Mesoscopic eigenvalue density correlations of Wigner matrices

Abstract

We investigate to what extent the microscopic Wigner–Gaudin–Mehta–Dyson (WGMD) (or sine kernel) statistics of random matrix theory remain valid on mesoscopic scales. To that end, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. In the mesoscopic regime, density correlations are much weaker than in the microscopic regime. Our result is an explicit formula for the two-point function. This formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that in the real symmetric case there are subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale. The proof is based on a hierarchy of Schwinger–Dyson equations for a sufficiently large class of polynomials in the entries of the Green function. The hierarchy is indexed by a tree, whose depth is controlled using stopping rules. A key ingredient in the derivation of the stopping rules is a new estimate on the density of states, which we prove to have bounded derivatives of all order on all mesoscopic scales.

Appendices

Appendix A: Comparison to Gustavsson’s theorem

In this section we compare our result to the ones in [24] and [34]. Roughly, the conclusion is that the question addressed in [24, 34] is independent from the one addressed in our work, and neither implies the other.

In Gustavsson’s work [24], the fluctuation of a single eigenvalue was first established to be Gaussian for GUE, and it was also showed that the joint limit distribution of eigenvalues is a Gaussian process, provided that the eigenvalue are separated by a mesoscopic distance. In [34], the result was extended to GOE and, by moment matching, to a class of Wigner matrices.

For the rest of the section let us adopt the assumptions in Theorem 1.2, and in addition we assume H is GOE. Recall the notations $\alpha :=-\log _N \eta$ and $\gamma :=-\log _N \omega$. We would like to compute $\int p_{E}(u,v) f_{+}(u)g_{-}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v$. By (5.2) we have

$$\begin{aligned}&\int p_{E}(u,v) f_{-}(u)g_{+}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v\nonumber \\&\quad =\sum _{i,j}\frac{1}{N^2\eta ^2}{\mathbb {E}} \Bigg \langle f\Bigg (\frac{(\lambda _i-E)\varrho _E-\omega }{\eta }\Bigg ) \Bigg \rangle \Bigg \langle g\Bigg (\frac{(\lambda _j-E)\varrho _E+\omega }{\eta }\Bigg ) \Bigg \rangle . \end{aligned}$$

(A.1)

Let $\lambda _1\leqslant \cdots \leqslant \lambda _N$ be the eigenvalues of H, and for $k=1,2,\ldots ,N$, let the quantile $\gamma _k$ be the typical location of $\lambda _k$, i.e. it satisfies $k/N=\int _{-2}^{\gamma _k} \varrho _x \mathrm {d}x$. By our assumptions $E \in [-2+\tau ,2-\tau ]$, $f,g \in C^{\infty }_c({\mathbb {R}})$, and a standard eigenvalue rigidity result (e.g. Theorem 2.2, [22]), we see that the sum in (A.1) can be reduced to

$$\begin{aligned}&\int p_{E}(u,v) f_{-}(u)g_{+}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v \\&\quad =\sum _{i\in A,j \in B}\frac{1}{N^2\eta ^2}{\mathbb {E}} \Bigg \langle f\Bigg (\frac{(\lambda _i-E)\varrho _E-\omega }{\eta }\Bigg ) \Bigg \rangle \Bigg \langle g\Bigg (\frac{(\lambda _j-E)\varrho _E+\omega }{\eta }\Bigg ) \Bigg \rangle +\mathrm {O}(N^{-10}), \end{aligned}$$

where $A :=\{i:|(\gamma _i-E)\varrho _E-\omega | \leqslant \eta \log N\}$, $B :=\{j:|(\gamma _j-E)\varrho _E+\omega | \leqslant \eta \log N\}$. Clearly for $i \in A$ and $j\in B$, we have $i-j \asymp N\omega = N^{1-\gamma }$, and Theorem 5 of [34] shows that

$$\begin{aligned} \bigg (\frac{\pi \varrho _{\gamma _i}N(\lambda _i-\gamma _i)}{\sqrt{\log N}}, \frac{\pi \varrho _{\gamma _j}N(\lambda _j-\gamma _j)}{\sqrt{\log N}}\bigg ) \end{aligned}$$

(A.2)

converges weakly to the Gaussian random vector $N({\varvec{\mu }},\Sigma )$ with

$$\begin{aligned} {\varvec{\mu }}= \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad \text{ and } \quad \Sigma =\bigg (\begin{array}{cc} 1 &{}\quad \gamma \\ \gamma &{}\quad 1 \\ \end{array}\bigg ), \end{aligned}$$

as $N \rightarrow \infty$. Also, by $|\gamma _i-E|=\mathrm {O}(\omega )$ we see that $\varrho _E=\varrho _{\gamma _i}+\mathrm {O}(\omega )$. Hence the random vector

$$\begin{aligned} \bigg (\frac{\pi \varrho _{E}N(\lambda _i-\gamma _i)}{\sqrt{\log N}}, \frac{\pi \varrho _{E}N(\lambda _j-\gamma _j)}{\sqrt{\log N}}\bigg ) \end{aligned}$$

(A.3)

has the same weak limit as the one in (A.2). With the rescaling $u_k:=(\lambda _k-E)N\varrho _E$ for $k=1,2,\ldots ,N$, we have

$$\begin{aligned} \int p_{E}(u,v) f_{-}(u)g_{+}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v= & {} \sum _{i\in A,j \in B}\frac{1}{N^2\eta ^2}{\mathbb {E}} \Big \langle f\Big (\frac{u_i-N\omega }{N\eta }\Big ) \Big \rangle \Big \langle g\Big (\frac{u_j+N\omega }{N\eta }\Big ) \Big \rangle \nonumber \\&+\mathrm {O}(N^{-10}), \end{aligned}$$

(A.4)

Now let us compute the right-hand side of (A.4) by using the weak limit of (A.3), i.e. that the random vector $(u_i,u_j)$ has asymptotically (as $N \rightarrow \infty$) the same law as the Gaussian vector $({\tilde{u}}_i, {\tilde{u}}_j) \overset{\text {{d}}}{=}N({\varvec{\mu }}', \Sigma ')$, where

$$\begin{aligned} {\varvec{\mu }}'= \begin{pmatrix} (\gamma _i-E)N\varrho _E \\ (\gamma _j-E)N\varrho _E \end{pmatrix} =:\begin{pmatrix} m_i \\ m_j \end{pmatrix} \ \ \text{ and } \ \ \ \Sigma '=\frac{\log N}{\pi ^2}\bigg (\begin{array}{cc} 1 &{}\quad \gamma \\ \gamma &{}\quad 1 \\ \end{array}\bigg ). \end{aligned}$$

By Taylor expansion, we have

$$\begin{aligned}&\sum _{i\in A,j \in B}\frac{1}{N^2\eta ^2}{\mathbb {E}} \Big \langle f\Big (\frac{u_i-N\omega }{N\eta }\Big ) \Big \rangle \Big \langle g\Big (\frac{u_j+N\omega }{N\eta }\Big ) \Big \rangle \nonumber \\&\quad =\sum _{i\in A,j \in B}\frac{1}{N^2\eta ^2}{\mathbb {E}} \bigg \langle f\Big (\frac{m_i-N\omega }{N\eta }\Big )\nonumber \\&\qquad +\frac{1}{N\eta }f'\Big (\frac{m_i-N\omega }{N\eta }\Big )(u_i-m_i) +\frac{1}{2N^2\eta ^2}f''\Big (\frac{m_i-N\omega }{N\eta }\Big )(u_i-m_i)^2+\cdots \bigg \rangle \nonumber \\&\qquad \cdot \bigg \langle g\Big (\frac{m_j+N\omega }{N\eta }\Big )+\frac{1}{N\eta }g'\Big (\frac{m_j+N\omega }{N\eta }\Big )(u_j-m_j)\nonumber \\&\qquad +\frac{1}{2N^2\eta ^2}g''\Big (\frac{m_j+N\omega }{N\eta }\Big )(u_j-m_j)^2+\cdots \bigg \rangle . \end{aligned}$$

(A.5)

The first non-vanishing term is the one proportional to $f'g'$. We now compute it asymptotically using Gustavsson’s theorem, by replacing $(u_i,u_j)$ with $({\tilde{u}}_i, {\tilde{u}}_j)$. The result is

$$\begin{aligned}&\sum _{i\in A,j \in B}\frac{1}{N^4\eta ^4}{\mathbb {E}} \bigg \langle f'\Big (\frac{m_i-N\omega }{N\eta }\Big )({\tilde{u}}_i-m_i) \bigg \rangle \bigg \langle g'\Big (\frac{m_j+N\omega }{N\eta }\Big )({\tilde{u}}_j-m_j) \bigg \rangle \nonumber \\&\quad =\sum _{i\in A,j \in B}\frac{\log N}{N^4\eta ^4\pi ^2} f'\Big (\frac{m_i-N\omega }{N\eta }\Big ) g'\Big (\frac{m_j+N\omega }{N\eta }\Big ) \nonumber \\&\quad =\sum _{j \in B}\frac{\log N}{N^3\eta ^3\pi ^2}g'\Big (\frac{m_j+N\omega }{N\eta }\Big )\int (1+\mathrm {O}(\omega ))f'(x)\, \mathrm {d}x+\mathrm {O}(N^{-10}) \nonumber \\&\quad =\frac{\log N}{N^2\eta ^2\pi ^2}\int (1+\mathrm {O}(\omega ))f'(x)g'(y) \,\mathrm {d}x\, \mathrm {d}y+\mathrm {O}(N^{-10})\prec N^{2\alpha -\gamma -2}, \end{aligned}$$

(A.6)

where in the second and third step we used the fact that $m_{k+1}-m_k=1+\mathrm {O}(\omega )$ for $k \in A\cup B$, and the Riemann sum of a smooth compactly supported function converges to its integral to any polynomial order (by the Poisson summation formula). Similarly, we can show that all other terms in (A.5) are bounded by $\mathrm {O}_{\prec }(N^{4\alpha -4})$. Thus, under the assumption that we can use Gustavsson’s theorem and the error from the replacement of $(u_i, u_j)$ with the Gaussian vector $({\tilde{u}}_i, {\tilde{u}}_j)$ is negligible, we get, together with (A.4),

$$\begin{aligned} \int p_{E}(u,v) f_{-}(u)g_{+}(v) \prec N^{2\alpha -\gamma -2}+N^{4\alpha -4}. \end{aligned}$$

We find that the right-hand side is much smaller than $N^{2\gamma -2}$ for $\alpha >3/4$ and $\gamma >2\alpha /3$. However, by Theorem 1.2 we know that the leading term in $\int p_{E}(u,v) f_{-}(u)g_{+}(v)$ is of order $N^{2\gamma -2}$. We conclude that the use of Gustavsson’s theorem leads to a wrong result in at least the regime $\alpha >3/4$ and $\gamma >2\alpha /3$. This argument shows that the computation of $\int p_{E}(u,v) f_{-}(u)g_{+}(v)$ using Gustavsson’s theorem leads to the wrong result in a rather flagrant fashion: the obtained quantity is much smaller than than the true value. In fact, we believe that more generally Gustavsson’s result cannot be used to recover the correct mesoscopic density–density correlations in any regime: the error made between (A.5) and (A.6) is never affordable.

The issue lies in the fact that, in order to compute sums of the form (A.5), even to leading order, one needs much stronger control on the joint distribution of the individual eigenvalues than is provided in [24] and [34]. This is a manifestation of the fact, mentioned in the introduction, that eigenvalue density–density correlations are much weaker than their location–location correlations. Going from the latter to the former consequently requires very precise asymptotics for the latter.

Appendix B: Comparison to results on macroscopic linear statistics

In this appendix we give the short calculation that shows how our main result, Theorem 1.2, recovers the well-known covariance formula for macroscopic linear statistics of Wigner matrices [30, 32]. This corresponds to the extreme case $\omega \asymp \eta \asymp 1$ in Theorem 1.2.

Let for shortness $f^{\eta }(x)=\eta ^{-1}f(((x-E)\varrho _E-\omega )/\eta )$ and $g^{\eta }(x)=\eta ^{-1}g(((x-E)\varrho _E+\omega )/\eta )$. It follows from (1.8), (3.16) and (5.2) that there exists $c=c(\tau )>0$ such that

$$\begin{aligned}&\,\frac{1}{N^2}\mathrm {Cov}({{\,\mathrm{Tr}\,}}f^\eta (H);g^{\eta }(H))\\&\quad =\int p_{E}(u,v) f_{+}(u)g_{-}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v\\&\quad =\,\int \Big (-\frac{1}{\pi ^2(u-v)^2}+\frac{F_1(u,v)}{N^2\kappa _E^2} +\frac{F_2(u,v)}{N^2\kappa _E^2}\sum _{i,j}{{\mathcal {C}}}_4(H_{ij})\Big ) f_{+}(u)g_{-}(v) \,{\mathrm {d}}u\,{\mathrm {d}}v\\&\qquad +\mathrm {O}(N^{-2-c})+\mathrm {O}\Big ({\frac{1}{N^4\omega ^4}}\Big )\\&\quad =\,\frac{1}{\pi ^2}\int \bigg (-\frac{1}{N^2(x_1-x_2)^2}\cdot \frac{4-x_1x_2}{\sqrt{4-x_1^2}\sqrt{4-x_2^2}} \\&\qquad +\frac{(x_1^2-2)(x_2^2-2)}{2N^2\sqrt{(4-x_1^2)(4-x_2^2)}}\sum _{i,j}{{\mathcal {C}}}_4(H_{ij})\bigg ) f^{\eta }(x_1)g^{\eta }(x_2)\\&\qquad +\mathrm {O}\Big (N^{-2-c}+{\frac{1}{(u-v)^4}}\Big ). \end{aligned}$$

Here $F_1,F_2$ are defined as in (3.16). We see that when $\omega \gg N^{-1/2}$ the integral RHS of the above equation dominates, and it matches Equation (VI.81) in [30].

Appendix C: Remark on the distribution of the diagonal entries

Theorems 1.2 and 2.1 can be extended to a more general case where we relax the assumption on the variance of $H_{ii}$ in Definition 1.1. Here we summarize the appropriate generalizations of Theorems 2.1 and 1.2, whose proofs are simple modifications of the arguments given in the previous sections. This generalization illustrates the great sensitivity of our results to the precise distribution of the matrix entries.

Let $\tilde{H}$ be a Wigner matrix where the condition ${\mathbb {E}} |\sqrt{N}H_{ii}|^2=2/\beta$ is replaced by ${\mathbb {E}}|\sqrt{N}\tilde{H}_{ii}|^2=\zeta _i$, and $\max _i \zeta _i = \mathrm {O}(1)$. Let $\tilde{G}(z):=(\tilde{H}-z)^{-1}$, and let ${\tilde{\Upsilon }}_{E,\beta }$ be the analogue of $\Upsilon _{E,\beta }$ defined in (1.8)–(1.9).

Theorem 2.1 is generalized as follows. For the real symmetric case ($\beta =1$), we have

$$\begin{aligned}&\mathrm {Cov}\big (\tilde{\underline{G} \!\,}(z_1), \tilde{\underline{G} \!\,}(z_2^*) \big ) \\&\quad =\mathrm {Cov}\big (\underline{G} \!\,(z_1), \underline{G} \!\,(z_2^*) \big ) + \bigg (\frac{-2+2\mathrm {i}{{\,\mathrm{Im}\,}}m(E)^2}{N^4(z_1-z_2^*)^2}+\frac{m(z_1)m(z_2^*)}{N^3\sqrt{z_1^2-4}\sqrt{z_2^{*2}-4}}\bigg )\sum _i(\zeta _i-2)+\mathcal {E}_1, \end{aligned}$$

and for the complex Hermitian case ($\beta =2$), we have

$$\begin{aligned}&\mathrm {Cov}\big (\tilde{\underline{G} \!\,}(z_1), \tilde{\underline{G} \!\,}(z_2^*) \big ) \\&\quad =\mathrm {Cov}\big (\underline{G} \!\,(z_1), \underline{G} \!\,(z_2^*) \big )+ \bigg (\frac{-1+\mathrm {i}{{\,\mathrm{Im}\,}}m(E)^2}{N^4(z_1-z_2^*)^2}+\frac{m(z_1)m(z_2^*)}{N^3\sqrt{z_1^2-4}\sqrt{z_2^{*2}-4}}\bigg )\sum _i(\zeta _i-1)+\mathcal {E}_1. \end{aligned}$$

Theorem 1.2 is generalized as follows. We have

$$\begin{aligned} {\tilde{\Upsilon }}_{E,1}(u,v)=\Upsilon _{E,1}(u,v)+\bigg (\frac{F_4(u,v)}{N^3\kappa _E^2}-\frac{1}{N^2\pi ^2(u-v)^2}\bigg )\sum _i(\zeta _i-2) +{\mathcal {E}} \end{aligned}$$

as well as

$$\begin{aligned} {\tilde{\Upsilon }}_{E,2}(u,v)=\Upsilon _{E,2}(u,v)+\bigg (\frac{F_4(u,v)}{N^3\kappa _E^2}-\frac{1}{2N^2\pi ^2(u-v)^2}\bigg )\sum _i(\zeta _i-1) +{\mathcal {E}}. \end{aligned}$$

where

$$\begin{aligned} F_4(u,v)=g_4\Big (\frac{u-E}{N\varrho _{E}},\frac{v-E}{N\varrho _{E}}\Big )\ \ \ \text{ and } \ \ \ g_4(x_1,x_2)=\frac{x_1x_2}{\sqrt{4-x_1^2}\sqrt{4-x_2^2}}. \end{aligned}$$