Eigenvector Distribution of Wigner Matrices

Introduction

The universality of random matrices can be roughly divided into the bulk universality in the interior of the spectrum and the edge universality near the spectral edge. Over the past two decades, spectacular progress on bulk and edge universality has been made for invariant ensembles, see e.g. BI ; DKMVZ1 ; DKMVZ2 ; PS and AGZ ; De1 ; De2 for a review. For non-invariant ensembles with i.i.d. matrix elements (Standard Wigner ensembles), edge universality can be proved via the moment method and its various generalizations; see e.g. SS ; Sosh ; So1 . In order to establish bulk universality, a new approach was developed in a series of papers ESY1 ; ESY2 ; ESY3 ; ESY4 ; ESYY ; EYY ; EYY2 ; EYYrigi based on three basic ingredients: (1) A local semicircle law – a precise estimate of the local eigenvalue density down to energy scales containing around $N^{\varepsilon}$ eigenvalues. (2) The eigenvalue distribution of Gaussian divisible ensembles via an estimate on the rate of decay to local equilibrium of the Dyson Brownian motion Dy . (3) A density argument which shows that for any probability distribution there exists a Gaussian divisible distribution with identical eigenvalue statistics down to scales $1/N$. In EYYrigi , edge universality is established as a corollary of this approach. It asserts that, near the spectral edge, the eigenvalue distributions of two generalized Wigner ensembles are the same provided the first two moments of the two ensembles match.

Another approach to both bulk and edge universality was developed in TV ; TV2 ; J1 . Using this approach, the authors show that the eigenvalue distributions of two standard Wigner ensembles are the same in the bulk, provided that the first four moments match. They also prove a similar result at the edge, assuming that the first two moments match and the third moments vanish.

In this paper, partly based on the approach of EYYrigi , we extend edge universality to eigenvectors associated with eigenvalues near the spectral edge, assuming the matching of the first two moments of the matrix entries. We prove that, under the same two-moment condition as in EYYrigi , the edge eigenvectors of Hermitian and symmetric Wigner matrices have the same joint distribution as those of the corresponding Gaussian ensembles. The joint distribution of the eigenvectors of Gaussian ensembles is well known and can be easily computed. More generally, we prove that near the spectral edge the joint eigenvector-eigenvalue distributions of two generalized Wigner matrix ensembles coincide provided that the first two moments of the ensembles match and one of the ensembles satisfies a level repulsion condition.

We also prove similar results in the bulk, under the stronger assumption that the first four moments of the two ensembles match. In particular, we extend the result of TV to cover the universality of bulk eigenvectors.

The law $\nu_{ij}$ and its variance $\sigma_{ij}^{2}$ may depend on $N$, but we omit this fact in the notation. We denote by $B\mathrel{\mathop{:}}=(\sigma^{2}_{ij})_{i,j=1}^{N}$ the matrix of the variances. We shall always make the following three assumptions on $H$.

Thus $B$ is symmetric and doubly stochastic and, in particular, satisfies $-1\leqslant B\leqslant 1$.

There exists constants $\delta_{-}>0$ and $\delta_{+}>0$, independent of $N$, such that $1$ is a simple eigenvalue of $B$ and

There exists a constant $C_{0}$, independent of $N$, such that $\sigma_{ij}^{2}\leqslant C_{0}N^{-1}$ for all $i,j=1,\dots,N$.

Examples of matrices satisfying Assumptions (A) – (C) include Wigner matrices, Wigner matrices whose diagonal elements are set to zero, generalized Wigner matrices, and band matrices whose band width is of order $cN$ for some $c>0$. See EYYrigi , Section 2, for more details on these examples.

Our analysis relies on a notion of high probability which involves logarithmic factors of $N$. The following definitions introduce convenient shorthands.

We set $L\equiv L_{N}\mathrel{\mathop{:}}=A_{0}\log\log N$ for some fixed $A_{0}$ as well as $\varphi\equiv\varphi_{N}\mathrel{\mathop{:}}=(\log N)^{\log\log N}$.

A key assumption for our result is the following level repulsion condition, which is in particular satisfied by the Gaussian ensembles (see Remark 1.5 below). Consider a spectral window whose size is much smaller than the typical eigenvalue separation. Roughly, the level repulsion condition says that the probability of finding more than one eigenvalue in this window is much smaller than the probability of finding precisely one eigenvalue. In order to state the level repulsion condition, we introduce the following counting function. For any $E_{1}\leqslant E_{2}$ we denote the number of eigenvalues in $[E_{1},E_{2}]$ by

The ensemble $H$ is said to satisfy level repulsion at the edge if, for any $C>0$, there is an $\alpha_{0}>0$ such that the following holds. For any $\alpha$ satisfying $0<\alpha\leqslant\alpha_{0}$ there exists a $\delta>0$ such that

for all $E$ satisfying $|E+2|\leqslant N^{-2/3}\varphi^{C}$.

The ensemble $H$ is said to satisfy level repulsion in the bulk if, for any $\kappa>0$, there is an $\alpha_{0}>0$ such that the following holds. For any $\alpha$ satisfying $0<\alpha\leqslant\alpha_{0}$ there exists a $\delta>0$ such that

Both the Gaussian Unitary Ensemble (GUE) and the Gaussian Orthogonal Ensemble (GOE) satisfy level repulsion in sense of Definitions 1.3 and 1.4. This can be established for instance as follows; see AGZ , Sections 3.5 and 3.7, and in particular Lemmas 3.5.1 and 3.7.2, for full details. For GUE and GOE, the correlation functions can be explicitly expressed in terms of Hermite polynomials. Using Laplace’s method, one may then derive the large-$N$ asymptotics of the correlation functions, from which (1.4) and (1.5) immediately follow. (Note that in AGZ , the exponent of $N$ in the error estimates was not tracked in order to simplify the presentation.)

In the more general case of Wigner matrices, level repulsion in the bulk, (1.5), was proved for matrices with smooth distributions in ESY3 and without a smoothness assumption in TV .

We shall use the level repulsion condition of Definition 1.3 to estimate the probability of finding two eigenvalues closer to each other than the typical eigenvalue separation. For definiteness, we formulate this estimate at the lower spectral edge $-2$. By partitioning the interval

into $O(\varphi^{C}N^{\alpha})$ subintervals of size $N^{-2/3-\alpha}$, we get from (1.4) that for any sufficiently small $\alpha$ there exists a $\delta>0$ such that

A similar result can be derived in the bulk using (1.5).

2 Results

Before stating our main results, we recall the definition of the classical eigenvalue locations. Let

be the integrated distribution function of the semicircle law. We use $\gamma_{\alpha}\equiv\gamma_{\alpha,N}$ to denote the classical location of the $\alpha$-th eigenvalue under the semicircle law, defined through

To avoid unnecessary technicalities in the presentation, we shall assume that the entries $h_{ij}$ of $H$ have uniform subexponential decay, i.e.

$\vartheta>0$ is some fixed constant. As observed in EKYY2 , Section 7, one may easily check that all of our results hold provided the subexponential condition (1.9) is replaced with the weaker assumption that there is a constant $C$ such that

where $C_{0}$ is a large universal constant.

Our main result on the distributions of edge eigenvectors is the following theorem.

Let $\rho$ be a positive constant. Then for any integer $k$ and any choice of indices $i_{1},\ldots i_{k}$, $j_{1},\ldots,j_{k}$, $\beta_{1},\ldots\beta_{k}$ and $\alpha_{1},\ldots\alpha_{k}$ with $\min(|\alpha_{l}|,|\alpha_{l}-N|)+\min(|\beta_{l}|,|\beta_{l}-N|)\leqslant\varphi_{N}^{\rho}$ for all $l$ we have

where $\theta$ is a smooth function that satisfies

The scaling in front of the arguments in (1.11) is the natural scaling near the spectral edge. Indeed, for e.g. GUE or GOE it is known (see e.g. AGZ ) that $(\lambda_{\beta}-\gamma_{\beta})\sim N^{-2/3}$ near the edge, and that $u_{\alpha}(i)\sim N^{1/2}$ (complete delocalization of eigenvectors).

Similarly, Theorem 1.6 and Remark 1.5 imply that the joint eigenvector-eigenvalue distribution of symmetric Wigner matrices agrees with that of GOE. Results similar to those outlined above on the eigenvector-eigenvalue distribution of GUE hold for GOE.

The universality of the eigenvalue distributions near the edge was already proved in EYYrigi under the assumption that the first two moments of the matrix entries match, and in TV2 under the additional assumption that the third moments vanish. Note that Theorem 1.6 holds in a stronger sense than the result in EYYrigi : it holds for probability density functions, not just the distribution functions.

In the bulk, a result similar to Theorem 1.6 holds under the stronger assumption that four, instead of two, moments of the matrix entries match.

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

Let $\rho>0$ be fixed. Then for any integer $k$ and any choice of indices $i_{1},\ldots i_{k}$, $j_{1},\ldots,j_{k}$, as well as $\rho N\leqslant\alpha_{1},\ldots\alpha_{k},\beta_{1},\ldots,\beta_{k}\leqslant(1-\rho)N$, we have

where $\theta$ is a smooth function that satisfies

The universality restricted to the bulk eigenvalues only has been previously established in several works. The following list provides a summary. Note that the small-scale statistics of the eigenvalues may be studied using correlation functions, which depend only on eigenvalue differences, or using joint distribution functions, as in (1.11) and (1.15), which in addition contain information about the eigenvalue locations.

In EYYrigi , bulk universality for generalized Wigner matrices was proved in the sense that correlation functions of bulk eigenvalues, averaged over a spectral window of size $N^{\varepsilon}$, converge to those of the corresponding Gaussian ensemble.

In TV the statement (1.15) on distribution functions, restricted to eigenvalues only, was proved for Hermitian and symmetric Wigner matrices for the case where the first four moments match as in (1.13).

For the case of Hermitian Wigner matrices with a finite Gaussian component, it was proved in J that the correlation functions converge to those of GUE.

In GEG1 , the joint distribution function of the eigenvalues of GUE was computed. This result was extended to cover GOE in GEG2 .

Note that (ii) and (iii) together imply the universality of the joint distribution of eigenvalues for Hermitian Wigner matrices, for which the first three moments match those of GUE and the distribution is supported on at least three points. Moreover, combining (ii) and (iv) allows one to compute the eigenvalue distribution of Hermitian and symmetric Wigner matrices, provided the four first moments match those of GUE/GOE.

Thus, Theorem 1.15 extends the results of EYYrigi to distribution functions of individual eigenvalues as well as to eigenvectors.

A while after this paper was posted online, a result similar to Theorem 1.10 appeared in TV3 . Its proof relies on a different method. The hypotheses of TV3 are similar to those of Theorem 1.10, with the two following exceptions. The result of TV3 is restricted to Wigner matrices instead of the generalized Wigner matrices defined by Assumptions (A) – (C). Moreover, in TV3 the derivatives of the observable $\theta$ are required to be uniformly bounded in $x$, where this uniform bound may grow slowly with $N$. This latter restriction allows the authors of TV3 to let $k$ grow slowly with $N$.

While the results of TV3 apply to eigenvectors near the spectral edge, the matching of four moments (as in Theorem 1.10) is also required for this case. As shown in Theorem 1.6, the universality of edge eigenvectors in fact only requires the first two moments to match.

3 Outline of the proof

The main idea behind our proof is to express the eigenvector components using matrix elements of the Green function $G(z)=(H-z)^{-1}$. To this end, we use the identity

where $\eta>0$. Using a good control on the matrix elements of $G(z)$, we may then apply a Green function comparison argument (similar to the Lindeberg replacement strategy) to complete the proof. For definiteness, let us consider a single eigenvalue $\lambda_{\alpha}$ located close to the spectral edge $-2$.

In a first step, we write $N\bar{u}_{\alpha}(i)u_{\alpha}(j)$ as an integral of (1.17) over an appropriately chosen (random) domain, up to a negligible error term. We choose $\eta$ in (1.17) to be much smaller than the typical eigenvalue separation, i.e. we set $\eta=N^{-2/3-\varepsilon}$ for some small $\varepsilon>0$. Note that the fraction on the left-hand side of (1.17) is an approximate delta function on the scale $\eta$. Then the idea is to integrate (1.17) over the interval $[\lambda_{\alpha}-\varphi^{C}\eta,\lambda_{\alpha}+\varphi^{C}\eta]$ for some large enough constant $C$. For technical reasons related to the Green function comparison (the third step below), it turns out to be advantageous to replace the above interval with $\mathcal{I}\mathrel{\mathop{:}}=[\lambda_{\alpha-1}+\varphi^{C}\eta,\lambda_{\alpha}+\varphi^{C}\eta]$. Using eigenvalue repulsion, we infer that, with sufficiently high probability, the eigenvalues $\lambda_{\alpha-1}$ and $\lambda_{\alpha+1}$ are located at a distance greater than $\varphi^{C}\eta$ from $\lambda_{\alpha}$. Therefore the $E$-integration over $\mathcal{I}$ of the right-hand side of (1.17) yields $N\bar{u}_{\alpha}(i)u_{\alpha}(j)$ up to a negligible error term.

The above proof may be easily generalized to multiple eigenvector components as well as to eigenvalues; this allows us to consider observables of the form given in (1.11). The necessary changes are given in Section 4.

The proof for bulk eigenvectors is similar, with two major differences. At the edge, the convolution integral on the right-hand side of (1.18) was over a domain of size $\varphi^{C}N^{-2/3}$. If the same expression were used in the bulk, this size would be $O(1)$ (since $E$ is separated from the spectral edge $-2$ by a distance of order $O(1)$), which is not affordable in the error estimates. Instead, a more refined multiscale approach using the Helffer-Sjöstrand functional calculus is required in order to rewrite the sharp indicator function on the left-hand side of (1.18) in terms of Green functions. The second major difference for bulk eigenvectors is the power counting in the Green function comparison argument, which is in fact easier than at the edge. The main reason for this is that the smallness associated with off-diagonal elements of $G$ is not available in the bulk. Hence we need to assume that four instead of two moments match, and the intricate bookkeeping of the number of off-diagonal resolvent elements is not required. Thus, thanks to the very strong assumption of four-moment matching, the proof of Theorem 1.10 is considerably simpler than that of Theorem 1.6. See Section 5 for a more detailed explanation as well the proof.

Conventions. We shall use the letters $C$ and $c$ to denote generic positive constants, which may depend on fixed quantities such as $\vartheta$ from (1.9), $\delta_{\pm}$ from Assumption (B), and $C_{0}$ from Assumption (C). We use $C$ for large constants and $c$ for small constants.

Acknowledgements. The authors would like to thank L. Erdős and H.T. Yau for many insights and helpful discussions.

Local semicircle law and rigidity of eigenvalues

In this preliminary section we collect the main tools we shall need for our proof. We begin by introducing some notation and by recalling the basic results from EYYrigi on the local semicircle law and the rigidity of eigenvalues.

Similarly, we define $m_{sc}(z)$ as the Stieltjes transform of the local semicircle law:

It is well known that $m_{sc}(z)$ can also be characterized as the unique solution of

with positive imaginary part for all $z$ with $\operatorname{Im}z>0$. Thus,

where the square root function is chosen with a branch cut in the segment $$so that asymptotically$\sqrt{z^{2}-4}\sim z$at infinity. This guarantees that the imaginary part of$m_{sc}$is non-negative for$\eta=\operatorname{Im}z>0$and in the limit$\eta\to 0$.

In order to state the local semicircle law, we introduce the control parameters

Let $H=(h_{ij})$ be a Hermitian or symmetric $N\times N$ random matrix satisfying Assumptions A – C. Suppose that the distributions of the matrix elements $h_{ij}$ have a uniformly subexponential decay in the sense of (1.9). Then there exist positive constants $A_{0}>1$, $C,c$, and $\tau<1$, such that the following estimates hold for $L$ as in Definition 1.1 and for $N\geqslant N_{0}(\vartheta,C_{0},\delta_{\pm})$ large enough.

The Stieltjes transform of the empirical eigenvalue distribution of $H$ satisfies

The individual matrix elements of the Green function satisfy

The norm of $H$ is bounded by $2+N^{-2/3}(\log N)^{9L}$ in the sense that

The local semicircle law implies that the eigenvalues are close to their classical locations with high probability. Recall that $\lambda_{1}\leqslant\lambda_{2}\leqslant\cdots\leqslant\lambda_{N}$ are the ordered eigenvalues of $H$. The classical location $\gamma_{\alpha}$ of the $\alpha$-th eigenvalue was defined in (1.7).

Under the assumptions of Theorem 2.1 there exist positive constants $A_{0}>1$, $C,c$, and $\tau<1$, depending only on $\vartheta$ in (1.9), $\delta_{\pm}$ in Assumption (B), and $C_{0}$ in Assumption (C), such that such that

A simple consequence of Theorem 2.1 is that the eigenvectors of $H$ are completely delocalized.

Under the assumptions of Theorem 2.1 we have

Choosing $\eta=N^{-1}(\log N)^{20L}$ yields the claim. ∎

The proofs of Propositions 2.4 and 2.5 are very similar. For definiteness, we give the details for the edge case (Proposition 2.4). The rest of this section is devoted to the proof of Proposition 2.4. The main tool is the following Green function comparison theorem, which was proved in EYYrigi , Theorem 6.3.

with some constant $C_{1}>0$. Then there exists a constant $\varepsilon_{0}>0$, depending only on $C_{1}$, such that for any $\varepsilon<\varepsilon_{0}$ and for any real numbers $E_{1}$ and $E_{2}$ satisfying

for some constant $C$ and large enough $N$, depending only on $C_{1}$, $\vartheta$ in (1.9), $\delta_{\pm}$ in Assumption (B), and $C_{0}$ in Assumption (C).

The basic idea behind the proof of Proposition 2.4 is to first cast the level repulsion estimate into an estimate in terms of Green functions and then use the Green function comparison theorem. Recalling $L$ from Definition 1.1, we set

be the characteristic function of the interval $[E_{L},E]$. For any $\eta>0$ we define the approximate delta function $\theta_{\eta}$ on the scale $\eta$ through

The following result provides a tool for estimating the number operator using Green functions. It is proved in EYYrigi , Lemma 6.1 and Corollary 6.2.

After these preparations we may complete the proof of Proposition 2.4.

Abbreviate $E_{\pm}=E\pm N^{-2/3-\alpha}$ and set $\varepsilon\mathrel{\mathop{:}}=2\alpha$. By using (2.19) for $E=E_{+}$ and $E=E_{-}$, and subtracting the resulting two inequalities, we get, with high probability,

Let $F$ be a nonnegative increasing smooth function satisfying $F(x)=1$ for $x\geqslant 2$ and $F(x)=0$ for $x\leqslant 3/2$. Then, using (2.20) and Lemma 2.6, we have

Proof of Theorem 1.6

To simplify presentation, in this section we prove Theorem 1.6 in the special case \theta=\theta\bigl{(}{N\bar{u}_{\alpha}(i)u_{\alpha}(j)}\bigr{)}, where $\alpha\leqslant\varphi^{\rho}$. The proof of the general case is analogous; see Section 4 for more details.

In a first step we convert the eigenvector problem into a problem involving the Green function $G_{ij}$. To that end, we define

where the second equality follows easily by spectral decomposition, $G_{ij}(z)=\sum_{\beta}\frac{\bar{u}_{\beta}(i)u_{\beta}(j)}{\lambda_{\beta}-z}$. Note that

as well as $\operatorname{Im}G_{ii}(z)=\widetilde{G}_{ii}(z)$. It is a triviality that all of the results from Section 2 hold with $z$ replaced with $\bar{z}$.

The following lemma expresses the eigenvector components as an integral of the Green function over an appropriate random interval.

Under the assumptions of Theorem 1.6, for any $\varepsilon>0$ there exist constants $C_{1}$, $C_{2}$ such that for $\eta=N^{-2/3-\varepsilon}$ we have

We shall fix $i,j$ and $\alpha\leqslant\varphi^{\rho}$; it is easy to check that all constants in the following are uniform in $i,j$, and $\alpha\leqslant\varphi^{\rho}$. We write

Using Theorem 2.3 it is easy to prove that for $C_{1}$ large enough we have

holds with high probability for some $c>0$, as long as

where we use the notation (3.3), i.e. $\lambda_{\alpha}^{\pm}\mathrel{\mathop{:}}=\lambda_{\alpha}\pm\varphi^{C_{1}}\eta$. We now choose

By the assumption on $\theta$ and using Theorem 2.3, we therefore find

for some constant $C_{0}$, where we used Theorem 2.3 and the assumption on $\theta$. Now the level repulsion estimate (1.6) implies that the second term of (3.8) is $o(1)$. We now observe that, by (2.10), we have $\lambda_{\alpha}^{+}\leqslant-2+N^{-2/3}\varphi^{C_{2}}$ and $\lambda_{\alpha-1}^{+}\geqslant-2-N^{-2/3}\varphi^{C_{2}}$ with high probability. It therefore easy to see that

In order to be able to apply the mean value theorem to $\theta$ with the decomposition (3.10), we need an upper bound on

where the inequality holds with high probability for any $C_{3}$; here we used Theorem 2.3. Using $\gamma_{\beta}\geqslant-2+c(\beta/N)^{2/3}$ as well as (2.10), we find with high probability for large enough $C_{3}$

Thus the left-hand side of (3.11) is bounded by $\varphi^{C_{0}+C_{3}+1}$.

Let us abbreviate $\chi(E)\mathrel{\mathop{:}}={\bf 1}(\lambda_{\alpha-1}\leqslant E^{-}\leqslant\lambda_{\alpha})$. Now, recalling the assumption on $\theta$, we may apply the mean value theorem as well as Theorem 2.3 to get

for some constant $\widetilde{C}\leqslant C(C_{0}+C_{3}+1)$ independent of $C_{1}$. We now estimate the right-hand side of (3.13). Exactly as in (3.12), one finds that there exists $C_{4}$ such that the contribution of $\beta\geqslant\varphi^{C_{4}}$ to the right-hand side of (3.13) vanishes in the limit $N\to\infty$. Next, we deal with the eigenvalues $\beta<\alpha$ (in the case $\alpha>1$). Using Theorem 2.3 we get

What remains is the estimate of the terms $\alpha<\beta\leqslant\varphi^{C_{4}}$ in (3.13). For a given constant $C_{5}>0$ we partition $I=I_{1}\cup I_{2}$ with $I_{1}\cap I_{2}=\emptyset$ and

It is easy to see that, for large enough $C_{5}$, we have

where $c>0$. Let us therefore consider the integral over $I_{1}$. One readily finds, for $\lambda_{\alpha}\leqslant\lambda_{\alpha+1}\leqslant\lambda_{\beta}$, that

From Theorem 2.3 we therefore find that there exists a constant $C_{6}$, depending on $C_{1}$, such that

In a second step we convert the cutoff function in lemma 3.1 into a function of $\widetilde{G}_{ij}$.

for $\varepsilon>0$. Then for $\varepsilon$ small enough we have (recall the definition (3.3))

with high probability for sufficiently large $N$. We therefore find that

Together with (3.2), the claim follows. ∎

In a third and final step, we use the Green function comparison method to show the following statement.

Under the assumptions of Lemma 3.2, we have

The rest of this section is devoted to the proof of Lemma 3.3.

The claimed uniformity in $i$ and $j$ is easy to check in our proof, and we shall not mention it anymore. Throughout the following we rename $i=\alpha$ and $j=\beta$ in order to use $i$ and $j$ as summation indices. We now fix $\alpha$ and $\beta$ for the whole proof. (Note that $\alpha$ and $\beta$ need not be different.)

We begin by dropping the diagonal terms in (3.22).

with high probability and, recalling that $q^{\prime}$ is bounded,

with high probability. Therefore the difference of the arguments of $\theta$ in (3.23) is bounded by $\varphi^{C}N^{-1/3-\varepsilon}$ with high probability. (Recall that $\lvert I\rvert\leqslant\varphi^{C}N^{-2/3}$.) Moreover, since $q$ is bounded, it is easy to see that both arguments of $\theta$ in (3.23) are bounded with high probability by

where we used Theorem 2.1. The claim now follows from the mean value theorem and the assumption on $\theta$. ∎

For the following we work on the product probability space of the ensembles $H^{\bf{v}}$ and $H^{\bf{w}}$. To distinguish them we denote the elements of $H^{\bf{v}}$ by $N^{1/2}v_{ij}$ and the elements of $H^{\bf{w}}$ by $N^{-1/2}w_{ij}$. We fix a bijective ordering map $\Phi$ on the index set of the independent matrix elements,

Let us now fix a $\gamma$ and let $(a,b)$ be determined by $\Phi(a,b)=\gamma$. Throughout the following we consider $\alpha,\beta,a,b$ to be arbitrary but fixed and often omit dependence on them from the notation. Our strategy is to compare $H_{\gamma-1}$ with $H_{\gamma}$ for each $\gamma$. In the end we shall sum up the differences in the telescopic sum (3.25).

Here $Q$ is the matrix obtained from $H_{\gamma}$ (or, equivalently, from $H_{\gamma-1}$) by setting the matrix elements indexed by $(a,b)$ and $(b,a)$ to zero. Next, we define the Green functions

For the estimates we need the following basic result, proved in EYYrigi (Equation (6.32)).

For any $\eta^{\prime}\mathrel{\mathop{:}}=N^{-2/3-\delta}$ we have with high probability

The same estimates hold for $S$ instead of $R$.

Our comparison is based on the resolvent expansion

Using Lemma 3.5 we easily get with high probability, for $i\neq j$,

we therefore have the trivial bound with high probability

The variable $s$ counts the maximum number of diagonal resolvent matrix elements in $\Delta X_{ij,k}$. The bookkeeping of $s$ will play a crucial role in our proof, since the smallness associated with off-diagonal elements (see Lemma 3.5) is needed to control the resolvent expansion (3.28) under the two-moment matching assumption.

For fixed $\alpha,\beta,a,b$ there exists exists a random variable $A$, which depends on the randomness only through $Q$ and the first two moments of $v_{ab}$, such that

where $t\mathrel{\mathop{:}}={|\{a,b\}\cap\{\alpha,\beta\}|}$ .

Before proving Lemma 3.6, we show how it implies Lemma 3.3.

It suffices to prove that each summand in (3.25) is bounded by o\bigl{(}{N^{-2+t}+N^{-2+{\bf 1}(a=b)}}\bigr{)}. This follows immediately by applying Lemma 3.6 to $S=(H_{\gamma-1}-z)^{-1}$ and $S^{\prime}\mathrel{\mathop{:}}=(H_{\gamma}-z)^{-1}$ and subtracting the statements; note that the random variables $A$ in the statement of Lemma 3.6 are by definition the same for $S$ and $S^{\prime}$. ∎

Throughout the proof of Lemma 3.6 we shall abbreviate $H\equiv H_{\gamma-1}=(h_{ij})$, as well as $S\equiv S(z)=(H-z)^{-1}$.

Since $E\in I$ (recall (3.3)) we get from Theorem 2.1 that with high probability

Next, in the definition of $x(E)$ and $y(E)$ we condition on the variable $s$ defined in (3.32) by introducing, for $s=0,1,2$,

As above, $s$ is a bookkeeping index that bounds the number of diagonal resolvent matrix elements appearing in the resolvent expansion.

We abbreviate $\Delta x_{s}(E)\mathrel{\mathop{:}}=x_{s}^{S}(E)-x_{s}^{R}(E)$ and $\Delta y_{s}(E)=y_{s}^{S}(E)-y_{s}^{R}(E)$. Recalling the definition $t={|\{a,b\}\cap\{\alpha,\beta\}|}$, we find with high probability

where we used Theorem 2.1 and the elementary inequality s+{\bf 1}\bigl{(}{s={\bf 1}(t>0)}\bigr{)}\leqslant t+1 which holds if $x_{s}(E)\neq 0$. Thus we get with high probability

Now we may argue similarly to (3.33). We find that, for any $E$-dependent random variable $f\equiv f(E)$ independent of $h_{ab}$, there exists a random variable $A_{2}$, which depends on the randomness only through $Q$, $f$, and the first two moments of $h_{ab}$, such that with high probability

where $\Omega$ is any event. Note that, as in (3.33), we find that (3.41) is suppressed by a factor $N^{-1}$ compared to (3.31). This may be easily understood, as the leading order error term in the resolvent expansion of (3.31) is of order $1$ in $H$, whereas the leading order error term in (3.41) is of order $3$ in $H$. These error terms have the same number of off-diagonal elements (estimated using Lemma 3.5), and the same entropy factor of the summation indices.

We may derive similar bounds for $y_{s}(E)$. As in (3.31), we have with high probability

Furthermore, we find that there exists an $E$-dependent random variable $A_{3}(E)$, which depends on the randomness only through $Q$ and the first two moments of $h_{ab}$, such that with high probability

here and in the following we omit the argument $E$ unless it is needed. Using (3.42) we have with high probability

The use of the mean value theorem for $\varepsilon$ small enough is easy to justify using the assumption on $\theta$ and the bounds (3.37) and (3.38). In the following we shall no longer mention such estimates of the argument of derivatives of $\theta$, which can always be easily checked in a similar fashion.

Recall that an error of order $o(N^{-2+t})$ is affordable in the error estimate. Thus, using the basic power counting given by (3.37), (3.38), (3.40), and (3.42), we find with high probability

We now start dealing with the individual terms on the right-hand side of (3.44).

First, we consider the terms containing $\Delta x_{1}$ and $\Delta y_{1}$. Applying (3.41) and (3.43) we find that there exists a random variable $A_{4}$, which depends on the randomness only through $Q$ and the first two moments of $h_{ab}$, such that

with high probability. Inserting this into (3.44), we find with high probability

Thus we only need to focus on the error terms $\Delta x_{0}$ and $\Delta y_{0}$. Note that we have

Recall that the $(i,j)$-component of the resolvent expansion (3.28) reads

Now we assume that $i\neq j$ and $|\{i,j\}\cap\{a,b\}|=0$. It is easy to see that this assumption holds for any matrix element in the formulas (3.47) and (3.48). Then we can use Lemma 3.5 to estimate the $m$-th term as follows:

Next, we apply the resolvent expansion to $X_{ij,k}$. Note that in our applications errors of size $O(N^{-8/3-c})$ are affordable in $\Delta X_{ij,k}$ for some $c>0$ independent of $\varepsilon$ (see (3.23) and (3.24)). Now let us assume that the indices $i,j,a,b,k$ satisfy the condition

$\{i,j\}\cap\{a,b\}=\emptyset$ and $k\neq i,j,a,b$.

In the applications we shall set $i=\alpha$ and $j=\beta$ in (3.47), and $i=j$ in (3.48). In both cases, it is easy to check that the condition $(*)$ is satisfied for nonvanishing summands.

We can therefore separate $\Delta X_{ij,k}$ into three parts, indexed according to how many $V$-matrix elements they contain,

where $[C]_{l}$, $l=1,2$, means the complex conjugate of the first $l$ terms on the right-hand side with $i$ and $j$ exchanged. Furthermore, it easy to see that the second term on the right-hand side of (3.54) is of order $O(N^{-17/6+C\varepsilon})$. Thus we find with high probability

where $Y$ is a finite sum of terms of the form

and terms obtained from (3.56) by (i) taking the complex conjugate and exchanging $i$ and $j$, and (ii) exchanging $a$ and $b$. Using Lemma 3.5 we find that (3.56) is equal to

with high probability. The splitting (3.51) induces a splitting

with high probability in self-explanatory notation. It is easy to see that

From (3.47) and (3.56), we find that $\Delta x_{0}^{(3)}$ is a finite sum of terms of the form

with high probability, where the other terms are obtained from (3.59) as described after (3.56).

Now we insert these bounds into (3.46). Recall that the upper index $l$ in $\Delta x_{0}^{(l)}$ and $\Delta y_{0}^{(l)}$ counts the number of $V$-matrix elements. Thus we find, recalling (3.46) and the power counting estimates (3.58) and (3.61), that there is a random variable $A_{5}$, depending on the randomness only through $Q$ and the two first moments of $h_{ab}$, such that

with high probability. Moreover, by the same power counting estimates we find that the second line of (3.62) is bounded by $o(N^{-1})$. We use this rough bound in the case $a=b$, and get

Hence Lemma 3.6 is proved if we can show that, for $a\neq b$, we have

with high probability. This is proved below. ∎

The other term on the left-hand side of (3.64) is estimated similarly. Let us abbreviate

From (3.36) and the assumption on $\theta$, we find that $\lvert B^{R}\rvert\leqslant N^{C\varepsilon}$ with high probability.

The proof of (3.64) is therefore complete if we can show that, assuming the sets $\{\alpha,\beta\},\{a\},\{b\},\{k\}$ are disjoint, we have

In order to prove (3.68), we first use a simple resolvent expansion to show that with high probability

where $B^{S}$ is defined analogously to (3.66) with $R$ replaced by $S$. Therefore it suffices to prove

In order to complete the proof, we introduce some notation. Recall that $H\equiv H_{\gamma-1}$ and $S=(H-z)^{-1}$. We define $H^{(a)}$ as the matrix obtained from $H$ by setting its $a$-th column and $a$-th row to be zero. For any function $F\equiv F(H)$ we define $F^{(a)}\mathrel{\mathop{:}}=F(H^{(a)})$. We now remove the $a$-th row and column from $H$ in (3.70), which we can do with a negligible error. The key identity is the following resolvent identity, proved in Lemma 4.2 of EYY : For $k\neq i,j$ we have

Next, we claim that the conditional expectation – with respect to the variables in the $a$-th column of $H$ – of $S_{\beta a}$ is much smaller than its typical size. To that end, we use the identities, valid for $i\neq j$,

proved in EKYY2 , Lemma 6.10. Now using (3.74) we find

The conditional expectation with respect to the variables in the $a$-th column of $H$ applied to the first term on the right-hand side of (3.75) vanishes; hence its contribution to the expectation of (3.73) also vanishes. In order to estimate the second term on the right-hand side of (3.73), we note that with high probability

by Lemma 3.5. Moreover, using the large deviation bound (3.9) in EYYrigi , we get with high probability

where in the last step we used (3.71) and Lemma 3.5. Putting everything together, we find that the expectation of (3.73) is bounded in absolute value by $N^{-4/3+C\varepsilon}$. By (3.72), this completes the proof of (3.70), and hence of (3.64). ∎

Extension to eigenvalues and several arguments

In this section we describe how the arguments of Section 3 extend to general functions $\theta$ as in (1.11).

Consider first the case of a single eigenvalue, $\lambda_{\beta}$, in which case the claim reads

where the first step follows from (2.18) and the second from Theorem 2.2.

Integrating by parts again, we find with high probability

Now we may apply the Green function comparison method from Section 3.1. In fact, in this case the analysis is easier as we have no fixed indices $i$ and $j$ to keep track of.

The general case, $\theta$ as in (1.11), is treated similarly. Repeating successively the above procedure for each argument $\lambda_{\beta_{1}},\dots,\lambda_{\beta_{k}}$, we find that there is a constant $C_{k}$, depending on $k$, such that

and set $\eta\mathrel{\mathop{:}}=N^{-2/3-\varepsilon}$; $q_{\alpha}$ is the function from Lemma 3.2. Here at each step we used the assumption on $\theta$, that $r^{\prime}_{\beta}$ is bounded, and the estimate

where in the first step we used (2.19), in the second Theorem 2.2, and in the third the definition (1.7) of $n_{sc}$.

The randomness on the right-hand side of (4.5) is expressed entirely in terms of Green functions; hence (4.5) is amenable to the Green function comparison method from Section 3.1. The complications are merely notational, as we now have $2k$ fixed indices $i_{1},j_{1},\dots,i_{k},j_{k}$ instead of just the two $i,j$.

Eigenvectors in the bulk: proof of Theorem 1.10

For any $|E|\leqslant 5$ and $0<\eta\leqslant 10$, we have with high probability

By Theorem 2.1, we only need to consider $\eta\leqslant y\mathrel{\mathop{:}}=\varphi^{C_{1}}N^{-1}$ for some $C_{1}>0$. We use the trivial bound

which follows by a simple dyadic decomposition; see EYY , Equation (4.9). Thus we get

The strategy behind the proof of Theorem 1.10 is very similar to that of Theorem 1.6, given in Section 3. In a first step, we express the eigenvector components using integrals involving resolvent matrix elements $G_{ij}$; in a second step, we replace the sharp indicator functions in the integrand by smoothed out functions which depend only on the resolvent; in a third step, we use the Green function comparison method to complete the proof.

For ease of presentation, we shall give the proof for the case $\theta=\theta(N\bar{u}_{\alpha}(i)u_{\alpha}(j))$; we show that

where $\rho N\leqslant\alpha\leqslant(1-\rho)N$. As outlined in Section 4, the extension to general functions $\theta$, as given in (1.15), is an easy extension which we sketch briefly at the end of this section.

We now spell out the three steps mentioned above.

Step 1. The analogue of Lemma 3.1 in the bulk is the following result whose proof uses (1.5) and Lemma 2.5, and is very similar to the proof of Lemma 3.1 (in fact somewhat easier). We omit further details.

Under the assumption of Theorem 1.10, for any $\varepsilon>0$ there exist constants $C_{1}$, $C_{2}$ such that for $\eta=N^{-1-\varepsilon}$ we have

Step 2. We choose $\eta=N^{-1-\varepsilon}$ for some small enough $\varepsilon>0$ and express the indicator function in

using Green functions (as before, we write $I_{\alpha}\equiv I$). Using Theorem 2.2, we know that

where $E_{L}\mathrel{\mathop{:}}=-2-\varphi^{C}N^{-2/3}$.

As explained in Section 1.3, the approach in Step 2 has to be modified slightly from the one employed in Section 3. The reason is that the size of the interval $[E_{L},E^{-}]$ is no longer small, but of order one.

Now we choose $\eta_{d}\mathrel{\mathop{:}}=N^{-1-d\varepsilon}$, for some fixed $d>2$. Then, using Lemma 5.1 and an argument similar to the proof of Lemma 3.2, we find that

To simplify notation, we follow the conventions of Section 3 in writing $I\equiv I_{\alpha}$, $q\equiv q_{\alpha}$ and $f_{E}\equiv f_{E_{L},E^{-},\eta_{d}}$, and set $\alpha=i$ and $\beta=j$. In this notation, we need to estimate

Now we express $f_{E}(H)$ in terms of Green functions using Helffer-Sjöstrand functional calculus. Let $\chi(y)$ be a smooth cutoff function with support in $$, with$\chi(y)=1$for$|y|\leqslant 1/2$ and with bounded derivatives. Then we have (see e.g. Equation (B.12) of ERSY )

Therefore the third term of (5.10) is bounded, with high probability, by

Step 3. We estimate (5.8) using a Green function comparison argument, similarly to Section 3.1. As in Section 3.1, we use the notation

Similarly to Lemma 3.4, we begin by dropping the diagonal terms. Using Lemma 5.1 we find

with high probability, so that it suffices to prove

Using (5.11) we find that it suffices to prove

By a telescopic expansion similar to (3.25), we find that (5.15) follows if we can prove, with high probability,

Now we prove (5.18). We use the resolvent expansion

We may now estimate the variables $x,y$, and $\widetilde{y}$. Let us first consider the variables $\widetilde{y}$. From the definition of $\chi$, we find that in the integrand of (5.17) we have $\sigma\geqslant c$ and therefore by Theorem 2.1 we have $\Lambda_{\sigma}\leqslant\varphi^{C}N^{-1/2}$ with high probability. Thus we get from (5.17)

In order to estimate the contributions of the variables $y$, we integrate by parts, first in $e$ and then in $\sigma$, to obtain

Using (5.22), it is easy to see that the sum of the two first terms of (5.24) is bounded by $N^{-p/2+C\varepsilon}$. In order to estimate the last term of (5.24), we use (5.22) and (5.1) to get the bound

with high probability. Moreover, we have the bound

with high probability. This concludes our estimate of the terms in the resolvent expansion of $x$,$y$, and $\widetilde{y}$.

Now using the power counting bounds from (5.23), (5.25), (5.26), and (5.27), we may easily complete the Green function comparison argument to prove (5.18), as in Section 3.1.

References