Random matrices: tail bounds for gaps between eigenvalues

Abstract

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications.

Appendices

Appendix A: Proof of Theorem 4.2

The statement for $A_n(p )$ follows from [9, Theorem 3.1]. For the Wigner model $X_n$ of subgaussian entries, we just need to recall from Lemma 5.3 that with probability at least $1-O(\exp (-\alpha _0n))$ the eigenvectors of $X_n$ belong to $\mathbf {Incomp}(c_0,c_1)$. On this event, by (8), each eigenvector has at least $c_0c_1^2n/2$ coordinates of absolute value at least $c_1/\sqrt{2n}$.

Appendix B: Proof of Theorem 5.9

The treatment here is based on [39]. For short, we will write $T_D$ instead of $T_{D,\kappa ,\gamma ,\alpha }$. The key ingredient is finding a fine net for $T_D$.

Lemma 11.1

Let $n^{-c}\le \alpha \le c'/4$. For every $D\ge 1$, the level set $T_D$ accepts a $O(\frac{\kappa }{\sqrt{\alpha }D})$-net ${\mathcal N}$ of size

$$\begin{aligned} |{\mathcal N}|\le \frac{(CD)^n}{\left( \sqrt{\alpha n}\right) ^{c'n/2}}D^{2/\alpha }, \end{aligned}$$

where C is an absolute constant.

Roughly speaking, the principle is similar to the proof of Lemmas 7.5 and 8.4. We will break $x\in T_D$ into disjoint subvectors of size roughly m, where $m=\lceil \alpha n \rceil$, and find appropriate nets for each. In what follows we will be working with $S^{m-1}$ first.

Definition 11.2

(Level sets) Let $D_0\ge c \sqrt{m}$. Define $S_{D_0}\subset S^{m-1}$ to be the level set

$$\begin{aligned} S_{D_0}:=\{x\in S^{m-1}: D_0 \le \mathbf {LCD}_{\kappa ,\gamma }(x) \le 2D_0 \}. \end{aligned}$$

Notice that $\kappa =o(\sqrt{m})$ because $\kappa =n^{2c}$ and $m\ge n^{1-c}$. We will invoke the following result from [26].

Lemma 11.3

[26, Lemma 4.7] There exists a $(2\kappa /D_0)$-net of $S_{D_0}$ of cardinality at most $(C_0D_0/\sqrt{m})^m$, where $C_0$ is an absolute constant.

As the proof of this result is short and uses the important notion of $\mathbf {LCD}$, we include it here for the reader’s convenience.

Proof of Lemma 11.3

For $x\in S_{D_0}$, denote

$$\begin{aligned} D(x):= \mathbf {LCD}_{\kappa ,\gamma }(x). \end{aligned}$$

By definition, $D_0\le D(x)\le 2D_0$ and there exists $p\in {\mathbf {Z}}^m$ with

$$\begin{aligned} \left\| x -\frac{p}{D(x)}\right\| \le \frac{\kappa }{D(x)} =O\left( \frac{n^{2c}}{n^{1-c}}\right) =o(1). \end{aligned}$$

As $\Vert x\Vert =1$, this implies that $\Vert p\Vert \approx D(x)$, more precisely

$$\begin{aligned} 1- \frac{\kappa }{D(x)} \le \left\| \frac{p}{D(x)}\right\| \le 1+ \frac{\kappa }{D(x)}. \end{aligned}$$

(55)

This implies that

$$\begin{aligned} \Vert p\Vert \le (1+o(1)) D(x) <3D_0. \end{aligned}$$

(56)

It also follows from (55) that

$$\begin{aligned} \left\| x -\frac{p}{\Vert p\Vert }\right\| \le \left\| x -\frac{p}{D(x)}\right\| + \left\| \frac{p}{\Vert p\Vert }(\frac{\Vert p\Vert }{D(x)} -1)\right\| \le 2\frac{\kappa }{D(x)} \le \frac{2\kappa }{D_0}. \end{aligned}$$

(57)

Now set

$$\begin{aligned} {\mathcal N}_0:=\left\{ \frac{p}{\Vert p\Vert }, p\in {\mathbf {Z}}^m \cap B(0,3D_0)\right\} . \end{aligned}$$

By (56) and (57), ${\mathcal N}_0$ is a $\frac{2\kappa }{D_0}$-net for $S_{D_0}$. On the other hand, it is known that the size of ${\mathcal N}_0$ is bounded by $(C_0\frac{D_0}{\sqrt{m}})^m$ for some absolute constant $C_0$.

In fact we can slightly improve the approximations in Lemma 11.3 as follows.

Lemma 11.4

Let $c \sqrt{m} \le D_0 \le D$. Then the set $S_{D_0}$ has a $(2\kappa /D)$-net of cardinality at most $(C_0D/\sqrt{m})^m$ for some absolute constant $C_0$ (probably different from that of Lemma 11.3).

Proof of Lemma 11.4 First, by Lemma 11.3 one can cover $S_{D_0}$ by $(C_0D_0/\sqrt{m})^m$ balls of radius $2\kappa /D_0$. We then cover these balls by smaller balls of radius $2\kappa /D$, the number of such small balls is at most $(O(D/D_0))^m$. Thus there are at most $(O(D/\sqrt{m}))^m$ balls in total.

Taking the union of these nets as $D_0$ ranges over powers of two, we thus obtain the following.

Lemma 11.5

Let $D\ge c \sqrt{m}$. Then the set $\{x\in S^{m-1}: c\sqrt{m} \le \mathbf {LCD}_{\kappa ,\gamma }(x) \le D \}$ has a $(2\kappa /D)$-net of cardinality at most

$$\begin{aligned} (C_0D/\sqrt{m})^m \log _2 D, \end{aligned}$$

for some absolute constant $C_0$ (probably different from that of Lemmas 11.3 and 11.4).

We can also update the net above for x without normalization.

Lemma 11.6

Let $D\ge c \sqrt{m}$. Then the set $\{x\in {\mathbb {R}}^m, \Vert x\Vert \le 1, c\sqrt{m} \le \mathbf {LCD}_{\kappa ,\gamma }(x/\Vert x\Vert ) \le D \}$ has a $(2\kappa /D)$-net of cardinality at most $(C_0D/\sqrt{m})^m D^2$ for some absolute constant $C_0$.

Proof of Lemma 11.6

Starting from the net obtained from Lemma 11.5, we just need to $2\kappa /D$-approximate the fiber of $span(x/\Vert x\Vert )$ in $B_2^m$.

We now justify our main lemma.

Proof of Lemma 11.1

We first write $x=x_{I_0}\cup \mathbf {spread}(x)$, where $\mathbf {spread}(x)= I_1\cup \cdots \cup I_{k_0} \cup J$ such that $|I_k| =\lceil \alpha n \rceil$ and $|J|\le \alpha n$. Notice that we trivially have

$$\begin{aligned} |\mathbf {spread}(x)| \ge |I_1\cup \cdots \cup I_{k_0}| = k_0 \lceil \alpha n \rceil \ge |\mathbf {spread}(x)|- \alpha n \ge c'n/2. \end{aligned}$$

Thus we have

$$\begin{aligned} \frac{c'}{2\alpha } \le k_0 \le \frac{2c'}{\alpha }. \end{aligned}$$

In the next step, we will construct nets for each $x_{I_j}$. For $x_{I_0}$, we construct trivially a (1 / D)-net ${\mathcal N}_0$ of size

$$\begin{aligned} |{\mathcal N}_0| \le (3D)^{|I_0|}. \end{aligned}$$

For each $I_k$, as

$$\begin{aligned} \mathbf {LCD}_{\kappa ,\gamma }(x_{I_k}/\Vert x_{I_k}\Vert )\le \widehat{\mathbf {LCD}}_{\kappa ,\gamma }(x) \le D, \end{aligned}$$

by Lemma 11.6 (where the condition $\mathbf {LCD}_{\kappa ,\gamma }(x_{I_k}/\Vert x_{I_k}\Vert )\gg \sqrt{|I_k|}$ follows from, say Theorem 7.2, because the entries of $x_{I_k}/\Vert x_{I_k}\Vert$ are all of order $\sqrt{\alpha n}$ while $\kappa =o(\sqrt{\alpha n})$), one obtains a $(2\kappa /D)$-net ${\mathcal N}_k$ of size

$$\begin{aligned} |{\mathcal N}_k|\le \left( \frac{C_0D}{\sqrt{|I_k|}}\right) ^{|I_k|} D^2. \end{aligned}$$

Combining the nets together, as $x=(x_{I_0},x_{I_1},\ldots ,x_{I_{k_0}},x_J)$ can be approximated by $y=(y_{I_0},y_{I_1},\ldots ,y_{I_{k_0}},y_J)$ with $\Vert x_{I_j}-y_{I_j}\Vert \le \frac{2\kappa }{D}$, we have

$$\begin{aligned} \Vert x-y\Vert \le \sqrt{k_0+1}\frac{2\kappa }{D} \ll \frac{\kappa }{\sqrt{\alpha } D}. \end{aligned}$$

As such, we have obtain a $\beta$-net ${\mathcal N}$, where $\beta =O(\frac{\kappa }{\sqrt{\alpha } D})$, of size

$$\begin{aligned} |{\mathcal N}| \le 2^n |{\mathcal N}_0| |{\mathcal N}_1| \cdots |{\mathcal N}_{k_0}| \le 2^n (3D)^{|I_0|} \prod _{k=1}^{k_0} \left( \frac{CD}{\sqrt{|I_k|}}\right) ^{|I_k|} D^2. \end{aligned}$$

This can be simplified to

$$\begin{aligned} |{\mathcal N}|\le \frac{(CD)^n}{\sqrt{\alpha n}^{c'n/2}}D^{O(1/\alpha )}. \end{aligned}$$

Before completing the proof of Lemma 5.9, we cite another important consequence of Lemma 5.8 (on the small ball estimate) and Lemma 7.7 (on the tensorization trick).

Lemma 11.7

Let $x\in \mathbf {Incomp}(c_0,c_1)$ and $\alpha \in (0,c')$. Then for any $\beta \ge \frac{1}{c_0}\sqrt{\alpha } (\widehat{\mathbf {LCD}}_{\kappa ,\gamma }(x,\alpha ))^{-1}$ one has

$$\begin{aligned} \rho _{\beta \sqrt{n}}((X_n-\lambda _0)x) \le \left( \frac{O(\beta )}{\sqrt{\alpha }} +e^{-\Theta (\kappa ^2)}\right) ^{n-\alpha n}. \end{aligned}$$

Proof of Lemma 5.9

It suffices to show the result for the level set $\{x\in T_D\backslash T_{D/2}\}$. With $\beta = \frac{\kappa }{\sqrt{\alpha } D}$, the condition on $\beta$ of Lemma 11.7 is clearly guaranteed,

$$\begin{aligned} \frac{\kappa }{\sqrt{\alpha } D} \gg \frac{\sqrt{\alpha }}{D}. \end{aligned}$$

This lemma then implies

$$\begin{aligned}&{\mathbf {P}}\left( \exists x\in T_D\backslash T_{D/2} : \Vert (X_n-\lambda _0) x -u\Vert =o (\beta \sqrt{n})\right) \\&\quad \le \left( \frac{O(\beta )}{\sqrt{\alpha }} +e^{-\Theta (\kappa ^2)}\right) ^{n-\alpha n} \times \frac{(CD)^n}{(\sqrt{\alpha n})^{c'n/2}}D^{O(1/\alpha )}\\&\quad \le \left( \frac{O(\kappa )}{\alpha D} +e^{-\Theta (\kappa ^2)}\right) ^{n-\alpha n} \times \frac{(CD)^n}{(\sqrt{\alpha n})^{c'n/2}}D^{O(1/\alpha )}\\&\quad \le n^{-c'n/8}, \end{aligned}$$

provided that $\kappa =n^{2c}$ with sufficiently small c compared to $c'$, and that $D \le n^{c/\alpha }$.