Mesoscopic Linear Statistics of Wigner Matrices of Mixed Symmetry Class

Abstract

We prove a central limit theorem for the mesoscopic linear statistics of $N\times N$ Wigner matrices H satisfying ${\mathbb {E}}|H_{ij}|^2=1/N$ and ${\mathbb {E}} H_{ij}^2= \sigma /N$, where $\sigma \in [-1,1]$. We show that on all mesoscopic scales $\eta$ ($1/N \ll \eta \ll 1$), the linear statistics of H have a sharp transition at $1-\sigma \sim \eta$. As an application, we identify the mesoscopic linear statistics of Dyson’s Brownian motion $H_t$ started from a real symmetric Wigner matrix $H_0$ at any nonnegative time $t \in [0,\infty ]$. In particular, we obtain the transition from the central limit theorem for GOE to the one for GUE at time $t \sim \eta$.

Appendices

Appendix A: Proof of Lemma 3.9

We proceed by induction. By Lemma 3.8 we see that (3.11) is true for $k=1$.

Suppose (3.11) is true for $k\leqslant n-1$, and we would like to prove it for $k=n$. We split according to the parity of n.

• (i) When n is odd, we write $n=2p+1$. Suppose for some $\lambda >0$ we have

  $$\begin{aligned} \big |\big (G^{(1)}\cdots G^{(2p+1)}\big )_{ij} \big | \prec \lambda \end{aligned}$$

  (A.1)

  uniformly in i, j and $G^{(1)},\dots ,G^{(2p+1)}\in {\mathcal {G}}$. Pick $G^{(1)},\dots ,G^{(2p+1)}\in {\mathcal {G}}$, and by Cauchy–Schwarz inequality we have

  $$\begin{aligned}&\big |\big (G^{(1)}\cdots G^{(2p+1)}\big )_{ij} \big |\nonumber \\&\quad \leqslant \sum _{l}\big |\big (G^{(1)}\cdots G^{(p+1)}\big )_{il} \big |\cdot \big |\big (G^{(p+2)}\cdots G^{(2p+1)}\big )_{lj} \big | \nonumber \\&\quad \leqslant \big [\big (G^{(1)}\cdots G^{(p+1)}G^{(p+1)*}\cdots G^{(1)*}\big )_{ii} \big (G^{(2p+1)*}\cdots G^{(p+2)*}G^{(p+2)}\cdots G^{(2p+1)}\big )_{jj}\big ]^{1/2},\nonumber \\ \end{aligned}$$

  (A.2)

  where we abbreviate $G^{(m)*}:=\big (G^{(m)}\big )^{*}$ . Note that resolvent identity and (A.1) shows

  $$\begin{aligned}&\big (G^{(1)}\cdots G^{(p+1)}G^{(p+1)*}\cdots G^{(1)*}\big )_{ii} \\&\quad = \frac{1}{2\eta } \big |\big (G^{(1)}\cdots G^{(p)}(G^{(p+1)}-G^{(p+1)*})G^{(p)*}\cdots G^{(1)*}\big )_{ii} \big | \prec \frac{\lambda }{\eta }\,, \end{aligned}$$

  and using (3.11) for $k=n-1=2p$ shows

  $$\begin{aligned} \big (G^{(2p+1)*}\cdots G^{(p+2)*}G^{(p+2)}\cdots G^{(2p+1)}\big )_{jj} \prec \frac{1}{\eta ^{2p-1}}\,. \end{aligned}$$

  Thus we have

  $$\begin{aligned} \big |\big (G^{(1)}\cdots G^{(2p+1)}\big )_{ij} \big | \prec \Big (\frac{\lambda }{\eta ^{2p}}\Big )^{1/2} \end{aligned}$$

  (A.3)

  provided $\big |\big (G^{(1)}\cdots G^{(2p+1)}\big )_{ij} \big | \prec \lambda$. The proof then follows from the trivial bound $\big |\big (G^{(1)}\cdots G^{(2p+1)}\big )_{ij} \big | \prec 1/\eta ^{2p+1}$ and iterating (A.3).

• (ii) When n is even, we write $n=2p$. Pick $G^{(1)},\dots ,G^{(2p)}\in {\mathcal {G}}$, and similar as in (A.2) we have

  $$\begin{aligned}&\big |\big (G^{(1)}\cdots G^{(2p)}\big )_{ij} \big | \leqslant \big [\big (G^{(1)}\cdots G^{(p)}G^{(p)*}\cdots G^{(1)*}\big )_{ii} \nonumber \\&\quad \big (G^{(2p)*}\cdots G^{(p+1)*}G^{(p+1)}\cdots G^{(2p)}\big )_{jj}\big ]^{1/2} \,. \end{aligned}$$

  (A.4)

Using resolvent identity and (3.11) for $k=n-1=2p-1$ we have

$$\begin{aligned}&\big (G^{(1)}\cdots G^{(p)}G^{(p)*}\cdots G^{(1)*}\big )_{ii}\\&\quad = \frac{1}{2\eta } \big |\big (G^{(1)}\cdots G^{(p-1)}(G^{(p)}-G^{(p)*})G^{(p-1)*}\cdots G^{(1)*}\big )_{ii} \big | \prec \frac{1}{\eta ^{2p-1}}\,, \end{aligned}$$

combining with a similar estimate of the last factor on RHS of (A.4) we get

$$\begin{aligned} \big |\big (G^{(1)}\cdots G^{(2p)}\big )_{ij} \big | \prec \frac{1}{\eta ^{2p-1}} \end{aligned}$$

as desired.

Appendix B: Proof of Lemma 3.11

The proof of (3.13) is similar to that of Lemma 4.8 in [16]. By the resolvent identity and Lemma 3.2, we arrive at

$$\begin{aligned} {{\mathbb {E}}}\underline{G^2} = \frac{1}{T} {{\mathbb {E}}}{\underline{G}} + \frac{2}{T} {{\mathbb {E}}}\langle {\underline{G}} \rangle \langle \underline{G^2} \rangle + \frac{2\sigma }{TN} {{\mathbb {E}}}\underline{G^2G^{\intercal }} -\frac{K^{(5)}}{T}- \frac{L^{(5)}}{T}\,, \end{aligned}$$

(B.1)

where $T :=-z-2{\mathbb {E}} \underline{G} \!\,$

$$\begin{aligned} K^{(2)}=N^{-2} \sum \limits _{i} {{\mathbb {E}}}\frac{\partial (G^2)_{ii}}{\partial H_{ii}}(\zeta _i-1-\sigma )\,, \end{aligned}$$

and

$$\begin{aligned} L^{(2)}= \frac{1}{N} \sum \limits _{i,j} \left[ \sum \limits _{k=2}^{\ell } \sum _{\begin{array}{c} p,q \geqslant 0,\\ p+q=k \end{array}}\frac{1}{p!\,q!} {\mathcal {C}}_{p+1,q}(H_{ji}) {{\mathbb {E}}}\frac{\partial ^k (G^2)_{ij}}{\partial H_{ji}^p \partial H_{ij}^q} +R_{\ell +1}^{(2,ji)} \right] \,. \end{aligned}$$

(B.2)

Here $R_{l+1}^{(2,ji)}$ is a remainder term defined analogously to $R_{\ell +1}^{(ji)}$ in (5.15). By Lemmas 3.8–3.10, we can argue similarly as in the proof of Lemma 4.8 in [16] and show that every term on the RHS of (B.1) is bounded by $O_{\prec }(N^{\alpha -\chi })$. This proves (3.13).

The proof of (3.14) is similar to that of Lemma 4.3 (ii) in [16]. By the resolvent identity and Lemma 3.2, we have

$$\begin{aligned} {{\mathbb {E}}}{\underline{G}} = \frac{1}{U} \Big ( 1+{{\mathbb {E}}}\langle {\underline{G}}\rangle ^2+\frac{\sigma }{N}{{\mathbb {E}}}\underline{G^2}-K^{(6)}-L^{(6)}\Big ), \end{aligned}$$

(B.3)

where $U:=-z-{{\mathbb {E}}}{\underline{G}}$,

$$\begin{aligned} K^{(6)}=N^{-2} \sum \limits _{i} {{\mathbb {E}}}\frac{\partial G_{ii}}{\partial H_{ii}}(\zeta _i-1-\sigma )\,, \end{aligned}$$

and

$$\begin{aligned} L^{(6)}= \frac{1}{N}\sum \limits _{i,j} \left[ \sum \limits _{k=2}^{\ell } \sum _{\begin{array}{c} p,q \geqslant 0,\\ p+q=k \end{array}}\frac{1}{p!\,q!} {\mathcal {C}}_{p+1,q}(H_{ji}) {{\mathbb {E}}}\frac{\partial ^k G_{ij}}{\partial H_{ji}^p \partial H_{ij}^q} +R_{\ell +1}^{(6,ji)} \right] . \end{aligned}$$

Here $R_{\ell +1}^{(6,ji)}$ is a remainder term defined analogously to $R_{\ell +1}^{(ji)}$ in (5.15). By Lemmas 3.8–3.10, we can argue similarly as in the proof of Lemma 4.3 (ii) in [16] and show that the last four terms on the RHS of (B.3) is bounded by $O(N^{\alpha -1-\chi })$. Thus

$$\begin{aligned} {{\mathbb {E}}}{\underline{G}}(z+{{\mathbb {E}}}{\underline{G}})+1=O_{\prec }(N^{\alpha -1-\chi })\,. \end{aligned}$$

(B.4)

Again by the argument in the proof of Lemma 4.3 (ii) in [16] we arrive at

$$\begin{aligned} |{{\mathbb {E}}}{\underline{G}} -m(z)|=O_{\prec }(N^{\alpha -1-\chi })\,, \end{aligned}$$

which completes the proof.