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Bulk universality for generalized Wigner matrices with few moments

Abstract

In this paper we consider \(N \times N\) real generalized Wigner matrices whose entries are only assumed to have finite \((2 + \varepsilon )\)th moment for some fixed, but arbitrarily small, \(\varepsilon > 0\). We show that the Stieltjes transforms \(m_N (z)\) of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when \(\eta = \mathfrak {I}(z)\) is almost of order \(N^{-1}\). As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large N limit.

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Acknowledgements

The author heartily thanks Horng-Tzer Yau for suggesting this question and for numerous valuable discussions. The author is also very grateful to Jiaoyang Huang and Benjamin Landon for many fruitful conversations and helpful explanations, and to Ji Oon Lee for several insightful comments.

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Correspondence to Amol Aggarwal.

Additional information

This work was funded by the NSF Graduate Research Fellowship under Grant Number DGE1144152 and partially by the Eric Cooper and Naomi Siegel Graduate Student Fellowship I.

Appendices

Large deviation estimates

In this section we state the large deviation results that were useful to us in Sects. 4, 5, and 6. The following proposition appears as the first part of Lemma 3.8 of [18].

Proposition 24

([18, Lemma 3.8]) Let \(N \in {\mathbb {Z}}_{> 1}\) and \(q = q_N > 1\). Let \(X_1, X_2, \ldots , X_N\); \(Y_1, Y_2, \ldots , Y_N\) be centered random variables such that there exists a \(C > 0\) satisfying

$$\begin{aligned} {\mathbb {E}} \big [ |X_i|^p \big ] \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{C}{q} \right) ^p; \qquad {\mathbb {E}} \big [ |Y_i|^p \big ] \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{C}{q} \right) ^p, \end{aligned}$$
(138)

for all \(2 \le p \le (\log N)^{\log \log N}\). Denote \(s_i = {\mathbb {E}} [|X_i|^2]\) for each \(1 \le i \le N\).

Let \(\{ R_i \}_{1 \le i \le N}\) and \(\{ R_{ij} \}_{1 \le i, j \le N}\) be sequences of real numbers. Then, there exists a constant \(\nu = \nu (C) > 0\), only dependent on the constant C in (138), such that the four estimates

$$\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{j = 1}^N R_j X_j \right| \ge (\log N)^{\xi } \left( q^{-1} \displaystyle \max _{1 \le i \le N} |R_i| + \left( \displaystyle \frac{1}{N} \displaystyle \sum _{j = 1}^N |R_j|^2 \right) ^{1 / 2} \right) \right] \nonumber \\&\quad \le \exp \big ( -\nu (\log N)^{\xi } \big ); \end{aligned}$$
(139)
$$\begin{aligned}&\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{j = 1}^N \big ( |X_j|^2 - s_j \big ) R_j \right| \ge (\log N)^{\xi } q^{-1} \displaystyle \max _{1 \le i \le N} |R_i| \right] \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned} \end{aligned}$$
(140)
$$\begin{aligned}&\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{1 \le i \ne j \le N } X_i R_{ij} X_j \right| \ge (\log N)^{2 \xi } \left( q^{-1} \displaystyle \max _{1 \le i \ne j \le N} |R_{ij}| + \left( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i \ne j \le N} |R_{ij}|^2 \right) ^{1 / 2} \right) \right] \\&\qquad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned} \end{aligned}$$
(141)
$$\begin{aligned}&\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{1 \le i, j \le N } X_i R_{ij} Y_j \right| \ge (\log N)^{2 \xi } \left( 2 q^{-1} \displaystyle \max _{1 \le i, j \le N} |R_{ij}| + \left( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i, j \le N} |R_{ij}|^2 \right) ^{1 / 2} \right) \right] \\&\qquad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned} \end{aligned}$$
(142)

all hold for any \(2 \le \xi \le \log \log N\).

In our setting, the random variables \(X_i\) are not centered but instead “almost centered.” The following corollary adapts the previous proposition to apply in this slightly more setting.

Corollary 25

Let \(N \in {\mathbb {Z}}_{> 1}\) and \(q = q_N \in (1, \sqrt{N})\). Let \(X_1, X_2, \ldots , X_N; Y_1, Y_2, \ldots , Y_N\) be random variables such that there exist \(\delta \in (0, 1)\) and \(C, C' > 1\) satisfying

$$\begin{aligned}&\big | {\mathbb {E}} [ X_i ] \big | \le C' N^{-1 - \delta }; \quad {\mathbb {E}} \big [ |X_i|^p \big ] \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{C}{q} \right) ^p; \quad \big | {\mathbb {E}} [ Y_i ] \big | \nonumber \\&\quad \le C' N^{-1 - \delta }; \quad {\mathbb {E}} \big [ |Y_i|^p \big ] \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{C}{q} \right) ^p, \end{aligned}$$
(143)

for each \(2 \le p \le (\log N)^{\log \log N}\). Denote \(s_i = {\mathbb {E}} [|X_i|^2]\) for each \(1 \le i \le N\).

Let \(\{ R_i \}_{1 \le i \le N}\) and \(\{ R_{ij} \}_{1 \le i, j \le N}\) be sequences of real numbers. There exists a constant \(\nu = \nu ( C, C' ) > 0\) (dependent on only C and \(C'\)) such that

$$\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{j = 1}^N R_j X_j \right| \ge (\log N)^{\xi } \left( (C' N^{- \delta } + q^{-1}) \displaystyle \max _{1 \le i \le N} |R_i| + \left( \displaystyle \frac{1}{N} \displaystyle \sum _{j = 1}^N |R_j|^2 \right) ^{1 / 2} \right) \right] \nonumber \\&\quad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned}$$
(144)
$$\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{j = 1}^N \big ( |X_j|^2 - s_j \big ) R_j \right| \ge (\log N)^{\xi } (5 C'^2 N^{- \delta } + q^{-1}) \displaystyle \max _{1 \le i \le N} |R_i| \right] \nonumber \\&\quad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned}$$
(145)
$$\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{1 \le i \ne j \le N } X_i R_{ij} X_j \right| \ge (\log N)^{2 \xi } \left( (5 C'^2 N^{- \delta } + q^{-1}) \displaystyle \max _{1 \le i \ne j \le N} |R_{ij}| \right. \right. \nonumber \\&\left. \left. \quad + \left( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i \ne j \le N} |R_{ij}|^2 \right) ^{1 / 2} \right) \right] \nonumber \\&\quad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned}$$
(146)
$$\begin{aligned}&{\mathbb {P}} \left[ \left| \displaystyle \sum _{1 \le i, j \le N } X_i R_{ij} Y_j \right| \ge (\log N)^{2 \xi } \left( (5 C'^2 N^{- \delta } + 2 q^{-1}) \displaystyle \max _{1 \le i, j \le N} |R_{ij}| \right. \right. \nonumber \\&\left. \left. \quad + \left( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i, j \le N} |R_{ij}|^2 \right) ^{1 / 2} \right) \right] \nonumber \\&\quad \le \exp \big ( - \nu (\log N)^{\xi } \big ); \end{aligned}$$
(147)

for all \(2 \le \xi \le \log \log N\).

Proof

The proof will follow from centering the \(X_j\) and then applying Proposition 24. To that end, let \(m_j = {\mathbb {E}} [X_j]\), for each \(j \in [1, N]\); by (143), we have that \(|m_j| \le C' N^{-1 - \delta }\).

Denote \(\widetilde{X}_i = X_i - m_j\). Then, the \(\widetilde{X}_i\) are centered and satisfy

$$\begin{aligned}&{\mathbb {E}} \big [ |\widetilde{X}_i|^p \big ] \le 2^p \Big ( {\mathbb {E}} \big [ |X_i|^p \big ] + |m_i|^p \Big ) \nonumber \\&\quad \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{2 C}{q} \right) ^p + \left( \displaystyle \frac{2 C'}{N^{1 + \delta }} \right) ^p \le \displaystyle \frac{q^2}{N} \left( \displaystyle \frac{2 (C + C')}{q} \right) ^p, \end{aligned}$$

for any \(2 \le p \le (\log N)^{\log \log N}\); in the first estimate, we used (143), and in the last estimate we used the fact that \(q, N > 1\).

Thus, Proposition 24 applies to the centered random variables \(\widetilde{X}_i\); the estimates (144), (145), (146), and (147) will follow from (139), (140), (141), and (142) respectively.

As an example, we only establish (146); the proofs of the other estimates are very similar and thus omitted. To that end, we apply (139) and (141) to obtain that

$$\begin{aligned} {\mathbb {P}} \left[ \bigg | \displaystyle \sum _{j = 1}^N R_{ij} \widetilde{X}_j \bigg | \ge (\log N)^{\xi } (q^{-1} + 1) \displaystyle \max _{1 \le i, j \le N} |R_{ij}| \right] \le \exp \big ( - \widetilde{\nu } (\log N)^{\xi } \big ), \end{aligned}$$
(148)

for each \(i \in [1, N]\), and

$$\begin{aligned} \begin{aligned} {\mathbb {P}} \Bigg [ \bigg | \displaystyle \sum _{1 \le i \ne j \le N} \widetilde{X}_i R_{ij} \widetilde{X}_j \bigg |&\ge (\log N)^{\xi } \bigg ( q^{-1} \displaystyle \max _{1 \le j \le N} |R_{ij}| + \Big ( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i \ne j \le N}^N |R_{ij}|^2 \Big )^{1 / 2} \bigg ) \Bigg ] \\&\quad \le \exp \big ( - \widetilde{\nu } (\log N)^{\xi } \big ), \end{aligned} \end{aligned}$$
(149)

where we set \(\widetilde{\nu } = \widetilde{\nu } (C, C') = \widetilde{\nu } \big ( 2 (C + C') \big )\).

Furthermore, observe that

$$\begin{aligned}&\left| \displaystyle \sum _{1 \le i \ne j \le N } X_i R_{ij} X_j - \displaystyle \sum _{1 \le i \ne j \le N} \widetilde{X}_i R_{ij} \widetilde{X}_j \right| \nonumber \\&\quad \le \displaystyle \sum _{j \ne i} m_j \left| \displaystyle \sum _{i = 1}^N \widetilde{X}_i R_{ij} \right| + \displaystyle \sum _{i \ne j} m_i \left| \displaystyle \sum _{j = 1}^N \widetilde{X}_j R_{ij} \right| + \left| \displaystyle \sum _{1 \le i \ne j \le N} m_i m_j R_{ij} \right| \nonumber \\&\quad \le C' N^{-\delta } \displaystyle \max _{1 \le i \le N} \left| \displaystyle \sum _{i = 1}^N \widetilde{X}_i R_{ij} \right| + C' N^{-\delta } \displaystyle \max _{1 \le j \le N} \left| \displaystyle \sum _{i = 1}^N \widetilde{X}_j R_{ij} \right| \nonumber \\&\qquad + C'^2 N^{-2 \delta } \displaystyle \max _{1 \le i, j \le N} |R_{ij}|, \end{aligned}$$
(150)

where we have used the fact that \(|m_i| < C' N^{-1 - \delta }\) for each \(i \in [1, N]\). Thus, applying (148) for all \(i \in [1, N]\), (149), (150), and a union estimate yields

$$\begin{aligned} {\mathbb {P}} \Bigg [ \bigg | \displaystyle \sum _{1 \le i \ne j \le N} \widetilde{X}_i R_{ij} \widetilde{X}_j \bigg |&\ge (\log N)^{2 \xi } \bigg ( \gamma \displaystyle \max _{1 \le j \le N} |R_{ij}| + \Big ( \displaystyle \frac{1}{N^2} \displaystyle \sum _{1 \le i \ne j \le N}^N |R_{ij}|^2 \Big )^{1 / 2} \bigg ) \Bigg ] \nonumber \\&\le (2 N + 1) \exp \big ( - \widetilde{\nu } (\log N)^{\xi } \big ), \end{aligned}$$
(151)

where

$$\begin{aligned} \gamma = q^{-1} + 2 C' N^{-\delta } (q^{-1} + 1) + C'^2 N^{-2 \delta } \le q^{-1} + 5 C'^2 N^{-\delta }. \end{aligned}$$

In the second estimate above, we used the facts that \(q, C', N \ge 1\). Now select \(\nu = \nu (C, C') = \widetilde{\nu } / 2\), so that \((2 N + 1) \exp \big ( - \widetilde{\nu } (\log N)^{\xi } \big ) \le \exp \big ( - \nu (\log N)^{\xi } \big )\). Then, (146) follows from (149). \(\square \)

Continuity of \(G_{ij} (z)\)

In this section we establish continuity estimates on the entries of the resolvent, which allowed us to proceed with the multiscale argument in Sects. 5 and 6.

Lemma 26

Fix \(E \in {\mathbb {R}}\); \(\eta , \eta ' \in {\mathbb {R}}_{> 0}\); \(N \in {\mathbb {Z}}_{> 0}\); and an \(N \times N\) matrix \({\mathbf{H}}\). Denote \(z = E + {\mathbf{i}} \eta \), \(z' = E + {\mathbf{i}} (\eta + \eta ')\), \(G (z) = ({\mathbf{H}} - z)^{-1} = \{ G_{jk} \}\), and \(G' (z) = ( {\mathbf{H}} - z')^{-1} = \{ G_{jk}' \}\).

Fixing \(j, k \in [1, N]\), we have that

$$\begin{aligned} \big | G_{jk}' - G_{jk} \big | \le \displaystyle \frac{\eta '}{2 \eta } \Big ( \big | \mathfrak {I}G_{jj}' \big | + \big | \mathfrak {I}G_{kk} \big | \Big ). \end{aligned}$$
(152)

Proof

Applying the resolvent identity (13) with \(A = H - z' {{\mathrm{Id}}}\) and \(B = H - z {{\mathrm{Id}}}\), and comparing (jk)-entries, we find that

$$\begin{aligned} G_{jk}' - G_{jk} = - \mathbf i \eta ' \displaystyle \sum _{i = 1}^N G_{ji}' G_{ik}. \end{aligned}$$

Thus,

$$\begin{aligned} \big | G_{jk}' - G_{jk} \big | = \eta ' \left| \displaystyle \sum _{i = 1}^N G_{ji}' G_{ik} \right|&\le \eta ' \left( \displaystyle \sum _{i = 1}^N \big | G_{ji}' \big |^2 \right) ^{1 / 2} \left( \displaystyle \sum _{i = 1}^N \big | G_{ik} \big |^2 \right) ^{1 / 2} \\&= \displaystyle \frac{\eta ' \big ( \mathfrak {I}G_{jj}' \mathfrak {I}G_{jj} \big )^{1 / 2}}{\big ( \eta ( \eta + \eta ') \big )^{1 / 2}} \le \displaystyle \frac{\eta ' }{2 \eta } \Big ( \big | \mathfrak {I}G_{jj}' \big | + \big | \mathfrak {I}G_{jj} \big | \Big ), \end{aligned}$$

where the third identity was deduced from Ward’s identity (17); this implies the lemma.

Corollary 27

Adopt the notation of Lemma 26. Then,

$$\begin{aligned} \displaystyle \frac{\min \big \{ |G_{jj}'|, |G_{jj}| \big \} }{\max \big \{ |G_{jj}'|, G_{jj}| \big \}} > 1 - \displaystyle \frac{\eta '}{\eta }. \end{aligned}$$
(153)

Proof

Let \(a = |G_{jj} (e + \mathbf i \eta + \mathbf i \eta ')|\) and \(b = |G_{jj} (e + \mathbf i \eta )|\), and assume that \(a \ge b\); the case \(b > a\) is entirely analogous. Lemma 26 applied with \(j = k\) yields \(a - b < (a + b) \eta ' / 2 \eta \). Therefore,

$$\begin{aligned} \displaystyle \frac{b}{a}&> \displaystyle \frac{2 \eta - \eta '}{2 \eta + \eta '} = 1 - \displaystyle \frac{2 \eta '}{2 \eta + \eta '} > 1 - \displaystyle \frac{\eta '}{\eta }, \end{aligned}$$

from which we deduce the corollary. \(\square \)

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Aggarwal, A. Bulk universality for generalized Wigner matrices with few moments. Probab. Theory Relat. Fields 173, 375–432 (2019). https://doi.org/10.1007/s00440-018-0836-y

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Keywords

  • Random matrix
  • Local semicircle law
  • Universality

Mathematics Subject Classification

  • 15B52
  • 60B20