dots,u_{N}(t))$ be an eigenbasis of $H_{t}$. The initial condition $H_{0}$ is assumed to be a symmetric generalized Wigner matrix. Then for any polynomial $P$ in $m$ variables and any ${\varepsilon}>0$, for large enough $N$ we have

Since $H_{0}$ is a generalized Wigner matrices, the isotropic local semicircle law, Theorem 4.1, holds for all time with $\xi$ arbitrarily small. With ${\omega}=2\xi$, and noticing that Lemma 4.2 holds for arbitrary large $\nu>0$, (4.9) implies that (4.10) holds. ∎

Because of (4.8) and $A_{1}\subset A_{2}\cap A_{3}$, we can assume in this proof that the trajectory $(H_{t})_{0\leqslant t\leqslant 1}$ is in $A_{1}({\bf q},\omega,\xi,N)\cap A_{2}(\omega,N)\cap A_{3}(\omega,N)$; the complement of this set induces an additional error $\OO(N^{-\nu+\xi})$ in (4.9), negligible compared to $N^{n\omega+{\varepsilon}-2\delta}$.

We begin with the case $n=1$. Let $f_{s}(k)=f_{s}(\bm{\eta})$, where $\bm{\eta}$ is the configuration with one particle at the lattice point $k$. The equation (3.6) becomes

for some $k_{0}$ ($k_{0}$ is not unique in general). Clearly, we have

Together with (4.11), for any $\eta>0$ we have

From the definition of $A({\bf q},\omega,\xi,\nu,N)$, for $N^{-1+\omega}<\eta<1$ we therefore have

where the error $N^{\omega}/(N\eta)$ comes from the missing term $j=k_{0}$ and we have used that for $(H_{0},\lambda)\in A({\bf q},\omega,\xi,\nu,N)$, $N\langle{\bf q},u_{j}\rangle^{2}$ is bounded by $N^{\omega}$ with very high probability. For the same reason, we have

Moreover, from the definition of $A({\bf q},\omega,\xi,\nu,N)$, we know that $-2-N^{-\frac{2}{3}+\omega}\leqslant\lambda_{k_{0}}\leqslant 2+N^{-\frac{2}{3}+\omega}$. As our final choice of $\eta$ will satisfy $N^{-\frac{2}{3}+\xi}\leqslant\eta\leqslant 1$, this implies that

Let $S_{s}=\sup_{k}(f_{s}(k)-1).$ Note that there may be some $s$ for which $S_{s}$ is not differentiable (at times when the maximum is obtained for at least two distinct indices). But if we denote

We chose $\eta=N^{-\frac{1}{2}+2\delta-{\varepsilon}}$ for some small ${\varepsilon}\in(0,2\delta-\omega)$ and $t=N^{-\frac{1}{4}+\delta}$. The Gronwall inequality gives

We can do the same reasoning for the minimum of $f$. This concludes the proof for $n=1$.

For $n\geqslant 2$ the same argument works and we will proceed by induction. Let $\bm{\xi}$ satisfy

Assume $\bm{\xi}$ is associated to $j_{r}$ particles at site $k_{r}$, $1\leqslant r\leqslant m$ for some $m\leqslant n$, where the $k_{r}$’s are distinct and $j_{r}\geqslant 1$. Then

where $\bm{\xi}^{k_{r}j}$ is defined in Section 3.2. We now estimate the first term on the right hand side (the second term was estimated in the previous $n=1$ step). By (4.6), for $(H_{0},\lambda)\in A({\bf q},\omega,\xi,\nu,N)$, $N\langle{\bf q},u_{j}\rangle^{2}$ is bounded by $N^{\omega}$ with very high probability. Thus we have

Moreover, by definition the above sum can be estimated by

where $\bm{\xi}\backslash k_{r}$ stands for the configuration $\bm{\xi}$ with one particle removed from site $k_{r}$. By induction assumption, we can use (4.9) to estimate $f_{s}(\bm{\xi}\backslash i_{r})$ for $s\in(t/2,t)$. We have thus proved that

on $(t/2,t)$. Notice that by our assumptions on the parameters ${\omega},\delta,\eta$ and $\xi$, the first error term always dominates the second. One can now bound $|f_{s}(\bm{\xi})-1|$ in the same way as in the $n=1$ case. ∎

If $\omega$ can be chosen arbitrarily small (this is true for generalized Wigner matrices), Theorem 4.3 gives $\sup_{\bm{\eta}:\mathcal{N}(\bm{\eta})=n}|f_{t}(\bm{\eta})-1|\to 0$ for any $t=N^{-1/4+{\varepsilon}}$. This could be improved to $t=N^{-1/3+{\varepsilon}}$ by allowing $\eta$ to depend on $k_{0}$ in the previous reasoning (chose $\eta=N^{-2/3+{\varepsilon}}{\hat{k}_{0}}^{1/3}$).

More generally, our proof shows that the following equation (4.17) (with the convention 4.13) holds. Let

where all variables depend on $t$ (remember in particular that G(z)=(\mbox{\boldmathu}_{t}^{*}\bm{\lambda}_{t}\mbox{\boldmathu}_{t}-z)^{-1}). Then the following maximum inequality holds:

Similar inequalities for a general number of particles can be obtained.

Proof of the main results

Corollary 4.4 asserts the asymptotic normality of eigenvector components for Gaussian divisible ensembles for $t$ not too small. In order to prove Theorem 1.2, we need to remove the small Gaussian components of the matrix elements in this Gaussian divisible ensemble. Similar questions occurred in the proof of universality conjecture for Wigner matrices and several methods were developed for this purpose (see, e.g., and ). Both methods can be extended to yielding similar eigenvector comparison results. In this paper, we will use the Green function comparison theorem introduced in [15, Theorem 2.3] (the parallel result following the argument of was given in ). Roughly speaking, [21, Theorem 1.10] states that the distributions of eigenvectors for two generalized Wigner ensembles are identical provided the first four moments of the matrix elements are identical and a level repulsion estimate holds for one of the two ensembles. We note that the level repulsion estimates needed in are substantially different. We first recall the following definition.

Fix an energy $E$ such that $\gamma_{k}\leqslant E\leqslant\gamma_{k+1}$ for some $k\in\llbracket 1,N\rrbracket$. A generalized Wigner ensemble is said to satisfy the level repulsion at the energy $E$ if there exist $\alpha_{0}>0$ such that for any $0<\alpha<\alpha_{0}$, there exists $\delta>0$ such that

where $\hat{k}=\min(k,N-k+1)$. A matrix ensemble is said to satisfy the level repulsion estimate uniformly if this property holds for any energy $E\in(-2,2)$.

The following theorem is a slight extension of [21, Theorem 1.10] with the following modifications : (1) We slightly weaken the fourth moment matching condition. (2) The original theorem was only for components of eigenvectors; we allow the eigenvector to project to a fixed direction. (3) We state it for all energies in the entire spectrum. (4) We include an error bound for the comparison. (5) We state it only for eigenvectors with no involvement of eigenvalues. Theorem 5.2 can be proved using the argument in ; the only modification is to replace the local semicircle law used in by the isotropic local semicircle law, Theorem 4.1. Since this type of argument based on the Green function comparison theorem has been done several times, we will not repeat it here. Notice that near the edge, the four moment matching condition can be replaced by just two moments. But for applications in this paper, this improvement will not be used and so we refer the interested reader to .

and that the first two diagonal moments of $H^{\bf{v}}$ and $H^{\bf{w}}$ are the same, i.e.

Assume also that the fourth off-diagonal moments of $H^{\bf{v}}$ and $H^{\bf{w}}$ are almost the same, i.e., there is an $a>0$ such that

Then there is ${\varepsilon}>0$ depending on $a$ such that for any integer $k$, any ${\bf q}_{1},\ldots{\bf q}_{k}$ and any choice of indices $1\leqslant j_{1},\ldots,j_{k}\leqslant N$ we have

where $\Theta$ is a smooth function that satisfies

2 Proof of Theorem 1.2 .

We now summarize our situation: Given a generalized Wigner ensemble $\hat{H}$, we wish to prove that (1.5) holds for the eigenvectors of $\hat{H}$. We have proved in (4.10) that this estimate holds for any Gaussian divisible ensemble of type $H_{0}+\sqrt{t}\ U$, and therefore by simple rescaling for any ensemble of type

where $H_{0}$ is any initial generalized Wigner matrix and $U$ is an independent standard GOE matrix, as long as $t\geqslant N^{-1/4+\delta}$. We fix $\delta$ a small number, say, $\delta=1/8$. Now we construct a generalized Wigner matrix $H_{0}$ such that the first three moments of $H_{t}$ match exactly those of the target matrix $\hat{H}$ and the differences between the fourth moments of the two ensembles are less than $N^{-c}$ for some $c$ positive. This existence of such an initial random variable is guaranteed by, say, Lemma 3.4 of . By the eigenvector comparison theorem, Theorem 5.2, we have proved (1.5) and this concludes our proof of Theorem 1.2.

3 Proof of Corollary 1.3.

From (1.5), there exists ${\varepsilon}>0$ such that

4 Proof of Corollary 1.4.

From (1.6), the first term of the right hand side is bounded by $N^{-{\varepsilon}}$ and the second term is bounded by $1/|a_{N}|$. The Markov inequality then allows us to conclude the proof of Corollary 1.4.

Finite speed of propagation

In this section, we prove a finite speed of propagation estimate for the dynamics (3.6). This estimate will be a key ingredient for proving optimal relaxation time for eigenvectors in the bulk. Finite speed of propagation was first proved in [17, Section 9.6] for (3.6) when the number of particle $n=1$. But it requires a level repulsion estimate which is difficult to prove. Our estimate requires only the rigidity of eigenvalues (which holds with very high probability) and the speed of propagation obtained is nearly optimal. Our key observation is that we can construct weight functions used in the finite speed estimate depending on eigenvalues so that the singularities in the equation (3.6) are automatically cancelled by the choices of these weight functions.

We will follow the approach of by decomposing the dynamics into a long range part and a short range part. The long range part can be controlled by a general argument based on decay estimate; the main new idea is in the proof of a finite speed of propagation for the short range dynamics, which is the content of Lemma 6.2.

We assume that for some (small) fixed parameter $\xi>0$ there is a constant $C$ such that for any $|i-j|\geqslant N^{\xi}$ and $0\leqslant s\leqslant 1$ the quantity $c_{ij}$ defined in (2.7) satisfies the following estimate

If $M_{N}$ is distributed as a generalized Wigner matrix, then for any $\xi>0$, (6.1) holds with probability $1-e^{-c(\log N)^{2}}$ for some $c>0$. In this section $M_{N}$ is not assumed to be distributed as a generalized Wigner matrix. Instead, we assume that $(\ref{far1})$ holds.

Notice that $\mathscr{S}$ and $\mathscr{L}$ are time dependent. Moreover, $\mathscr{S}$ is also reversible with respect to $\pi$ (the proof of Proposition 3.2 applies to any symmetric $c_{ij}$’s). Denote by ${\rm U}_{\mathscr{S}}(s,t)$ the semigroup associated with $\mathscr{S}$ from time $s$ to time $t$, i.e.

where $C$ only depends on $\xi$ (in particular not on $\bm{\eta}$).

Since ${\rm U}_{\mathscr{B}}$ and ${\rm U}_{\mathscr{S}}$ are contractions in ${\rm L}^{1}$, this yields

2 Finite speed of propagation for the short range dynamics.

Suppose that $\bm{\eta}$ is a configuration with $n$ particles. We denote the particles in nondecreasing order by ${\bf{x}}(\bm{\eta})=(x_{1}(\bm{\eta}),\ldots,x_{n}(\bm{\eta}))$ with $\alpha N\leqslant x_{1}\leqslant\dots\leqslant x_{n}\leqslant(1-\alpha)N$. We will drop the dependence on $\bm{\eta}$ and simply use $(x_{1},\ldots,x_{n})$. In the same way, we also denote the configuration $\bm{\xi}$ by ${\bf{y}}$ with $1\leqslant y_{1}\leqslant\dots\leqslant y_{n}\leqslant N$ where we have dropped the dependence of $\bm{\xi}$ in $y_{\alpha}(\bm{\xi})$. This convention will be followed for the rest of this paper.

We define the following distance on the set of configurations with $n$ particles:

For the second equality, observe that for any $x\leqslant y$ and $a\leqslant b$, we have $|x-a|+|y-b|\leqslant|x-b|+|y-a|$.

Before stating our finite speed result, we also need the notation $r_{s}(\bm{\eta},\bm{\xi})=({\rm U}_{\mathscr{S}}(0,s)\delta_{\bm{\eta}})(\bm{\xi})$.

We first consider the case (i) corresponding to $\bm{\eta}$ supported in the bulk, but the reader may want to read first the proof of (ii), written for the simpler case $n=1$ for the sake of simplicity.

For any configuration $\bm{\xi}$ with $n$ particles we define

similarly to (6.3). For the second equality, observe that if $\alpha\leqslant\beta$ and $a\leqslant b$, then $\psi_{\alpha}(a)+\psi_{\beta}(b)\leqslant\psi_{\alpha}(b)+\psi_{\beta}(a)$ (the function $a\mapsto\psi_{\alpha}(a)-\psi_{\beta}(a)$ is nondecreasing).

The coefficients $c_{1}$ and $c_{2}$ are non-random positive combinatorial factors depending on the locations of $i$, $\bm{\eta},\bm{\xi}$, but we will only need that they are uniformly bounded in $N$. We will adopt the convention to use indices $1\leqslant\alpha,\beta\leqslant n,1\leqslant i,j,k\leqslant N$. We define

where $\pi=\pi^{(s)}$ is the reversible measure defined in (3.9) for the symmetric eigenvector moment flow (which is also reversible w.r.t its short-range cutoff version). Then

Second step: bound on (6.7) and (6.9). Using reversibility with respect to $\pi$, the first term can be written

Here the equality (6.10) is a direct application of the reversibility property, while (6.11) also follows from the reversibility as follows. Notice that

One can check that (6.11) follows from $\langle g,\mathscr{S}r\rangle_{\pi}=\langle\mathscr{S}g,r\rangle_{\pi}$.

We now estimate the term $\frac{\phi(\bm{\xi}^{kj})}{\phi(\bm{\xi})}+\frac{\phi(\bm{\xi})}{\phi(\bm{\xi}^{kj})}-2$ in (6.11). If it is nonzero (and we assume first that $j<k$) then there exists $1\leqslant p<q\leqslant n$ such that $y_{p}\leqslant j<y_{p+1}$, $y_{q-1}<k=y_{q}$ (recall $y_{q}=y_{q}(\bm{\xi})$) and

Moreover, the bracket term (6.9) is easily bounded by

Third step: bound on (6.8). We can bound (6.8) by

Denote $\bar{\bm{\xi}}$ the configuration exchanging all particles from sites $i$ and $j$, i.e. $\bar{\xi}_{i}=\xi_{j}$, $\bar{\xi}_{j}=\xi_{i}$ and $\bar{\xi}_{k}=\xi_{k}$ if $k\neq i,j$. Using $\pi(\bm{\xi})=\pi(\bar{\bm{\xi}})$, we can bound the sum over $\bm{\xi}$ in (6.15) by

where the constant $c_{q}=0$ if $p$ is even and $q=p/2$, and $c_{q}=1$ otherwise. Remember that for any $a\leqslant b$, we have $\psi^{\prime}_{a}\geqslant\psi^{\prime}_{b}$. This implies that $\sum_{\alpha:y_{\alpha}=i}\psi^{\prime}_{x_{\alpha}}(\lambda_{i})\geqslant\sum_{\alpha:\bar{y}_{\alpha}=j}\psi^{\prime}_{x_{\alpha}}(\lambda_{i})$ and $\sum_{\alpha:\bar{y}_{\alpha}=i}\psi^{\prime}_{x_{\alpha}}(\lambda_{i})\geqslant\sum_{\alpha:y_{\alpha}=j}\psi^{\prime}_{x_{\alpha}}(\lambda_{i})$ so that

Equations (6.16) and (6.17) together with $\lambda_{i}<\lambda_{j}$ give

where we used, in the second inequality, $\|\psi_{x_{\alpha}}^{\prime}\|_{\infty}\leqslant 1$. Note that transforming $\bm{\xi}$ into $\bar{\bm{\xi}}$ can be achieved by transferring a particle for $i$ to $j$ (or $j$ to $i$) one by one at most $n$ times, so that

for any $M>0$. We finally proved that the drift term from (6.8) is bounded above by

Fourth step: conclusion. All together, the above estimates give

where $\tau$ is defined by (6.5). Then by definition of the dynamics and the Itô formula, we have (here we drop the time parameter $s$ whenever it is obvious)

Thus if we define $X_{s}=\sum_{k=1}^{N}v_{s}(k)^{2}$, we obtain

Relaxation to equilibrium for t≳N−1{t\,{\gtrsim}\,N^{-1}}

assuming the optimal isotropic local semicircle law with a tiny error $N^{\xi}/\sqrt{N\eta}$. Choosing $\eta=N^{-1+{\varepsilon}}$ for some small ${\varepsilon}>0$ then gives, by Gronwall, a relaxation time of order $\gtrsim N^{-1}$. The purpose of this section is to make this argument rigorous by using the finite speed of propagation for the eigenvector moment flow, i.e., Lemma 6.2.

The initial matrix is denoted $M_{N}=M_{N}(0)$, it satisfies the local semicircle law, and its eigenvalues follow the usual Dyson Brownian motion dynamics.

Let $G^{(s)}_{N}$ (resp. $G^{(h)}_{N}$) be a sequence of $N\times N$ random matrices from the Gaussian orthogonal (resp. unitary) ensemble (normalized with limiting spectral measure supported on $(-2,2)$, for example). Note that in this section $G$ stands for a Gaussian matrix, not its Green function.

Let ${\varepsilon}$ be any arbitrarily small positive constant and $t=N^{-1+{\varepsilon}}$. Assume that, for a deterministic sequence of matrices $(M_{N})_{N\geqslant 1}$ and a sequence of unit vectors ${\bf q}={\bf q}_{N}$, we have, for any $\omega>\xi>0,D>0$ and $N$ large enough (depending on these parameters),

If moreover (7.1) holds for any given sequence $({\bf q}_{N})_{N\geqslant 1}$, then any bulk eigenvector of $M_{N}+\sqrt{t}\,G^{(s)}_{N}$ have asymptotically independent normal entries (the analogue of Corollary 1.3) and each eigenvector satisfy local quantum unique ergodicity (the analogue of Corollary 1.4).

Similar results hold for the Hermitian matrices $M_{N}+\sqrt{t}\,G^{(h)}_{N}$.

The Green function in $A_{1}$ appearing in (7.1) is with respect to the matrix $(M_{N}(s))_{s\geqslant 0}$ with $M_{N}=M_{N}(0)$ being the initial matrix and $M_{N}(s)$ the value at time $t$ of a (matrix) Dyson Brownian Motion.

Theorem 7.1 means that, the initial structure of bulk eigenvectors completely disappears with the addition of a small noise, provided that the initial matrix satisfies a strong form of semicircle law. If the initial condition is a generalized Wigner matrix, the matrix Dyson Brownian motion is again a generalized Wigner ensemble after rescaling. In this case, the asymptotic normality of the eigenvectors was already proved in Theorem 1.2 and therefore the conclusion of Theorem 7.4 was proved as well. The key point of Theorem 7.1 lies in that it holds for deterministic initial matrices, provided that the local isotropic semicircle law holds.

Note that by standard perturbation theory Theorem 7.1 in general does not hold for $t\ll N^{-1}$. Recall that Dyson’s conjecture states that the relaxation time to local equilibrium for bulk eigenvalues under the DBM is $t\sim N^{-1}$. Thus Theorem 7.1 is the analogue of this conjecture in the context of bulk eigenvectors.

Theorem 7.1 gives optimal relaxation speed for dynamics of bulk eigenvectors provided that the local law holds along the whole spectrum, i.e. condition (7.1) holds. One may be interested in the dynamics relaxation only locally, i.e. proving QUE only for certain eigenvectors with corresponding energy $\lambda_{i}$ around $E_{0}=\gamma_{k_{0}}\in(-2.2)$. Then as an input, the local law is only needed in a small window around $E$.

More precisely, let $c>\varepsilon>0$ be fixed (remember $t=N^{-1+{\varepsilon}}$). Assume that (7.1) holds for any $\omega>\xi>0$ in the smaller domain (replacing the original domain defined in (4.1))

Then the conclusion (7.2) holds after restricting the $\sup$ to $I\subset\llbracket k_{0}-N^{c}/10,k_{0}+N^{c}/10\rrbracket$.

To summarize, the optimal time relaxation result, Theorem 7.1, can be made local in the spectrum, because the key input in this result, the finite speed of propagation Lemma 6.2, holds locally. The modifications needed to prove these local versions are obvious and we leave them to interested readers.

We will prove Theorem 7.1 by using the maximum principle locally. For this purpose, we will use the finite speed of propagation estimate, Lemma 6.2. This will be explained in the next subsections.

2 Flattening of initial condition at the edge.

Let $\alpha>0$ be a fixed small number. We define the following flattening and averaging operators on the space of functions of configurations with $n$ points: any $a\in\llbracket 1,N/2\rrbracket$,

for some coefficient $a_{\bm{\eta}}\in$ ($a_{\bm{\eta}}=0$ if $\bm{\eta}\not\subset\llbracket\alpha N,(1-\alpha)N\rrbracket$, $1$ if $\bm{\eta}\subset\llbracket 2\alpha N,(1-2\alpha)N\rrbracket$). We will only use the elementary property

We will abbreviate $g_{\bm{\lambda},t}(\bm{\eta})$ by $g_{t}(\bm{\eta})$, and $f_{\bm{\lambda},t}(\bm{\eta})$ by $f_{t}(\bm{\eta})$ (for $n=1$, we write these functions as $f_{t}(k)$ and $g_{t}(k)$ where $\bm{\eta}$ is the configuration with $1$ particle at $k$). We remind the reader that $f_{t}(\bm{\eta})$ can be define either by (3.5) or by the solution of the equation (3.6). In particular, $f_{t}(k)$ is the conditional expectation of $|\langle{\bf q},u_{k}(t)\rangle|^{2}$ given $\bm{\lambda}$, i.e.,

where ${\bf q}$ is a fixed unit vector. In all our application, the initial data $f_{\bm{\lambda},0})(\bm{\eta})$ is independent of $\bm{\lambda}$ and given by (3.5) with $t=0$. For $g_{\bm{\lambda},t}$, we can only understand it as the solution to (7.6).

For small time $t$, by finite speed of propagation we will prove that $g=1$ (up to exponentially small corrections) close to the edge, so that the maximum principle for the dynamics (7.6) can be localized in the bulk.

We first prove that for these modified dynamics, the isotropic law holds in the following sense. The following result is deterministic.

where $\bm{\eta}\backslash k_{0}$ stands for the configuration $\bm{\eta}$ with one particle removed from site $k_{0}$.

We first show that the difference between $g_{t}(k)=({\rm U}_{\mathscr{S}}(0,t){\rm Av}f_{0})(k)$ and $({\rm Av}{\rm U}_{\mathscr{B}}(0,t)f_{0})(k)$ is small. More precisely, we can bound the left hand side of (7.8) by $|{\rm(i)}|+|{\rm(ii)}|+|{\rm(iii)}|$ where

The term (i) will be controlled by finite speed of propagation; (ii) will be controlled by Lemma 6.1, and (iii) by the isotropic local semicircle law.

To bound the term (ii), define the reversed dynamics ${\rm U}_{\mathscr{S}}^{*}$ by

and $s$ is always set to be $=0$ ] and similarly for ${\rm U}_{\mathscr{B}}^{*}$. Notice that Lemma 6.1 holds for these time-reversed dynamics, the proof is unchanged. Thus we have

Concerning the error term (iii), we proceed as follows. Let $m_{0}$ be the index such that $|\Re(z)-\gamma_{m_{0}}|=\inf_{1\leqslant i\leqslant N}\{|\Re(z)-\gamma_{i}|\}$. Then

where we use that $\|f_{0}\|_{\infty}\leqslant N^{\omega}$ (which follows from the condition (4.6)). For any function $f$ we write ${\rm(Av)}f(k)=a_{k}f(k)+(1-a_{k})$ with the notation from (7.4). We obtain

Moreover, the first sum above is equal to

where we used $(M_{N},\bm{\lambda})\in A({\bf q},\omega,\xi,\nu,N)$. From (7.5), we have $|a_{k}-a_{m_{0}}|\leqslant\sqrt{\eta}N^{\omega}$ and the second sum in (7.15) can be bounded by $\OO(N^{\omega}\sqrt{\eta})$, which is smaller than $N^{\omega}/(N\eta)$ for $\eta\leqslant N^{-3/4}$. Gathering all estimates, we obtain that (7.8) holds.

In the case of general $n$, to prove (7.9), we proceed in the same way. As the term of type (i) is also bounded by finite speed of propagation, we just need to prove that

Thanks to Lemma 6.1 it is sufficient to prove the above estimate replacing ${\rm U}_{\mathscr{S}}$ by ${\rm U}_{\mathscr{B}}$. We also can restrict the summation to $|k-k_{0}|\leqslant N\sqrt{\eta}$. Then, similarly to the $n=1$ case, we write

Using (7.5) and $|k-k_{0}|\leqslant N\sqrt{\eta}$ to bound the above second term, we are left with proving that

The second sum above is properly estimated by $m(z)$ because we are in a good set. Concerning the first sum, its contribution is not trivial if $a_{\bm{\eta}}\neq 0$, in particular $k_{0}\in\llbracket\alpha N,(1-\alpha)N\rrbracket$. Then $\Im m(z)\sim 1$ and this first sum can be estimated exactly as in (4.15), (4.16). This concludes the proof. ∎

3 Localized maximum principle.

The following result states that, for a typical initial conditions and a generic eigenvalue path, the relaxation time of the bulk eigenvectors is of order at most $N^{-1+{\varepsilon}}$ for any small ${\varepsilon}>0$.

Assume that for some $0<\xi<\omega<\omega_{0}$, $(M_{N}(s))_{0\leqslant s\leqslant 1}$ is in $A_{1}({\bf q},\omega,\xi,N)$. Assume moreover that $(\ref{eqn:finite})$ holds. Let $f$ be a solution of the eigenvector moment flow (3.6) with initial matrix $M_{N}$ and path $\bm{\lambda}$. Then there exists $c>0$ such that for large enough $N$ we have

As $\alpha$ is arbitrary we just need to prove the result for $\alpha$ replaced by $3\alpha$. Moreover, we only need to prove (7.16) with $f_{t}(\bm{\eta})$ replaced by $g_{t}(\bm{\eta})$ solving the cutoff dynamics (7.6). Indeed, we have

We will prove that such an estimate holds for any $\alpha>0$ by induction on $n$. Assume there is just one particle. Following the idea from the proof of Theorem 4.3, for a given $0\leqslant s\leqslant t$ let $k_{0}$ be an index such that $g_{s}(k_{0})=\sup_{k}\{g_{s}(k)\}$. We consider two possible cases: if $g_{s}(k_{0})-1\leqslant N^{-10}$ then there is nothing to prove. If $g_{s}(k_{0})-1\geqslant N^{-10}$, then from the finite speed of propagation assumption (i.e., we are in the set $\mathcal{A}$), $k_{0}$ is in the bulk, i.e., $k_{0}\in\llbracket\frac{\alpha}{2}N,(1-\frac{\alpha}{2})N\rrbracket$ (the reason is that if $k_{0}$ were near the edges, then $g_{s}(k_{0})-1$ is exponentially small). We then have

By Gronwall’s lemma, we obtain $S_{t}=\OO(N^{-{\varepsilon}/4})$. This concludes the proof for $n=1$.

For general $n$, as in the 1-particle case we can assume that $\sup_{\bm{\eta}}g_{t}(\bm{\eta})$ is achieved for some $\bm{\xi}\subset\llbracket\frac{\alpha}{2}N,(1-\frac{\alpha}{2})N\rrbracket$. Then the analogue of (4.14) holds with $f$ replaced by $g$. The first sum in (4.14) then can be evaluated using (7.9):

From the result at rank $n-1$ with $\alpha$ replaced by $\alpha/10$, we know that for $s\in[t/2,t]$ we have

One now can conclude the proof as in the $n=1$ case. ∎

Under the above two assumptions, we apply Theorem 7.4, which proves the first statement of Theorem 7.1. The last two statements of Theorem 7.1 easily follow by the arguments used in Sections 5.3 and 5.4.

Appendix A Appendix A Continuity estimate for t≲N−1/2{t\,{\lesssim}\,N^{-1/2}}

The main result in Section 7, Theorem 7.4, asserts the asymptotic normality of eigenvector components for Gaussian divisible ensembles for $t\gtrsim N^{-1}$. To prove Theorem 1.2 for bulk eigenvectors, in this appendix we remove the small Gaussian components of the matrix elements. As we saw in Section 5, one way to proceed consists in a Green function comparison theorem. Here, we proceed in a different way: the Dyson Brownian motion preserves the local structure of generalized Wigner matrices up to time $N^{-1/2}$ (see the lemma hereafter). This approach is much more direct and there is no need to construct moment matching matrices. It provides a completely dynamical proof of Theorem 1.2 for bulk eigenvectors.

We remark that although this proof is very simple, the fact that the Dyson Brownian motion preserves the detailed behaviour of eigenvalues and eigenvectors is surprising and even contradictory. Consider for example the eigenvalue flow. It was proved that this spectral dynamics take very general initial data to local equilibrium for any time $t\gtrsim N^{-1}$. So how can we prove that the changes of the eigenvalues up to time $N^{-1/2}$ is less than the accuracy $N^{-1}$? The answer is that we only prove the preservation of the Dyson Brownian motion for matrix models. In other words, the matrix structure gives this preservation of the local structure.

We start with the following matrix stochastic differential equation which is an Ornstein-Uhlenbeck version of the Dyson Brownian motion. Let $H_{t}=(h_{ij}(t))$ be a symmetric $N\times N$ matrix. The dynamics of the matrix entries are given by the stochastic differential equations

Suppose that we have $c/N\leqslant s_{ij}\leqslant C/N$ for some fixed constants $c$ and $C$, uniformly in $i$ and $j$. Denote $\partial_{ij}=\partial_{h_{ij}}$. Suppose that $F$ is a smooth function of the matrix elements $(h_{ij})_{i\leqslant j}$ satisfying

Together with the condition $c/N\leqslant s_{ij}\leqslant C/N$, we have

Integration over time finishes the proof. ∎

The previous lemma implies the following eigenvalues and eigenvectors continuity estimate for the dynamics (A.1).

for some $C>0$. Denote by $(u_{1}(t),\dots,u_{N}(t))$ the eigenvectors of $H_{t}$ associated with the eigenvalues $\lambda_{1}(t)\leqslant\dots\leqslant\lambda_{N}(t)$. Then there exists ${\varepsilon}>0$ (depending only on $\Theta,\delta$ and $\alpha$) such that, for large enough $N$,