The least singular value of a random symmetric matrix

Abstract

Let A be an $n \times n$ symmetric matrix with $(A_{i,j})_{i\leqslant j}$ independent and identically distributed according to a subgaussian distribution. We show that

$$ \begin{align*}\mathbb{P}(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon + e^{-cn},\end{align*} $$

where $\sigma _{\min }(A)$ denotes the least singular value of A and the constants $C,c>0 $ depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability that A has a repeated eigenvalue is $e^{-\Omega (n)}$, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777–816].

1 Introduction

Let A be an $n\times n$ random symmetric matrix whose entries on and above the diagonal $(A_{i,j})_{i\leqslant j}$ are independent and identically distributed (i.i.d.) with mean $0$ and variance $1$ . This matrix model, sometimes called the Wigner matrix ensemble, was introduced in the 1950s in the seminal work of Wigner [50], who established the famous “semicircular law” for the eigenvalues of such matrices.

In this paper, we study the extreme behavior of the least singular value of A, which we denote by $\sigma _{\min }(A)$ . Heuristically, we expect that $\sigma _{\min }(A) = \Theta (n^{-1/2})$ , and thus it is natural to consider

(1.1) $$ \begin{align} \mathbb{P}( \sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ), \end{align} $$

for all $\varepsilon \geqslant 0$ (see Section 1.2). In this paper, we prove a bound on this quantity which is optimal up to constants, for all random symmetric matrices with i.i.d. subgaussian entries. This confirms the folklore conjecture, explicitly stated by Vershynin in [46].

Theorem 1.1. Let $\zeta $ be a subgaussian random variable with mean $0$ and variance $1$ , and let A be an $n \times n$ random symmetric matrix whose entries above the diagonal $(A_{i,j})_{i\leqslant j}$ are independent and distributed according to $\zeta $ . Then for every $\varepsilon \geqslant 0$ ,

(1.2) $$ \begin{align} \mathbb{P}_A(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2}) \leqslant C \varepsilon + e^{-cn}, \end{align} $$

where $C,c>0$ depend only on $\zeta $ .

This conjecture is sharp up to the value of the constants $C,c>0$ and resolves the “up-to-constants” analogue of the Spielman–Teng [38] conjecture for random symmetric matrices (see Section 1.2). Also note that the special case $\varepsilon = 0 $ tells us that the singularity probability of any random symmetric A with subgaussian entry distribution is exponentially small, generalizing our previous work [4] on the $\{-1,1\}$ case.

1.1 Repeated eigenvalues

Before we discuss the history of the least singular value problem, we highlight one further contribution of this paper: a proof that a random symmetric matrix has no repeated eigenvalues with probability $1-e^{-\Omega (n)}$ .

In the 1980s, Babai [43] conjectured that the adjacency matrix of the binomial random graph $G(n,1/2)$ has no repeated eigenvalues with probability $1-o(1)$ (see [43]). Tao and Vu [43] proved this conjecture in 2014 and, in subsequent work on the topic with Nguyen [24], went on to conjecture the probability that a random symmetric matrix with i.i.d. subgaussian entries has no repeated eigenvalues is $1-e^{-\Omega (n)}$ . In this paper, we prove this conjecture en route to proving Theorem 1.1, our main theorem.

Theorem 1.2. Let $\zeta $ be a subgaussian random variable with mean $0$ and variance $1$ , and let A be an $n \times n$ random symmetric matrix, where $(A_{i,j})_{i\leqslant j}$ are independent and distributed according to $\zeta $ . Then A has no repeated eigenvalues with probability at least $1-e^{-cn}$ , where $c>0$ is a constant depending only on $\zeta $ .

Theorem 1.2 is easily seen to be sharp whenever $A_{i,j}$ is discrete: consider the event that three rows of A are identical; this event has probability $e^{-\Theta (n)}$ and results in two $0$ eigenvalues. Also note that the constant in Theorem 1.2 can be made arbitrary small; consider the entry distribution $\zeta $ , which takes value $0$ with probability $1-p$ and each of $\{-p^{-1/2},p^{-1/2}\}$ with probability $p/2$ . Here, the probability of $0$ being a repeated root is $\geqslant e^{-(3+o(1))pn}$ .

We in fact prove a more refined version of Theorem 1.2, which gives an upper bound on the probability that two eigenvalues of A fall into an interval of length $\varepsilon $ . This is the main result of Section 7. For this, we let $\lambda _1(A)\geqslant \ldots \geqslant \lambda _n(A)$ denote the eigenvalues of the $n\times n$ real symmetric matrix A.

Theorem 1.3. Let $\zeta $ be a subgaussian random variable with mean $0$ and variance $1$ , and let A be an $n \times n$ random symmetric matrix, where $(A_{i,j})_{i\leqslant j}$ are independent and distributed according to $\zeta $ . Then for each $\ell < cn$ and all $\varepsilon \geqslant 0$ , we have

$$ \begin{align*}\max_{k \leqslant n-\ell} \, \mathbb{P}\big( |\lambda_{k+\ell}(A) - \lambda_{k}(A)| \leqslant \varepsilon n^{-1/2} \big) \leqslant \left(C\varepsilon \right)^{\ell} + 2e^{-cn} \, ,\end{align*} $$

where $C,c>0$ are constants, depending only on $\zeta $ .

In the following subsection, we describe the history of the least singular value problem. In Section 1.3, we discuss a technical theme which is developed in this paper, and then, in Section 2, we go on to give a sketch of Theorem 1.1.

1.2 History of the least singular value problem

The behavior of the least singular value was first studied for random matrices $B_n$ with i.i.d. coefficients, rather than for symmetric random matrices. For this model, the history goes back to von Neumann [48], who suggested that one typically has

$$\begin{align*}\sigma_{\min}(B_n) \approx n^{-1/2},\end{align*}$$

while studying approximate solutions to linear systems. This was then more rigorously conjectured by Smale [36] and proved by Szarek [39] and Edelman [8] in the case that $B_n = G_n$ is a random matrix with i.i.d. standard gaussian entries. Edelman found an exact expression for the density of the least singular value in this case. By analyzing this expression, one can deduce that

(1.3) $$ \begin{align} \mathbb{P}( \sigma_{\min}(G_n) \leqslant \varepsilon n^{-1/2} ) \leqslant \varepsilon, \end{align} $$

for all $\varepsilon \geqslant 0$ (see, e.g. [38]). While this gives a very satisfying understanding of the gaussian case, one encounters serious difficulties when trying to extend this result to other distributions. Indeed, Edelman’s proof relies crucially on an exact description of the joint distribution of eigenvalues that is available in the gaussian setting. In the last 20 or so years, intense study of the least singular value of i.i.d. random matrices has been undertaken with the overall goal of proving an appropriate version of (1.3) for different entry distributions and models of random matrices.

An important and challenging feature of the more general problem arises in the case of discrete distributions, where the matrix $B_n$ can become singular with nonzero probability. This singularity event will affect the quantity (1.1) for very small $\varepsilon $ and thus estimating the probability that $\sigma _{\min }(B_n) = 0$ is a crucial aspect of generalizing (1.3). This is reflected in the famous and influential Spielman–Teng conjecture [37] which proposes the bound

(1.4) $$ \begin{align} \mathbb{P}( \sigma_{\min}(B_n) \leqslant \varepsilon n^{-1/2} ) \leqslant \varepsilon + 2e^{-cn}, \end{align} $$

where $B_n$ is a Bernoulli random matrix. Here, this added exponential term “comes from” the singularity probability of $B_n$ . In this direction, a key breakthrough was made by Rudelson [30], who proved that if $B_n$ has i.i.d. subgaussian entries, then

$$ \begin{align*} \mathbb{P}(\sigma_{\min}(B_n) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon n + n^{-1/2}\,. \end{align*} $$

This result was extended in a series of works [32, 40, 44, 49], culminating in the influential work of Rudelson and Vershynin [31], who showed the “up-to-constants” version of Spielman-Teng:

(1.5) $$ \begin{align} \mathbb{P}( \sigma_{\min}(B_n) \leqslant \varepsilon n^{-1/2} ) \leqslant C\varepsilon + e^{-cn}, \end{align} $$

where $B_n$ is a matrix with i.i.d. entries that follow any subgaussian distribution and $C,c>0$ depend only on $\zeta $ . A key ingredient in the proof of (1.5) is a novel approach to the “inverse Littlewood-Offord problem,” a perspective pioneered by Tao and Vu [44] (see Section 1.3 for more discussion).

Another very different approach was taken by Tao and Vu [41], who showed that the distribution of the least singular value of $B_n$ is identical to the least singular value of the Gaussian matrix $G_n$ , up to scales of size $n^{-c}$ . In particular, they prove that

(1.6) $$ \begin{align} \big| \mathbb{P}( \sigma_{\min}(B_n) \leqslant \varepsilon n^{-1/2} ) - \mathbb{P}( \sigma_{\min}(G_n) \leqslant \varepsilon n^{-1/2}) \big| = O(n^{-c_0}), \end{align} $$

thus resolving the Spielman-Teng conjecture for $\varepsilon \geqslant n^{-c_0}$ , in a rather strong form. While falling just short of the Spielman-Teng conjecture, the work of Tao and Vu [41], Rudelson and Vershynin [31], and subsequent refinements by Tikhomirov [45] and Livshyts et al. [22] (see also [21, 29]) leave us with a very strong understanding of the least singular value for i.i.d. matrix models. However, progress on the analogous problem for random symmetric matrices, or Wigner random matrices, has come somewhat more slowly and more recently: In the symmetric case, even proving that $A_n$ is nonsingular with probability $1-o(1)$ was not resolved until the important 2006 paper of Costello et al. [7].

Progress on the symmetric version of Spielman–Teng continued with Nguyen [25, 26] and, independently, Vershynin [46]. Nguyen proved that for any $B>0$ , there exists an $A>0$ for which ¹

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_n) \leqslant n^{-A}) \leqslant n^{-B}. \end{align*}$$

Vershynin [46] proved that if $A_n$ is a matrix with subgaussian entries then, for all $\varepsilon>0$ , we have

(1.7) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_n) \leqslant \varepsilon n^{-1/2}) \leqslant C_\eta\varepsilon^{1/8 -\eta} + 2e^{-n^c}, \end{align} $$

for all $\eta>0$ , where the constants $C_\eta ,c> 0$ may depend on the underlying subgaussian random variable. He went on to conjecture that $\varepsilon $ should replace $\varepsilon ^{1/8 - \varepsilon }$ as the correct order of magnitude, and that $e^{-cn}$ should replace $e^{-n^{c}}$ .

After Vershynin, a series of works [3, 5, 16, 17, 19] made progress on singularity probability (i.e., the $\varepsilon = 0$ case of Vershynin’s conjecture), and we, in [4], ultimately showed that the singularity probability is exponentially small, when $A_{i,j}$ is uniform in $\{-1,1\}$ :

$$\begin{align*}\mathbb{P}( \det(A_n) = 0 ) \leqslant e^{-cn}, \end{align*}$$

which is sharp up to the value of $c>0$ .

However, for general $\varepsilon $ , the state of the art is due to Jain et al. [19], who improved on Vershynin’s bound (1.7) by showing

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_n) \leqslant \varepsilon n^{-1/2} ) \leqslant C\varepsilon^{1/8} + e^{-\Omega(n^{1/2})}\,, \end{align*}$$

under the subgaussian hypothesis on $A_n$ .

For large $\varepsilon $ , for example, $\varepsilon \geqslant n^{-c}$ , another very different and powerful set of techniques have been developed, which in fact apply more generally to the distribution of other “bulk” eigenvalues and additionally give distributional information on the eigenvalues. The works of Tao and Vu [40, 42], Erdős, Schlein and Yau [10, 11, 13], Erdős et al. [9], and specifically, Bourgade et al. [2] tell us that

(1.8) $$ \begin{align} \mathbb{P}( \sigma_{\min}(A_n) \leqslant \varepsilon n^{-1/2} ) \leqslant \varepsilon + o(1), \end{align} $$

thus obtaining the correct dependence ² on $\varepsilon $ when n is sufficiently large compared to $\varepsilon $ . These results are similar in flavor to (1.6) in that they show the distribution of various eigenvalue statistics is closely approximated by the corresponding statistics in the gaussian case. We note, however, that it appears these techniques are limited to these large $\varepsilon $ and different ideas are required for $\varepsilon < n^{-C}$ , and certainly for $\varepsilon $ as small as $e^{-\Theta (n)}$ .

Our main theorem, Theorem 1.1, proves Vershynin’s conjecture and thus proves the optimal dependence on $\varepsilon $ for all $\varepsilon> e^{-cn}$ , up to constants.

1.3 Approximate negative correlation

Before we sketch the proof of Theorem 1.1, we highlight a technical theme of this paper: the approximate negative correlation of certain “linear events.” While this is only one of several new ingredients in this paper, we isolate these ideas here, as they seem to be particularly amenable to wider application. We refer the reader to Section 2 for a more complete overview of the new ideas in this paper.

We say that two events $A,B$ in a probability space are negatively correlated if

$$\begin{align*}\mathbb{P}(A\cap B) \leqslant \mathbb{P}(A) \mathbb{P}(B). \end{align*}$$

Here, we state and discuss two approximate negative correlation results: one of which is from our paper [4], but is used in an entirely different context, and one of which is new.

We start by describing the latter result, which says that a “small ball” event is approximately negatively correlated with a large deviation event. This complements our result from [4], which says that two “small ball events,” of different types, are negatively correlated. In particular, we prove something in the spirit of the following inequality, though in a slightly more technical form.

(1.9) $$ \begin{align} \mathbb{P}_X\big( |\langle X, v \rangle| \leqslant \varepsilon \text{ and } \langle X, u \rangle>t \big) \lesssim \mathbb{P}_X(|\langle X, v \rangle| \leqslant \varepsilon )\mathbb{P}_X( \langle X, u \rangle >t ), \end{align} $$

where $u, v$ are unit vectors and $t,\varepsilon>0$ and $X = (X_1,\ldots ,X_n)$ with i.i.d. subgaussian random variables with mean $0$ and variance $1$ .

To state and understand our result, it makes sense to first consider, in isolation, the two events present in (1.9). The easier of the two events is $\langle X, u \rangle>t$ , which is a large deviation event for which we may apply the essentially sharp and classical inequality (see Chapter 3.4 in [47])

$$\begin{align*}\mathbb{P}_X( \langle X, u \rangle>t ) \leqslant e^{-ct^2},\end{align*}$$

where $c>0$ is a constant depending only on the distribution of X.

We now turn to understand the more complicated small-ball event $|\langle X , v \rangle | \leqslant \varepsilon $ appearing in (1.9). Here, we have a more subtle interaction between v and the distribution of X, and thus we first consider the simplest possible case: when X has i.i.d. standard gaussian entries. Here, one may calculate

(1.10) $$ \begin{align} \mathbb{P}_X(|\langle X, v \rangle| \leqslant \varepsilon) \leqslant C\varepsilon , \end{align} $$

for all $\varepsilon>0$ , where $C>0$ is an absolute constant. However, as we depart from the case when X is gaussian, a much richer behavior emerges when the vector v admits some “arithmetic structure.” For example, if $v = n^{-1/2}(1,\ldots ,1)$ and the $X_i$ are uniform in $\{-1,1\}$ , then

$$\begin{align*}\mathbb{P}_X( |\langle X, v \rangle| \leqslant \varepsilon ) = \Theta(n^{-1/2}),\end{align*}$$

for any $0< \varepsilon < n^{-1/2}$ . This, of course, stands in contrast to (1.10) for all $\varepsilon \ll n^{-1/2}$ and suggests that we employ an appropriate measure of the arithmetic structure of v.

For this, we use the notion of the “least common denominator” of a vector, introduced by Rudelson and Vershynin [31]. For parameters $\alpha ,\gamma \in (0,1)$ define the least common denominator (LCD) of $v \in \mathbb {R}^n$ to be

(1.11) $$ \begin{align} D_{\alpha,\gamma}(v):=\inf\bigg\{\phi>0:~\|\phi v\|_{\mathbb{T}}\leqslant \min\left\{\gamma\phi\|v\|_2, \sqrt{\alpha n}\right\}\bigg\}, \end{align} $$

where $ \| v \|_{\mathbb {T}} : = \mathrm {dist}(v,\mathbb {Z}^n)$ , for all $v \in \mathbb {R}^n$ . What makes this definition useful is the important “inverse Littlewood-Offord theorem” of Rudelson and Vershynin [31], which tells us (roughly speaking) that one has (1.10) whenever $D_{\alpha ,\gamma }(v) = \Omega (\varepsilon ^{-1})$ . This notion of least common denominator is inspired by Tao and Vu’s introduction and development of “inverse Littlewood-Offord theory,” which is a collection of results guided by the meta-hypothesis: “If $\mathbb {P}_X( \langle X,v\rangle = 0 )$ is large then v must have structure.” We refer the reader to the paper of Tao and Vu [44] and the survey of Nguyen and Vu [28] for more background and history on inverse Littlewood-Offord theory and its role in random matrix theory. We may now state our version of (1.9), which uses $D_{\alpha ,\gamma }(v)^{-1}$ as a proxy for $\mathbb {P}(|\langle X, v \rangle | \leqslant \varepsilon )$ .

Theorem 1.4. For $n \in \mathbb {N}$ , $\varepsilon ,t>0$ and $\alpha ,\gamma \in (0,1)$ , let $v \in {\mathbb {S}}^{n-1}$ satisfy $D_{\alpha ,\gamma }(v)> C/\varepsilon $ and let $u \in {\mathbb {S}}^{n-1}$ . Let $\zeta $ be a subgaussian random variable, and let $X \in \mathbb {R}^n$ be a random vector whose coordinates are i.i.d. copies of $\zeta $ . Then

$$\begin{align*}\mathbb{P}_X\left( |\langle X,v \rangle| \leqslant \varepsilon \text{ and } \langle X, u \rangle> t \right) \leqslant C \varepsilon e^{-ct^2} + e^{-c(\alpha n + t^2)}, \end{align*}$$

where $C,c>0$ depend only on $\gamma $ and the distribution of $\zeta $ .

In fact, we need a significantly more complicated version of this result (Lemma 5.2), where the small-ball event $|\langle X,v\rangle | \leqslant \varepsilon $ is replaced with a small-ball event of the form

$$\begin{align*}|f(X_1,\ldots,X_n)| \leqslant \varepsilon, \end{align*}$$

where f is a quadratic polynomial in variables $X_1,\ldots ,X_n$ . The proof of this result is carried out in Section 5 and is an important aspect of this paper. Theorem 1.4 is stated here to illustrate the general flavor of this result, and is not actually used in this paper. We do provide a proof in Appendix 9 for completeness and to suggest further inquiry into inequalities of the form (1.9).

We now turn to discuss our second approximate negative dependence result, which deals with the intersection of two different small ball events. This was originally proved in our paper [4], but is put to a different use here. This result tells us that the events

(1.12) $$ \begin{align} |\langle X, v\rangle| \leqslant \varepsilon \qquad \text{ and } \qquad |\langle X, w_1 \rangle| \ll 1 , \ldots , |\langle X, w_k \rangle| \ll 1 \end{align} $$

are approximately negatively correlated, where $X = (X_1,\ldots ,X_n)$ is a vector with i.i.d. subgaussian entries and $w_1,\ldots ,w_k$ are orthonormal. That is, we prove something in the spirit of

$$\begin{align*}\mathbb{P}_X\bigg(\{ |\langle X, v\rangle| \leqslant \varepsilon \} \cap \bigcap_{i=1}^k \{ |\langle X, w_i \rangle| \ll 1 \}\bigg) \lesssim \mathbb{P}_X\big( |\langle X, v \rangle| \leqslant \varepsilon \big)\mathbb{P}_X\bigg( \bigcap_{i=1}^k \{ |\langle X, w_i \rangle| \ll 1 \}\bigg),\end{align*}$$

though in a more technical form.

To understand our result, again, it makes sense to consider the two events in (1.12) in isolation. Since we have already discussed the subtle event $|\langle X, v \rangle | \leqslant \varepsilon $ , we consider the event on the right of (1.12). Returning to the gaussian case, we note that if X has independent standard gaussian entries, then one may compute directly that

(1.13) $$ \begin{align} \mathbb{P}_X\left(|\langle X, w_1 \rangle| \ll 1 , \ldots , |\langle X, w_k \rangle| \ll 1\right) = \mathbb{P}( |X_1| \ll 1,\ldots |X_k| \ll 1 ) \leqslant e^{-\Omega(k)}\, , \end{align} $$

by rotational invariance of the gaussian. Here, the generalization to other random variables is not as subtle, and the well-known Hanson-Wright [18] inequality tells us that (1.13) holds more generally when X has general i.i.d. subgaussian entries.

Our innovation in this line is our second “approximate negative correlation theorem,” which allows us to control these two events simultaneously. Again, we use $D_{\alpha ,\gamma }(v)^{-1}$ as a proxy for $\mathbb {P}(|\langle X,v \rangle | \leqslant \varepsilon )$ .

Here, for ease of exposition, we state a less general version for $X = (X_1,\ldots ,X_n) \in \{-1,0,1\}$ with i.i.d. c-lazy coordinates, meaning that $\mathbb {P}(X_i = 0) \geqslant 1-c$ . Our theorem is stated in full generality in Section 9 (see Theorem 9.2).

Theorem 1.5. Let $\gamma \in (0,1)$ , $d \in \mathbb {N}$ , $\alpha \in (0,1)$ , $0\leqslant k \leqslant c_1 \alpha d$ , and $\varepsilon \geqslant \exp (-c_1\alpha d)$ . Let $v \in {\mathbb {S}}^{d-1}$ , let $w_1,\ldots ,w_k \in {\mathbb {S}}^{d-1}$ be orthogonal, and let W be the matrix with rows $w_1,\ldots ,w_k$ .

If $X \in \{-1,0,1 \}^d$ is a $1/4$ -lazy random vector and $D_{\alpha ,\gamma }(v)> 16/\varepsilon $ , then

$$ \begin{align*} \mathbb{P}_X\left( |\langle X, v \rangle| \leqslant \varepsilon\, \text{ and }\, \|W X \|_2 \leqslant c_2\sqrt{k} \right) \leqslant C \varepsilon e^{- c_1 k}, \end{align*} $$

where $C,c_1,c_2>0$ are constants, depending only on $\gamma $ .

In this paper, we will put Theorem 1.5 to a very different use than to that in [4], where we used it to prove a version of the following statement.

Let $v \in {\mathbb {S}}^{d-1}$ be a vector on the sphere, and let H be an $n \times d$ random $\{-1,0,1\}$ -matrix conditioned on the event $\|Hv\|_2 \leqslant \varepsilon n^{1/2}$ , for some $\varepsilon> e^{-cn}$ . Here, $d = cn$ and $c>0$ is a sufficiently small constant. Then the probability that the rank of H is $n-k$ is $\leqslant e^{-ckn}$ .

In this paper, we use (the generalization of) Theorem 1.5 to obtain good bounds on quantities of the form

$$\begin{align*}\mathbb{P}_X( \|BX\|_2 \leqslant \varepsilon n^{1/2} ), \end{align*}$$

where B is a fixed matrix with an exceptionally large eigenvalue (possibly as large as $e^{cn}$ ), but is otherwise pseudo-random, meaning (among other things) that the rest of the spectrum does not deviate too much from that of a random matrix. We use Theorem 1.5 to decouple the interaction of X with the largest eigenvector of B, from the interaction of X with the rest of B. We refer the reader to (2.10) in the sketch in Section 2 and to Section 9 for more details.

The proof of Theorem 9.2 follows closely along the lines of the proof of Theorem 1.5 from [4], requiring only technical modifications and adjustments. So as not to distract from the new ideas of this paper, we have sidelined this proof to the Appendix.

Finally, we note that it may be interesting to investigate these approximate negative correlation results in their own right, and investigate to what extent they can be sharpened.

2 Proof sketch

Here, we sketch the proof of Theorem 1.1. We begin by giving the rough “shape” of the proof, while making a few simplifying assumptions, (2.2) and (2.3). We shall then come to discuss the substantial new ideas of this paper in Section 2.2, where we describe the considerable lengths we must go to in order to remove our simplifying assumptions. Indeed, if one were to only tackle these assumptions using standard tools, one cannot hope for a bound much better than $\varepsilon ^{1/3}$ in Theorem 1.1 (see Section 2.2.2).

2.1 The shape of the proof

Recall that $A_{n+1}$ is a $(n+1)\times (n+1)$ random symmetric matrix with subgaussian entries. Let ${X := X_1,\ldots ,X_{n+1}}$ be the columns of $A_{n+1}$ , let

$$\begin{align*}V = \mathrm{Span}\{ X_2,\ldots,X_{n+1}\},\end{align*}$$

and let $A_n$ be the matrix $A_{n+1}$ with the first row and column removed. We now use an important observation from Rudelson and Vershynin [31] that allows for a geometric perspective on the least singular value problem ³

$$\begin{align*}\mathbb{P}( \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} ) \lesssim \mathbb{P}( \mathrm{dist}(X,V) \leqslant \varepsilon ). \end{align*}$$

Here, our first significant challenge presents itself: X and V are not independent, and thus the event $\mathrm {dist}(X,V) \leqslant \varepsilon $ is hard to understand directly. However, one can establish a formula for $\mathrm {dist}(X,V)$ that is a rational function in the vector X with coefficients that depend only on V. This brings us to the useful inequality ⁴ due to Vershynin [46],

(2.1) $$ \begin{align} \mathbb{P}( \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} ) \lesssim \sup_{r \in \mathbb{R}} \mathbb{P}_{A_n,X}\big( |\langle A_n^{-1}X, X \rangle - r| \leqslant \varepsilon \|A_n^{-1}X\|_2 \big) ,\end{align} $$

where we are ignoring the possibility of $A_n$ being singular for now. We thus arrive at our main technical focus of this paper, bounding the quantity on the right-hand side of (2.1).

We now make our two simplifying assumptions that shall allow us to give the overall shape of our proof without any added complexity. We shall then layer-on further complexities as we discuss how to remove these assumptions.

As a first simplifying assumption, let us assume that the collection of X that dominates the probability at (2.1) satisfies

(2.2) $$ \begin{align} \|A_n^{-1}X\|_2 \approx \|A_n^{-1}\|_{\mathrm{HS}}. \end{align} $$

This is not, at first blush, an unreasonable assumption to make as $\mathbb {E}_X\, \|A_n^{-1}X\|_2^2 = \|A_n^{-1}\|_{\mathrm {HS}}^2 $ . Indeed, the Hanson-Wright inequality tells us that $ \|A_n^{-1}X\|_2 $ is concentrated about its mean, for all reasonable $A_n^{-1}$ . However, as we will see, this concentration is not strong enough for us here.

As a second assumption, we assume that the relevant matrices $A_n$ in the right-hand side of (2.1) satisfy

(2.3) $$ \begin{align} \|A_n^{-1}\|_{\mathrm{HS}} \approx cn^{1/2}. \end{align} $$

This turns out to be a very delicate assumption, as we will soon see, but is not entirely unreasonable to make for the moment: for example, we have $\|A_n^{-1}\|_{\mathrm {HS}} = \Theta _{\delta }(n^{1/2})$ with probability $1-\delta $ . This, for example, follows from Vershynin’s theorem [46] along with Corollary 8.4, which is based on the work of [13].

With these assumptions, we return to (2.1) and obverse our task has reduced to proving

(2.4) $$ \begin{align} \min_r \mathbb{P}_{X}\big( |\langle A^{-1}X, X \rangle - r| \leqslant \varepsilon n^{1/2} \big) \lesssim \varepsilon , \end{align} $$

for all $\varepsilon> e^{-cn}$ , where we have written $A^{-1} = A_{n}^{-1}$ and think of $A^{-1}$ as a fixed (pseudo-random) matrix.

We observe, for a general fixed matrix $A^{-1}$ , there is no hope in proving such an inequality: Indeed, if $A^{-1} = n^{-1/2}J$ , where J is the all-ones matrix, then the left-hand side of (2.4) is $\geqslant cn^{-1/2}$ for all $\varepsilon>0$ , falling vastly short of our desired (2.4).

Thus, we need to introduce a collection of fairly strong “quasi-randomness properties” of A that hold with, probably $1-e^{-cn}$ . These will ensure that $A^{-1}$ is sufficiently “non-structured” to make our goal (2.4) possible. The most important and difficult of these quasi-randomness conditions is to show that the eigenvectors v of A satisfy

$$\begin{align*}D_{\alpha,\gamma}(v)> e^{cn}, \end{align*}$$

for some appropriate $\alpha ,\gamma $ , where $D_{\alpha ,\gamma }(v)$ is the least common denominator of v defined at (1.11). Roughly, this means that none of the eigenvectors of A “correlate” with a rescaled copy of the integer lattice $t\mathbb {Z}^n$ , for any $e^{-cn} \leqslant t \leqslant 1$ .

To prove that these quasi-randomness properties hold with probability $1-e^{-cn}$ is a difficult task and depends fundamentally on the ideas in our previous paper [4]. Since we don’t want these ideas to distract from the new ideas in this paper, we have opted to carry out the details in the Appendix. With these quasi-randomness conditions in tow, we can return to (2.4) and apply Esseen’s inequality to bound the left-hand side of (2.4) in terms of the characteristic function ${\varphi }({\theta })$ of the random variable $\langle A^{-1}X, X \rangle $ ,

$$\begin{align*}\min_r \mathbb{P}_{X}\big( |\langle A^{-1}X, X \rangle - r| \leqslant \varepsilon n^{1/2} \big) \lesssim \varepsilon \int_{-1/\varepsilon}^{1/\varepsilon} |{\varphi}({\theta})| \, d\theta. \end{align*}$$

While this maneuver has been quite successful in work on characteristic functions for (linear) sums of independent random variables, the characteristic function of such quadratic functions has proved to be a more elusive object. For example, even the analogue of the Littlewood-Offord theorem is not fully understood in the quadratic case [6, 23]. Here, we appeal to our quasi-random conditions to avoid some of the traditional difficulties: we use an application of Jensen’s inequality to decouple the quadratic form and bound ${\varphi }({\theta })$ pointwise in terms of an average over a related collection of characteristic functions of linear sums of independent random variables

$$\begin{align*}|{\varphi}({\theta})|^2 \leqslant \mathbb{E}_{Y} |{\varphi}( A^{-1}Y; {\theta})| , \end{align*}$$

where Y is a random vector with i.i.d. entries and ${\varphi }(v; {\theta })$ denotes the characteristic function of the sum $\sum _{i} v_iX_i$ , where $X_i$ are i.i.d. distributed according to the original distribution $\zeta $ . We can then use our pseudo-random conditions on A to bound

$$\begin{align*}|{\varphi}(A^{-1}Y; {\theta})| \lesssim \exp\left( -c{\theta}^{2} \right), \end{align*}$$

for all but exponentially few Y, allowing us to show

$$\begin{align*}\int_{-1/\varepsilon}^{1/\varepsilon} |{\varphi}({\theta})| \, d\theta \leqslant \int_{-1/\varepsilon}^{1/\varepsilon} \left[ \mathbb{E}_{Y} |{\varphi}(A^{-1}Y; {\theta})| \right]^{1/2} \leqslant \int_{-1/\varepsilon}^{1/\varepsilon} \left(\exp\left( -c{\theta}^{2} \right) + e^{-cn}\right)\, d{\theta} = O(1) \end{align*}$$

and thus completing the proof, up to our simplifying assumptions.

2.2 Removing the simplifying assumptions

While this is a good story to work with, the challenge starts when we turn to remove our simplifying assumptions (2.2), (2.3). We also note that if one only applies standard methods to remove these assumptions, then one would get stuck at the “base case” outlined below. We start by discussing how to remove the simplifying assumption (2.3), whose resolution governs the overall structure of the paper.

2.2.1 Removing the assumption (2.3)

What is most concerning about making the assumption $\|A_n^{-1}\|_{\mathrm {HS}} \approx n^{-1/2}$ is that it is, in a sense, circular: If we assume the modest-looking hypothesis $\mathbb {E}\, \|A^{-1}\|_{\mathrm {HS}} \lesssim n^{1/2}$ , we would be able to deduce

$$\begin{align*}\mathbb{P}( \sigma_{\min}(A_n) \leqslant \varepsilon n^{-1/2} ) = \mathbb{P}( \sigma_{\max}(A^{-1}_n) \geqslant n^{1/2}/\varepsilon) \leqslant \mathbb{P}( \|A^{-1}_n\|_{\mathrm{HS}} \geqslant n^{1/2}/\varepsilon) \lesssim \varepsilon, \end{align*}$$

by Markov. In other words, showing that $\|A^{-1}\|_{\mathrm {HS}}$ is concentrated about $n^{-1/2}$ (in the above sense) actually implies Theorem 1.1. However, this is not as worrisome as it appears at first. Indeed, if we are trying to prove Theorem 1.1 for $(n+1) \times (n+1)$ matrices using the above outline, we only need to control the Hilbert-Schmidt norm of the inverse of the minor $A_n^{-1}$ . This suggests an inductive or (as we use) an iterative “bootstrapping argument” to successively improve the bound. Thus, in effect, we look to prove

$$\begin{align*}\mathbb{E}\, \|A_n^{-1}\|^{\alpha}_{\mathrm{HS}}{\mathbf{1}}( \sigma_{\min}(A_n) \geqslant e^{-cn} ) \lesssim n^{\alpha/2}, \end{align*}$$

for successively larger $\alpha \in (0,1]$ . Note, we have to cut out the event of A being singular from our expectation, as this event has nonzero probability.

2.2.2 Base case

In the first step of our iteration, we prove a “base case” of

(2.5) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_n) \leqslant \varepsilon n^{-1/2} ) \lesssim \varepsilon^{1/4} + e^{-cn}\,\end{align} $$

without the assumption (2.3) which is equivalent to

$$\begin{align*}\mathbb{E} \, \|A_n^{-1}\|^{1/4}_{\mathrm{HS}}{\mathbf{1}}( \sigma_{\min}(A_n) \geqslant e^{-cn} ) \lesssim n^{1/8}.\end{align*}$$

To prove this “base case,” we upgrade (2.1) to

(2.6) $$ \begin{align} \mathbb{P}\left(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \right) \lesssim \varepsilon + \sup_{r \in \mathbb{R}}\, \mathbb{P}\left(\frac{|\langle A_n^{-1}X, X\rangle - r|}{ \|A_n^{-1} X \|_2} \leqslant C \varepsilon , \|A_{n}^{-1}\|_{\mathrm{HS}} \leqslant \frac{n^{1/2}}{\varepsilon} \right) \,.\end{align} $$

In other words, we can intersect with the event

(2.7) $$ \begin{align} \| A_n^{-1} \|_{\mathrm{HS}} \leqslant n^{1/2}/\varepsilon \end{align} $$

at a loss of only $C\varepsilon $ in probability.

We then push through the proof outlined in Section 2.1 to obtain our initial weak bound of (2.5). For this, we first use the Hanson-Wright inequality to give a weak version of (2.2), and then use (2.7) as a weak version of our assumption (2.3). We note that this base step (2.5) already improves the best known bounds on the least singular value problem for random symmetric matrices.

2.2.3 Bootstrapping

To improve on this bound, we use a “bootstrapping” lemma which, after applying it three times, allows us to improve (2.5) to the near-optimal result

(2.8) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n}) \leqslant \varepsilon n^{-1/2}) \lesssim \varepsilon\sqrt{\log 1/\varepsilon} + e^{-cn}\,. \end{align} $$

Proving this bootstrapping lemma essentially reduces to the problem of getting good estimates on

(2.9) $$ \begin{align} \mathbb{P}_X\left( \|A^{-1}X\|_2 \leqslant s \right) \qquad \text{ for } \qquad s \in (\varepsilon,n^{-1/2}), \end{align} $$

where A is a matrix with $\|A^{-1}\|_{op} = \delta ^{-1}$ and $ \delta \in (\varepsilon , c n^{-1/2})$ but is “otherwise pseudo-random.” Here, we require two additional ingredients.

To start unpacking (2.9), we use that $\|A^{-1}\|_{op} = \delta ^{-1}$ to see that if v is a unit eigenvector corresponding to the largest eigenvalue of $A^{-1}$ , then

$$\begin{align*}\|A^{-1}X\|_2 \leqslant s \qquad \text{ implies that } \qquad |\langle X, v\rangle| < \delta s.\end{align*}$$

While this leads to a decent first bound of $O(\delta s)$ on the probability (2.9) (after using the quasi-randomness properties of A), however, this is not enough for our purposes, and in fact, we have to use the additional information that X must also have small inner product with many other eigenvectors of A (assuming s is sufficiently small). Working along these lines, we show that (2.9) is bounded above by

(2.10) $$ \begin{align} \mathbb{P}_X\bigg( |\langle X, v_1 \rangle| \leqslant s \delta \text{ and } |\langle X, v_i\rangle| \leqslant \sigma_i s \text{ for all } i =2,\dots, n-1 \bigg), \end{align} $$

where $w_i$ is a unit eigenvector of A corresponding to the singular value $\sigma _i = \sigma _i(A)$ . Now, appealing to the quasi-random properties of the eigenvectors of $A^{-1}$ , we may apply our approximate negative correlation theorem (Theorem 1.5) to see that (2.10) is at most

(2.11) $$ \begin{align} O(\delta s) \exp( - c N_{A}(-c/s,c/s)) ,\end{align} $$

where $c>0$ is a constant and $N_{A}(a,b)$ denotes the number of eigenvalues of the matrix A in the interval $(a,b)$ . The first $O(\delta s)$ factor comes from the event $|\langle X, v_1 \rangle | \leqslant s\delta $ , and the second factor comes from approximating

(2.12) $$ \begin{align} \mathbb{P}_X\Big( |\langle X,w_i\rangle| < c \text{ for all } i \text{ s.t. } s\sigma_i < c \Big) = \exp\big(-\Theta(N_{A}(-c/s,c/s))\, \big)\,. \end{align} $$

This bound is now sufficiently strong for our purposes, provided the spectrum of A adheres sufficiently closely to the typical spectrum of $A_n$ . This now leads us to understand the rest of the spectrum of $A_n$ and, in particular, the next smallest singular values $\sigma _{n-1},\sigma _{n-2},\ldots $ .

Now, this might seem like a step in the wrong direction, as we are now led to understand the behavior of many singular values and not just the smallest. However, this “loss” is outweighed by the fact that we need only to understand these eigenvalues on scales of size $\Omega ( n^{-1/2} )$ , which is now well understood due to the important work of Erdős et al. [13].

These results ultimately allow us to derive sufficiently strong results on quantities of the form (2.9), which, in turn, allow us to prove our “bootstrapping lemma.” We then use this lemma to prove the near-optimal result

(2.13) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n}) \leqslant \varepsilon n^{-1/2}) \lesssim \varepsilon\sqrt{\log 1/\varepsilon} + e^{-cn}\,. \end{align} $$

2.2.4 Removing the assumption (2.2) and the last jump to Theorem 1.1

We now turn to discuss how to remove our simplifying assumption (2.2), made above, which will allow us to close the gap between (2.13) and Theorem 1.1.

To achieve this, we need to consider how $\|A^{-1}X\|_2$ varies about $\|A^{-1}\|_{\mathrm {HS}}$ , where we are, again, thinking of $A^{-1} = A_{n}^{-1}$ as a sufficiently quasi-random matrix. Now, the Hanson-Wright inequality tells us that, indeed, $\|A^{-1}X \|_2$ is concentrated about $\|A^{-1} \|_{\mathrm {HS}}$ , on a scale $ \lesssim \|A^{-1}\|_{op}$ . While this is certainly useful for us, it is far from enough to prove Theorem 1.1. For this, we need to rule out any “macroscopic” correlation between the events

(2.14) $$ \begin{align} \{|\langle A^{-1}X,X\rangle -r| < K \varepsilon \|A^{-1}\|_{\mathrm{HS}} \} \text{ and } \{ \|A^{-1}X\|_2> K\|A^{-1}\|_{\mathrm{HS}} \} \end{align} $$

for all $K> 0$ . Our first step toward understanding (2.14) is to replace the quadratic large deviation event $\|A^{-1}X\|_2> K\|A^{-1}\|_{\mathrm {HS}} $ with a collection of linear large deviation events:

$$\begin{align*}\langle X, w_i \rangle> K\log(i+1) ,\end{align*}$$

where $w_n,w_{n-1},\ldots ,w_1$ are the eigenvectors of A corresponding to singular values $\sigma _n \leqslant \sigma _{n-1} \leqslant \ldots \leqslant \sigma _1$ , respectively, and the $\log (i+1)$ factor should be seen as a weight function that assigns more weight to the smaller singular values.

Interestingly, we run into a similar obstacle as before: If the “bulk” of the spectrum of $A^{-1}$ is sufficiently erratic, this replacement step will be too lossy for our purposes. Thus, we are led to prove another result, showing that we may assume that the spectrum of $A^{-1}$ adheres sufficiently to the typical spectrum of $A_n$ . This reduces to proving

$$\begin{align*}\mathbb{E}_{A_n}\, \left[\frac{ \sum_{i=1}^n \sigma_{n-i-1}^{-2} (\log i )^2}{ \sum_{i=1}^n \sigma_{n-i-1}^{-2} } \right] = O(1) ,\end{align*}$$

where the left-hand side is a statistic which measures the degree of distortion of the smallest singular values of $A_n$ . To prove this, we again lean on the work of Erdős et al. [13].

Thus, we have reduced the task of proving the approximate independence of the events at (2.14) to proving the approximate independence of the collection of events

$$\begin{align*}\{|\langle A^{-1}X,X\rangle -r| < K \varepsilon \|A^{-1}\|_{\mathrm{HS}} \} \text{ and } \{ \langle v_i, X \rangle> K\log(i+1) \}. \end{align*}$$

This is something, it turns out, that we can handle on the Fourier side by using a quadratic analogue of our negative correlation inequality, Theorem 1.4. The idea, here, is to prove an Esseen-type bound of the form

(2.15) $$ \begin{align} \mathbb{P}( |\langle A^{-1} X, X \rangle - t| < \delta, \langle X,u \rangle \geqslant s ) \lesssim \delta e^{-s}\int_{-1/\delta}^{1/\delta} \left|\mathbb{E} e^{2\pi i \theta \langle A^{-1} X, X \rangle + \langle X,u \rangle }\right|\,d\theta\,.\end{align} $$

Which introduces this extra “exponential tilt” to the characteristic function. From here, one can carry out the plan sketched in Section 2.1 with this more complicated version of Esseen, then integrate over s to upgrade (2.13) to Theorem 1.1.

2.3 Outline of the rest of the paper

In the next short section, we introduce some key definitions, notation, and preliminaries that we use throughout the paper. In Section 4, we establish a collection of crucial quasi-randomness properties that hold for the random symmetric matrix $A_n$ with probability $1-e^{-\Omega (n)}$ . We shall condition on these events for most of the paper. In Section 5, we detail our Fourier decoupling argument and establish an inequality of the form (2.15). This allows us to prove our new approximate negative correlation result Lemma 5.2. In Section 6, we prepare the ground for our iterative argument by establishing (2.6), thereby switching our focus to the study of the quadratic form $\langle A_n^{-1}X, X\rangle $ . In Section 7, we prove Theorem 1.2 and Theorem 1.3, which tell us that the eigenvalues of A cannot “crowd” small intervals. In Section 8, we establish regularity properties for the bulk of the spectrum of $A^{-1}$ . In Section 9, we deploy the approximate negative correlation result (Theorem 1.5) in order to carry out the portion of the proof sketched between (2.9) and (2.12). In Section 10, we establish our base step (2.5) and bootstrap this to prove the near optimal bound (2.13). In the final section, Section 11, we complete the proof of our main Theorem 1.1.

3 Key definitions and preliminaries

We first need a few notions out of the way, which are related to our paper [4] on the singularity of random symmetric matrices.

3.1 Subgaussian and matrix definitions

Throughout, $\zeta $ will be a mean $0$ , variance $1$ random variable. We define the subgaussian moment of $\zeta $ to be

$$ \begin{align*}\| \zeta \|_{\psi_2} := \sup_{p \geqslant 1} p^{-1/2} (\mathbb{E}\, |\zeta|^p)^{1/p}\, .\end{align*} $$

A mean $0$ , variance $1$ random variable is said to be subgaussian if $ \| \zeta \|_{\psi _2}$ is finite. We define $\Gamma $ to be the set of subgaussian random variables and, for $B>0$ , we define $\Gamma _B \subseteq \Gamma $ to be a subset of $\zeta $ with $\| \zeta \|_{\psi _2} \leqslant B$ .

For $\zeta \in \Gamma $ , define $\mathrm {Sym\,}_{n}(\zeta )$ to be the probability space on $n \times n$ symmetric matrices A for which $(A_{i,j})_{i \geqslant j}$ are independent and distributed according to $\zeta $ . Similarly, we write $X \sim \mathrm {Col\,}_n(\zeta )$ if $X \in \mathbb {R}^n$ is a random vector whose coordinates are i.i.d. copies of $\zeta $ .

We shall think of the spaces $\{\mathrm {Sym\,}_n(\zeta )\}_{n}$ as coupled in the natural way: The matrix $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ can be sampled by first sampling $A_n \sim \mathrm {Sym\,}_n(\zeta )$ , which we think of as the principal minor $(A_{n+1})_{[2,n+1] \times [2,n+1]}$ , and then generating the first row and column of $A_{n+1}$ by generating a random column $X \sim \mathrm {Col\,}_n(\zeta )$ . In fact, it will make sense to work with a random $(n+1)\times (n+1)$ matrix, which we call $A_{n+1}$ throughout. This is justified, as much of the work is done with the principal minor $A_n$ of $A_{n+1}$ , due to the bound (2.1) as well as Lemma 6.1.

3.2 Compressible vectors

We shall require the now-standard notions of compressible vectors, as defined by Rudelson and Vershynin [31].

For parameters $\rho ,\delta \in (0,1)$ , we define the set of compressible vectors $\mathrm {Comp\,}(\delta ,\rho )$ to be the set of vectors in ${\mathbb {S}}^{n-1}$ that are distance at most $\rho $ from a vector supported on at most $\delta n$ coordinates. We then define the set of incompressible vectors to be all other unit vectors, that is $\mathrm {Incomp\,}(\delta ,\rho ) := {\mathbb {S}}^{n-1} \setminus \mathrm {Comp\,}(\delta ,\rho ).$ The following basic fact about incompressible vectors from [31] will be useful throughout:

Fact 3.1. For each $\delta ,\rho \in (0,1)$ , there is a constant $c_{\rho ,\delta } \in (0,1)$ , so that for all $v \in \mathrm {Incomp\,}(\delta ,\rho )$ , we have that $|v_j|n^{1/2} \in [ c_{\rho ,\delta }, c_{\rho ,\delta }^{-1}]$ for at least $c_{\rho ,\delta } n$ values of j.

Fact 3.1 assures us that for each incompressible vector, we can find a large subvector that is “flat.” Using the work of Vershynin [46], we will safely be able to ignore compressible vectors. In particular, [46, Proposition 4.2] implies the following lemma. We refer the reader to Appendix XII for details.

Lemma 3.2. For $B>0$ and $\zeta \in \Gamma _B$ , let $A \sim \mathrm {Sym\,}_n(\zeta )$ . Then there exist constants $\rho ,\delta ,c \in (0,1) $ , depending only on B, so that

$$ \begin{align*}\sup_{u \in \mathbb{R}^n} \mathbb{P}\big(\exists x \in \mathrm{Comp\,}(\delta,\rho) , \exists t \in \mathbb{R} : Ax = tu\big) \leqslant 2e^{-cn}\end{align*} $$

and

$$ \begin{align*}\mathbb{P}\big(\exists u \in \mathrm{Comp\,}(\delta,\rho), \exists t \in \mathbb{R}: Au = tu\big) \leqslant 2e^{-cn}\,.\end{align*} $$

The first statement says, roughly, that $A^{-1} u$ is incompressible for each fixed u; the second states that all unit eigenvectors are incompressible.

Remark 3.3 (Choice of constants, $\rho ,\delta ,c_{\rho ,\delta }$ ).

Throughout, we let $\rho ,\delta $ denote the constants guaranteed by Lemma 3.2 and $c_{\rho ,\delta }$ the corresponding constant from Fact 3.1. These constants shall appear throughout the paper and shall always be considered as fixed.

Lemma 3.2 follows easily from [46, Proposition 4.2] with a simple net argument.

3.3 Notation

We quickly define some notation. For a random variable X, we use the notation $\mathbb {E}_X$ for the expectation with respect to X and we use the notation $\mathbb {P}_X$ analogously. For an event $\mathcal {E}$ , we write ${\mathbf {1}}_{\mathcal {E}}$ or ${\mathbf {1}} \{ \mathcal {E}\}$ for the indicator function of the event $\mathcal {E}$ . We write $\mathbb {E}^{\mathcal {E}}$ to be the expectation defined by $\mathbb {E}^{\mathcal {E}}[\, \cdot \, ] = \mathbb {E}[\, \cdot \, {\mathbf {1}}_{\mathcal {E}}]$ . For a vector $v \in \mathbb {R}^{n}$ and $J \subset [n]$ , we write $v_J$ for the vector whose ith coordinate is $v_i$ if $i \in J$ and $0$ otherwise.

We shall use the notation $X \lesssim Y$ to indicate that there exists a constant $C>0$ for which $X \leqslant CY$ . In a slight departure from convention, we will always allow this constant to depend on the subgaussian constant B, if present. We shall also let our constants implicit in big-O notation to depend on B, if this constant is relevant in the context. We hope that we have been clear as to where the subgaussian constant is relevant, and so this convention is to just reduce added clutter.

4 Quasi-randomness properties

In this technical section, we define a list of “quasi-random” properties of $A_n$ that hold with probability $1-e^{-\Omega (n)}$ . This probability is large enough that we can assume that these properties hold for all the principal minors of $A_{n+1}$ . Showing that several of these quasi-random properties hold with probability $1-e^{-\Omega (n)}$ will prove to be a challenging task, and our proof will depend deeply on ideas from our previous paper [4], on the singularity probability of a random symmetric matrix. So as not to distract from the new ideas in this paper, we do most of this work in the Appendix.

4.1 Defining the properties

It will be convenient to assume throughout that every minor of $A_{n+1}$ is invertible, and so we will perturb the matrix slightly so that we may assume this. If we add to $A_{n+1}$ an independent random symmetric matrix whose upper triangular entries are independent gaussian random variables with mean $0$ and variance $n^{-n}$ , then with probability $1 - e^{-\Omega (n)}$ , the singular values of $A_{n+1}$ move by at most, say, $n^{-n/3}$ . Further, after adding this random gaussian matrix, every minor of the resulting matrix is invertible with probability $1$ . Thus, we will assume without loss of generality throughout that every minor of $A_{n+1}$ is invertible.

In what follows, we let $A=A_n \sim \mathrm {Sym\,}_n(\zeta )$ and let $X \sim \mathrm {Col\,}_{n}(\zeta )$ be a random vector, independent of A. Our first quasi-random property is standard from the concentration of the operator norm of a random symmetric matrix. We define $\mathcal {E}_{1}$ by

(4.1) $$ \begin{align} \mathcal{E}_1 = \{\|A\|_{op} \leqslant 4 \sqrt{n} \}. \end{align} $$

For the next property, we need a definition. Let $X,X' \sim \mathrm {Col\,}_n(\zeta )$ , and define the random vector in $\mathbb {R}^n$ as $\tilde {X} := X_J - X^{\prime }_J$ , where $J \subseteq [n]$ is a $\mu $ -random subset, that is, for each $j \in [n]$ , we have $j \in J$ independently with probability $\mu $ . The reason behind this definition is slightly opaque at present, but will be clear in the context of Lemma 5.2 in Section 5. Until we get there, it is reasonable to think of $\tilde {X}$ as being essentially X; in particular, it is a random vector with i.i.d. subgaussian entries with mean $0$ and variance $\mu $ . We now define $\mathcal {E}_{2}$ to be the event in A defined by

(4.2) $$ \begin{align} \mathcal{E}_2 = \left\{\mathbb{P}_{\widetilde{X}}\left( A^{-1} \widetilde{X} / \|A^{-1} \widetilde{X}\|_2 \in \mathrm{Comp\,}(\delta,\rho) \right) \leqslant e^{-c_2 n} \right\}. \end{align} $$

We remind the reader that $\mathrm {Comp\,}(\delta ,\rho )$ is defined in Section 3.2, and $\delta ,\rho \in (0,1)$ are constants, fixed throughout the paper, and chosen according to Lemma 3.2. In the (rare) case that $\widetilde {X} = 0$ , we interpret $\mathbb {P}_{\widetilde {X}}( A^{-1} \widetilde {X} / \|A^{-1} \widetilde {X}\|_2 \in \mathrm {Comp\,}(\delta ,\rho ) ) = 1$ .

Recalling the least common denominator defined at (1.11), we now define the event $\mathcal {E}_3$ by

(4.3) $$ \begin{align}\mathcal{E}_3 = \{ D_{\alpha,\gamma}(u) \geqslant e^{c_3 n} \text{ for every unit eigenvector }u \text{ of }A\}\,. \end{align} $$

The next condition tells us that the random vector $A^{-1}\widetilde {X}$ is typically unstructured. We will need a slightly stronger notion of structure than just looking at the LCD, in that, we will need all sufficiently large subvectors to be unstructured. For $\mu \in (0,1)$ , define the subvector least common denominator, as

$$ \begin{align*}\hat{D}_{\alpha,\gamma,\mu}(v) :=\min_{\substack{I\subset [n]\\|I|\geqslant (1-2\mu)n}}D_{\alpha,\gamma}\left(v_I/\|v_I\|_2\right)\,.\end{align*} $$

We note that this is closely related to the notion of “regularized least common denominator” introduced by Vershynin in [46].

Now, if we define the random vector $v = v(\widetilde {X}) := A^{-1} \widetilde {X}$ , then we define $\mathcal {E}_4$ to be the event that A satisfies

(4.4) $$ \begin{align} \mathcal{E}_4 = \left\{\mathbb{P}_{\widetilde{X}}\left( \hat{D}_{\alpha,\gamma,\mu}\left(v \right) < e^{c_4 n} \right) \leqslant e^{-c_4n} \right\}\,. \end{align} $$

As is the case for $\mathcal {E}_2$ , under the event that $\widetilde {X} = 0$ , we interpret $\mathbb {P}_{\widetilde {X}}( \hat {D}_{\alpha ,\gamma ,\mu }(v ) < e^{c_4 n} ) = 1$ .

We now define our main quasi-randomness event $\mathcal {E}$ to be the intersection of these events:

(4.5) $$ \begin{align} \mathcal{E}:= \mathcal{E}_1 \cap \mathcal{E}_2 \cap \mathcal{E}_3 \cap \mathcal{E}_4\,. \end{align} $$

The following lemma essentially allows us to assume that $\mathcal {E}$ holds in what follows.

Lemma 4.1. For $B>0$ , $\zeta \in \Gamma _{B}$ , and all sufficiently small $\alpha ,\gamma ,\mu \in (0,1)$ , there exist constants $c_2,c_3,c_4 \in (0,1)$ appearing in (4.2), (4.3), and (4.4) so that

(4.6) $$ \begin{align} \mathbb{P}_A(\mathcal{E}^c) \leqslant 2e^{-\Omega(n)}. \end{align} $$

Remark 4.2 (Choice of constants, $\alpha ,\gamma , \mu $ ).

We take $\alpha ,\gamma \in (0,1)$ to be sufficiently small so that Lemma 4.1 holds. For $\mu $ , we will choose it to be sufficiently small so that (1) Lemma 4.1 holds; (2) we have $\mu \in (0,2^{-15})$ ; and so that (3) $\mu>0$ is small enough to guarantee that every set $I \subseteq [n]$ with $|I| \geqslant (1-2\mu )n$ satisfies

(4.7) $$ \begin{align} \|w\|_2 \leqslant c^{-2}_{\rho,\delta} \|w_I\|_2, \end{align} $$

for every $w \in \mathrm {Incomp\,}(\delta ,\rho )$ . This is possible by Fact 3.1. These constants $\alpha ,\gamma ,\mu $ will appear throughout the paper and will always be thought of as fixed according to this choice.

4.2 Statement of our master quasi-randomness theorem and the deduction of Lemma 4.1

We will deduce Lemma 4.1 from a “master quasi-randomness theorem” together with a handful of now-standard results in the area.

For the purposes of the following sections, we shall informally consider a vector as “structured” if

$$\begin{align*}\hat{D}_{\alpha,\gamma,\mu}(v) \leqslant e^{c_\Sigma n}, \end{align*}$$

where $c_\Sigma \in (0,1)$ is a small constant, to be chosen shortly. Thus, it makes sense to define the set of “structured directions” on the sphere

(4.8) $$ \begin{align} \Sigma = \Sigma_{\alpha,\gamma,\mu} := \{ v \in {\mathbb{S}}^{n-1} : \hat{D}_{\alpha,\gamma,\mu}(v) \leqslant e^{c_{\Sigma} n} \}\,. \end{align} $$

We now introduce our essential quasi-randomness measure of a random matrix. For $\zeta \in \Gamma $ , ${A \sim \mathrm {Sym\,}_n(\zeta )}$ , and a given vector $w \in \mathbb {R}^n$ , define

(4.9) $$ \begin{align} q_n(w) = q_n(w;\alpha,\gamma,\mu) := \mathbb{P}_A\left(\exists v\in \Sigma \text{ and } \exists s,t\in [-4\sqrt{n}, 4\sqrt{n}]:~Av=sv+tw \right) \end{align} $$

and set

(4.10) $$ \begin{align} q_n = q_n(\alpha,\gamma,\mu) := \sup_{w\in {\mathbb{S}}^{n-1}} q_n(w)\,. \end{align} $$

We now state our “master quasi-randomness theorem,” from which we deduce Lemma 4.1.

Theorem 4.3 (Master quasi-randomness theorem).

For $B>0$ and $\zeta \in \Gamma _B$ , there exist constants $\alpha ,\gamma ,\mu ,c_{\Sigma },c \in (0,1)$ depending only on B so that

$$\begin{align*}q_{n}(\alpha, \gamma ,\mu) \leqslant 2e^{-cn}\,. \end{align*}$$

The proof of Theorem 4.3 is quite similar to the main theorem of [4], albeit with a few technical adaptations, and is proved in the Appendix. Note that $q_n(\alpha ,\gamma ,\mu )$ is monotone decreasing as $\alpha ,\gamma $ , and $\mu $ decrease. As such, Theorem 4.3 implies that its conclusion holds for all sufficiently small $\alpha ,\gamma ,\mu $ as well.

We now prove that our pseudo-random event $\mathcal {E} = \mathcal {E}_1 \cap \mathcal {E}_2 \cap \mathcal {E}_3 \cap \mathcal {E}_4$ holds with probability $1-e^{-\Omega (n)}$ .

Proof of Lemma 4.1.

The event $\mathcal {E}_1$ : From [15], we may deduce ⁵ the following concentration bound

(4.11) $$ \begin{align} \mathbb{P}\big(\, \|A\|_{op} \geqslant (3 + t)\sqrt{n} \big) \lesssim e^{-ct^{3/2}n}, \end{align} $$

which holds ⁶ for all $t \geqslant 0$ . Thus, by (4.11), the event $\mathcal {E}_1$ at (4.1) fails with probability $\lesssim e^{-\Omega (n)}$ .

The event $\mathcal {E}_2$ : By Lemma 3.2, there is a $c> 0$ so that for each $u \neq 0$ , we have

$$ \begin{align*}\mathbb{P}_A(A^{-1} u / \|A^{-1}u \|_2 \in \mathrm{Comp\,}(\delta,\rho)) \leqslant e^{-cn}\, .\end{align*} $$

Applying Markov’s inequality shows

$$ \begin{align*}\mathbb{P}_A\left(\mathbb{P}_{\widetilde{X}}\left( A^{-1}\widetilde{X} / \|A^{-1}\widetilde{X}\|_2 \in \mathrm{Comp\,}(\delta,\rho), \widetilde{X} \neq 0 \right)> e^{-cn/2}\right) \leqslant e^{-cn/2}\,,\end{align*} $$

and so the event in (4.2) fails with probability at most $O\left (e^{-\Omega (n)}\right )$ , under the event $\widetilde {X} \neq 0$ . By Theorem 3.1.1, in [47], we have that

(4.12) $$ \begin{align} \mathbb{P}_{\widetilde{X}}(\widetilde{X} = 0) \leqslant e^{-\Omega(\mu n)}\,. \end{align} $$

Choosing $c_2$ small enough shows an exponential bound on $\mathbb {P}(\mathcal {E}_2^c)$ .

The event $\mathcal {E}_3$ : If $D_{\alpha ,\gamma }(u) \leqslant e^{c_3n}$ , for an u an eigenvector $Au = {\lambda } v$ , we have that

$$\begin{align*}\hat{D}_{\alpha,\gamma,\mu}(u) \leqslant D_{\alpha,\gamma}(u) \leqslant e^{c_3 n} , \end{align*}$$

where the first inequality is immediate from the definition. Now, note that if $\mathcal {E}_1$ holds, then ${\lambda } \in [-4\sqrt {n},4\sqrt {n}]$ , and so

$$\begin{align*}\mathbb{P}(\mathcal{E}^{c}_3) \leqslant \mathbb{P}\big( \exists u \in \Sigma, {\lambda} \in [-4\sqrt{n},4\sqrt{n}] : Au = {\lambda} u \big) + \mathbb{P}(\mathcal{E}_1^{c}) \leqslant q_n(0) + e^{-\Omega(n)},\end{align*}$$

where the first inequality holds if we choose $c_3\leqslant c_\Sigma $ . We now apply Theorem 4.3 to see $q_n(0) \leqslant q_n \lesssim e^{-\Omega (n)}$ , yielding the desired result.

The event $\mathcal {E}_4$ : Note first that, by (4.12), we may assume $\widetilde {X} \neq 0$ . For a fixed instance of $\widetilde {X} \not = 0 $ , we have

(4.13) $$ \begin{align} \mathbb{P}_A\left( \hat{D}_{\alpha,\gamma,\mu}\left( A^{-1}\tilde{X}/\|\tilde{X}\|_2 \right) < e^{c_4n} \right) \leqslant \mathbb{P}_A\big( \exists v \in \Sigma : Av = \tilde{X}/\|\tilde{X}\|_2 \big) \leqslant q_n\left(\tilde{X}/\|\tilde{X}\|_2 \right), \end{align} $$

which is at most $e^{-\Omega (n)}$ , by Theorem 4.3. Here, the first inequality holds when $c_4 \leqslant c_{\Sigma }$ .

We now write $v = A^{-1}\tilde {X}/\|\tilde {X}\|_2$ and apply Markov’s inequality

$$\begin{align*}\mathbb{P}(\mathcal{E}_4^c) = \mathbb{P}_{A}\left( \mathbb{P}_{\tilde{X}}\left( \hat{D}_{\alpha,\gamma,\mu}(v) < e^{c_4n} \right) \geqslant e^{-c_4n} \right)\leqslant e^{c_4n} \mathbb{E}_{\tilde{X}} \mathbb{P}_{A}( \hat{D}_{\alpha,\gamma,\mu}(v) < e^{c_4}n) = e^{-\Omega(n)}, \end{align*}$$

where the last line follows when $c_4$ is taken small relative to the implicit constant in the bound on the right-hand side of (4.13).

Since we have shown that each of $\mathcal {E}_1,\mathcal {E}_2,\mathcal {E}_3,\mathcal {E}_4$ holds with probability $1-e^{-\Omega (n)}$ , the intersection fails with exponentially small probability.

5 Decoupling quadratic forms

In this section, we will prove our Esseen-type inequality that will allow us to deal with a small ball event and a large deviation event simultaneously.

Lemma 5.1. For $B>0$ , let $\zeta \in \Gamma _B$ and $X \sim \mathrm {Col\,}_n(\zeta )$ . Let M be an $n \times n $ symmetric matrix, $u\in \mathbb {R}^n$ , $t \in \mathbb {R}$ , and $s, \delta \geqslant 0$ . Then

(5.1) $$ \begin{align} \mathbb{P}( |\langle M X, X \rangle - t| < \delta, \langle X,u \rangle \geqslant s ) \lesssim \delta e^{-s}\int_{-1/\delta}^{1/\delta} \left|\mathbb{E}\, e^{2\pi i \theta \langle M X, X \rangle + \langle X,u \rangle }\right|\,d\theta\,. \end{align} $$

We will then bound the integrand (our so-called “titled” characteristic function) with a decoupling maneuver, somewhat similar to a “van der Corput trick” in classical Fourier analysis. This amounts to a clever application of Cauchy-Schwarz, inspired by Kwan and Sauermann’s work on Costello’s conjecture [20] (a similar technique appears in [1] and [25]). We shall then be able to mix in our quasi-random conditions on our matrix A to ultimately obtain Lemma 5.2, which gives us a rather tractable bound on the left-hand side of (5.1). To state this lemma, let us recall that $\mathcal {E}$ (defined at (4.5)) is the set of symmetric matrices satisfying the quasi-randomness conditions in the previous section, Section 4. Also recall that the constant $\mu \in (0,2^{-15})$ is defined in Section 4 so that Lemma 4.1 holds and is treated as a fixed constant throughout this paper.

Lemma 5.2. For $B>0$ , let $\zeta \in \Gamma _B$ , $X \sim \mathrm {Col\,}_n(\zeta )$ and let A be a real symmetric $n\times n$ matrix with $A \in \mathcal {E}$ and set $\mu _1 := \sigma _{\max }(A^{-1})$ . Also let $s \geqslant 0, \delta> e^{-c n}$ and $u \in {\mathbb {S}}^{n-1}$ . Then

$$\begin{align*}\mathbb{P}_X\left(\left|\langle A^{-1} X,X\rangle - t \right| \leqslant \delta \mu_1, \langle X, u \rangle \geqslant s \right) \lesssim \delta e^{-s} \int_{-1/\delta}^{1/\delta} I(\theta) ^{1/2}\,d\theta + e^{-\Omega(n)}\, , \end{align*}$$

where

$$\begin{align*}I(\theta) := \mathbb{E}_{J,X_J,X_J^{\prime }} \, \exp\bigg( \langle (X + X')_J,u \rangle -c\theta^2 \mu_1^{-2} \|A^{-1}(X - X')_J \|_2^2\bigg)\,, \end{align*}$$

$X' \sim \mathrm {Col\,}_n(\zeta )$ is independent of X, and $J \subseteq [n]$ is a $\mu $ -random set. Here, $c> 0$ is a constant depending only on B.

While the definition of $I({\theta })$ (and therefore the conclusion of the lemma) is a bit mysterious at this point, we assure the reader that this is a step in the right direction.

All works bounding the singularity probability for random symmetric matrices contain a related decoupling step [3, 4, 5, 16, 25, 46], starting with Costello et al.’s breakthrough [7], building off of Costello’s earlier work [6] on anticoncentration of bilinear and quadratic forms. A subtle difference in the decoupling approach from [20] used here is that the quadratic form is decoupled after bounding a small ball probability in terms of the integral of a characteristic function rather than on the probability itself; the effect of this approach is that we do not lose a power of $\delta $ , but only lose by a square root “under the integral” on the integrand $I(\theta )$ .

5.1 Proofs

We now dive in and prove our Esseen-type inequality. For this, we shall appeal to the classical Esseen inequality [14]: If Z is a random variable taking values in $\mathbb {R}$ with characteristic function ${\varphi }_Z({\theta }):= \mathbb {E}_Z\, e^{2\pi i \theta Z}$ , then for all $t \in \mathbb {R}$ , we have

$$\begin{align*}\mathbb{P}_X( |Z - t| \leqslant \delta ) \lesssim \delta \int_{-1/\delta}^{1/\delta}\, |{\varphi}_Z( {\theta} )| \, d{\theta}. \end{align*}$$

We shall also use the following basic fact about subgaussian random vectors (see, for example, [47, Proposition 2.6.1]): If $\zeta \in \Gamma _B$ and $Y \sim \mathrm {Col\,}_n(\zeta )$ , then for every vector $u \in \mathbb {R}^n$ , we have

(5.2) $$ \begin{align} \mathbb{E}_Y e^{\langle Y, u \rangle } \leqslant \exp(2B^2\|u\|_2^2)\,. \end{align} $$

Proof of Lemma 5.1.

Since ${\mathbf {1}}\{ x \geqslant s \} \leqslant e^{x - s}$ , we may bound

(5.3) $$ \begin{align} \mathbb{P}_X( |\langle M X, X \rangle - t| < \delta, \langle X,u \rangle \geqslant s ) \leqslant e^{-s}\mathbb{E}\left[{\mathbf{1}}\{|\langle M X, X \rangle - t| < \delta\} e^{\langle X,u\rangle } \right]\,. \end{align} $$

Define the random variable $Y \in \mathbb {R}^n$ by

(5.4) $$ \begin{align} \mathbb{P}(Y \in U) = (\mathbb{E}\, e^{\langle X,u\rangle })^{-1} \mathbb{E} [{\mathbf{1}}_U e^{\langle X,u \rangle}], \end{align} $$

for all open $U \subseteq \mathbb {R}^n$ . Note that the expectation $\mathbb {E}_X e^{\langle X, u \rangle }$ is finite by (5.2). We now use this definition to rewrite the expectation on the right-hand side of (5.3),

$$ \begin{align*} \mathbb{E}_X\left[{\mathbf{1}}\{|\langle M X, X \rangle - t| < \delta\} e^{\langle X,u\rangle } \right] = \left( \mathbb{E}\, e^{\langle X ,u \rangle } \right) \mathbb{P}_Y( |\langle MY,Y\rangle - t| \leqslant \delta )\,.\end{align*} $$

Thus, we may apply Esseen’s lemma to the random variable Y to obtain

$$ \begin{align*}\mathbb{P}_Y( |\langle MY,Y\rangle - t| \leqslant \delta ) \lesssim \delta \int_{-1/\delta}^{1/\delta} |\mathbb{E}_Y\, e^{2\pi i\theta \langle MY,Y\rangle}| \, d\theta\,.\end{align*} $$

By the definition of Y, we have

$$ \begin{align*}\mathbb{E}_Y\, e^{2\pi i\theta \langle MY,Y\rangle} = \left(\mathbb{E}_X\, e^{\langle X,u\rangle}\right) ^{-1} \mathbb{E}\, e^{2\pi i\theta\langle MX,X\rangle + \langle X,u\rangle},\end{align*} $$

completing the lemma.

To control the integral on the right-hand side of Lemma 5.1, we will appeal to the following decoupling lemma, which is adapted from Lemma 3.3 from [20].

Lemma 5.3 (Decoupling with an exponential tilt).

Let $\zeta \in \Gamma $ , let $X,X' \sim \mathrm {Col\,}_n(\zeta )$ be independent, and let $J\cup I = [n]$ be a partition of $[n]$ . Let M be an $n \times n$ symmetric matrix and let $u\in \mathbb {R}^n$ . Then

$$ \begin{align*} \left|\mathbb{E}_X\, e^{2\pi i \theta \langle MX,X\rangle + \langle X,u \rangle } \right|^2 \leqslant \mathbb{E}_{X_J,X_J^{\prime}}\, e^{\langle (X + X')_J,u\rangle} \cdot \left|\mathbb{E}_{X_I} e^{4\pi i\theta \langle M(X - X')_J, X_I \rangle + 2\langle X_I,u\rangle } \right|. \end{align*} $$

Proof. After partitioning the coordinates of X according to J and writing $\mathbb {E}_X = \mathbb {E}_{X_I}\mathbb {E}_{X_J}$ , we apply Jensen’s inequality to obtain

$$ \begin{align*} E := \left|\mathbb{E}_X\, e^{2\pi i \theta \langle MX,X\rangle + \langle X,u \rangle } \right|^2 = \left|\mathbb{E}_{X_I} \mathbb{E}_{X_J}\, e^{2\pi i \theta \langle MX,X\rangle + \langle X,u \rangle } \right|^2 \leqslant \mathbb{E}_{X_I} \left|\mathbb{E}_{X_J}e^{2\pi i \theta \langle MX,X\rangle + \langle X,u \rangle } \right|^2. \end{align*} $$

We now expand the square $\left |\mathbb {E}_{X_J}e^{2\pi i \theta \langle MX,X\rangle + \langle X,u \rangle } \right |{}^2$ as

$$ \begin{align*} &\mathbb{E}_{X_J,X_J^{\prime}} e^{2\pi i \theta \langle M(X_I + X_J),(X_I + X_J)\rangle + \langle (X_I + X_J),u\rangle - 2\pi i \theta \langle M(X_I + X_J^{\prime}),(X_I + X_J^{\prime})\rangle + \langle (X_I + X_J^{\prime}),u\rangle } \\ &= \mathbb{E}_{X_J,X_J^{\prime}} e^{4\pi i \theta \langle M(X_J - X_J^{\prime}),X_I\rangle + \langle X_J + X_J^{\prime}, u \rangle + 2\langle X_I,u\rangle + 2\pi i \langle M X_J, X_J \rangle - 2\pi i \langle M X_J^{\prime},X_J^{\prime}\rangle }, \end{align*} $$

where we used the fact that M is symmetric. Thus, swapping expectations yields

$$ \begin{align*} E &\leqslant \mathbb{E}_{X_J,X_J^{\prime}} \mathbb{E}_{X_I} e^{4\pi i \theta \langle M(X_J - X_J^{\prime}),X_I\rangle + \langle X_J + X_J^{\prime}, u \rangle + 2\langle X_I,u\rangle + 2\pi i \langle M X_J, X_J \rangle - 2\pi i \langle M X_J^{\prime},X_J^{\prime}\rangle } \\ &\leqslant \mathbb{E}_{X_J,X_J^{\prime}} \left|\mathbb{E}_{X_I} e^{4\pi i \theta \langle M(X_J - X_J^{\prime}),X_I\rangle + \langle X_J + X_J^{\prime}, u \rangle + 2\langle X_I,u\rangle + 2\pi i \langle M X_J, X_J \rangle - 2\pi i \langle M X_J^{\prime},X_J^{\prime}\rangle } \right| \\ & = \mathbb{E}_{X_J,X_J^{\prime}}\, e^{\langle X_J + X_J^{\prime},u \rangle } \left|\mathbb{E}_{X_I} e^{4\pi i\theta \langle M(X - X')_J, X_I \rangle + 2\langle X_I,u\rangle } \right|,\, \end{align*} $$

as desired. Here, we could swap expectations, since all expectations are finite, due to the subgaussian assumption on $\zeta $ .

We need a basic bound that will be useful for bounding our tilted characteristic function. This bound appears in the proof of Theorem 6.3 in Vershynin’s paper [46].

Fact 5.4. For $B>0$ , let $\zeta \in \Gamma _B$ , let $\zeta '$ be an independent copy of $\zeta $ , and set $\xi = \zeta - \zeta '$ . Then for all $a \in \mathbb {R}^n$ , we have

$$ \begin{align*}\prod_{j}\mathbb{E}_{\xi}\, |\cos(2\pi \xi a_j)| \leqslant \exp\left(-c \min_{r \in [1,c^{-1}]} \| r a\|_{\mathbb{T}}^2\right)\,,\end{align*} $$

where $c>0$ depends only on B.

A simple symmetrization trick along with Cauchy-Schwarz will allow us to prove a similar bound for the tilted characteristic function.

Lemma 5.5. For $B>0$ , let $\zeta \in \Gamma _B$ , $X \sim \mathrm {Col\,}_n(\zeta )$ and let $u ,v \in \mathbb {R}^n$ . Then

(5.5) $$ \begin{align} \left|\mathbb{E}_X e^{2\pi i \langle X,v \rangle + \langle X,u \rangle} \right| \leqslant \exp\left(-c\min_{r \in [1,c^{-1}]} \|rv\|_{\mathbb{T}}^2 + c^{-1} \|u \|_2^2\right)\, , \end{align} $$

where $c \in (0,1)$ depends only on B.

Proof. Let $\zeta '$ be an independent copy of $\zeta $ , and note that

$$ \begin{align*}\left|\mathbb{E}_\zeta\, e^{2\pi i \zeta v_j + \zeta u_j} \right|^2 = \mathbb{E}_{\zeta,\zeta'}\, e^{2\pi i (\zeta-\zeta')v_j + (\zeta+\zeta')u_j } = \mathbb{E}_{\zeta,\zeta'}\left[ e^{ (\zeta + \zeta')u_j } \cos(2\pi(\zeta - \zeta')v_j)\right]\,.\end{align*} $$

Let $\widetilde {X} = (\widetilde {X}_i)_{i=1}^n$ , $\widetilde {Y} = (Y_i)_{i=1}^n$ denote vectors with i.i.d. coordinates distributed as $\xi :=\zeta - \zeta '$ and $\zeta + \zeta '$ , respectively. We have

(5.6) $$ \begin{align} \left|\mathbb{E}_X e^{2\pi i \langle X,v \rangle + \langle X,u \rangle} \right|^2 \leqslant \mathbb{E}\, e^{\langle \widetilde{Y},u\rangle} \prod_{j } \cos (2\pi\widetilde{X}_jv_j) \leqslant \left(\mathbb{E}_{\widetilde{Y}} e^{2\langle \widetilde{Y},u\rangle }\right)^{1/2} \left( \prod_{j} \mathbb{E}_{\xi} |\cos(2\pi \xi v_j)| \right)^{1/2}, \end{align} $$

where we have applied the Cauchy-Schwarz inequality along with the bound $|\cos (x)|^2 \leqslant |\cos (x)|$ to obtain the last inequality. By (5.2), the first expectation on the right-hand side of (5.6) is at most $\exp (O(\|u\|_2^2))$ . Applying Fact 5.4 completes the lemma.

5.2 Quasi-random properties for triples $(J,X_J,X^{\prime }_J)$

We now prepare for the proof of Lemma 5.2 by introducing a quasi-randomness notion on triples $(J,X_J,X^{\prime }_J)$ . Here, $J \subseteq [n]$ and $X,X' \in \mathbb {R}^n$ . For this, we fix an $n\times n$ real symmetric matrix $A \in \mathcal {E}$ and define the event $\mathcal {F} = \mathcal {F}(A)$ as the intersection of the events $\mathcal {F}_1,\mathcal {F}_2,\mathcal {F}_3$ , and $\mathcal {F}_4$ , which are defined as follows. Given a triple $(J,X_J,X^{\prime }_J)$ , we write $\tilde {X} := X_J - X_J^{\prime }$ .

Define events $\mathcal {F}_1,\mathcal {F}_2,\mathcal {F}_3(A)$ by

(5.7) $$ \begin{align} \mathcal{F}_1 &:= \left\{ |J| \in [\mu n/2, 2\mu n] \right\}\qquad\qquad\quad\kern-1pt\end{align} $$

(5.8) $$ \begin{align} \mathcal{F}_2 &:= \{ \|\widetilde{X} \|_2 n^{-1/2} \in [c , c^{-1}]\}\qquad\qquad\!\!\end{align} $$

(5.9) $$ \begin{align} \mathcal{F}_3(A) &:= \{ A^{-1 } \widetilde{X} / \|A^{-1}\widetilde{X}\|_2 \in \mathrm{Incomp\,}(\delta,\rho) \}\,. \end{align} $$

Finally, we write $v = v(\tilde {X}) := A^{-1} \widetilde {X}$ and $I := [n] \setminus J$ and then define $\mathcal {F}_4(A)$ by

(5.10) $$ \begin{align} \mathcal{F}_4(A) := \left\{D_{\alpha,\gamma}\left(\frac{v_I}{\|v_I\|} \right)> e^{c n} \right\}\,. \end{align} $$

We now define $\mathcal {F}(A) := \mathcal {F}_1 \cap \mathcal {F}_2 \cap \mathcal {F}_3(A) \cap \mathcal {F}_4(A)$ and prove the following basic lemma that will allow us to essentially assume that (5.7),(5.8),(5.9),(5.10) hold in all that follows. We recall that the constants $\delta ,\rho ,\mu ,\alpha ,\gamma $ were chosen in Lemmas 3.2 and 4.1 as a function of the subgaussian moment B. Thus, the only new parameter in $\mathcal {F}$ is the constant c in lines (5.8) and (5.10).

Lemma 5.6. For $B>0$ , let $\zeta \in \Gamma _B$ , let $X,X' \sim \mathrm {Col\,}_n(\zeta )$ be independent, and let $J \subseteq [n]$ be a $\mu $ -random subset. Let A be an $n \times n$ real symmetric matrix with $A \in \mathcal {E}$ . We may choose the constant $c \in (0,1)$ appearing in (5.8) and (5.10) as a function of B and $\mu $ so that

$$\begin{align*}\mathbb{P}_{J,X_J,X^{\prime}_J}(\mathcal{F}^c) \lesssim e^{-cn}\,.\end{align*}$$

Proof. For $\mathcal {F}_1$ , we use Hoeffding’s inequality to see $\mathbb {P}(\mathcal {F}_1^c) \lesssim e^{-\Omega (n)}$ . To bound $\mathbb {P}(\mathcal {F}_2^c)$ , we note that the entries of $\widetilde {X}$ are independent, subgaussian, and have variance $2\mu $ , and so $\widetilde {X}/(\sqrt {2\mu })$ has i.i.d. entries with mean zero, variance $1$ and subgaussian moment bounded by $B/\sqrt {2\mu }$ . Thus, from Theorem 3.1.1 in [47], we have

$$\begin{align*}\mathbb{P}\big( \, |\|\widetilde{X}\|_2 - \sqrt{2n\mu}|> t \big) < \exp( -c\mu t^2/B^4 ). \end{align*}$$

For $\mathcal {F}_3(A), \mathcal {F}_4(A)$ , recall that $A \in \mathcal {E}$ means that (4.2) and (4.4) hold, thus exponential bounds on $\mathbb {P}(\mathcal {F}_3^c)$ and $\mathbb {P}(\mathcal {F}_4^c)$ follow from Markov’s inequality.

5.3 Proof of Lemma 5.2

We now prove Lemma 5.2 by applying the previous three lemmas in sequence.

Proof of Lemma 5.2.

Let $\delta \geqslant e^{-c_1n}$ , where we will choose $c_1>0$ to be sufficiently small later in the proof. Apply Lemma 5.1 to write

(5.11) $$ \begin{align} \mathbb{P}_X\left(\left|\langle A^{-1} X,X\rangle - t \right| \leqslant \delta \mu_1, \langle X, u \rangle \geqslant s \right) \lesssim \delta e^{-s} \int_{-1/\delta}^{1/\delta}\left| \mathbb{E}_X \, e^{2\pi i \theta \frac{\langle A^{-1} X,X \rangle}{\mu_1} + \langle X,u\rangle } \right| \,d\theta\, , \end{align} $$

where we recall that $\mu _1 = \sigma _{\max }(A^{-1})$ . We now look to apply our decoupling lemma, Lemma 5.3. Let J be a $\mu $ -random subset of $[n]$ , define $I:=[n] \setminus J$ , and let $X'$ be an independent copy of X. By Lemma 5.3, we have

(5.12) $$ \begin{align} &\left|\mathbb{E}_X\, e^{2\pi i \theta \frac{\langle A^{-1} X,X \rangle}{\mu_1} + \langle X,u\rangle } \right|^2 \leqslant \mathbb{E}_{J}\mathbb{E}_{X_J, X^{\prime}_J}\, e^{\langle (X + X')_J,u \rangle } \cdot \left| \mathbb{E}_{X_I}\, e^{4\pi i \theta\left\langle\frac{A^{-1}\widetilde{X}}{\mu_1} ,X_I\right\rangle+2\langle X_I,u\rangle}\right| \,, \end{align} $$

where we recall that $\widetilde {X}=(X-X')_J$ .

We first consider the contribution to the expectation on the right-hand side of (5.12) from triples $(J,X_J,X_J^{\prime }) \not \in \mathcal {F}$ . For this, let Y be a random vector, such that $Y_j=X_j+X^{\prime }_j$ , if $j\in J$ , and $Y_j=2X_j$ , if $j\in I$ . Applying the triangle inequality, we have

$$ \begin{align*} \mathbb{E}_{J, X_J, X_J^{\prime}}^{\mathcal{F}^c}\, e^{\langle (X + X')_J,u \rangle } \cdot \left|\mathbb{E}_{X_I}\, e^{4\pi i \theta\langle \frac{A^{-1} \widetilde{X} }{\mu_1},X_I\rangle+2\langle X_I,u\rangle} \right| &\leqslant \mathbb{E}_{J, X_J, X_J^{\prime}}^{\mathcal{F}^c}\, e^{\langle (X + X')_J,u \rangle }\mathbb{E}_{X_I}\, e^{2\langle X_I,u\rangle} \\ &= \mathbb{E}_{J,X,X'}^{\mathcal{F}^c}e^{\langle Y,u \rangle } .\end{align*} $$

By Cauchy-Schwarz, (5.2), and Lemma 5.6, we have

(5.13) $$ \begin{align} \mathbb{E}_{J,X,X'}^{\mathcal{F}^c}\,e^{\langle Y,u \rangle } \leqslant \mathbb{E}_{J, X, X'}\left[e^{\langle Y, 2u \rangle}\right]^{1/2} \mathbb{P}_{J, X_J, X_J^{\prime}}(\mathcal{F}^c)^{1/2}\lesssim e^{-\Omega(n)}\,. \end{align} $$

We now consider the contribution to the expectation on the right-hand side of (5.12) from triples $(J,X_J,X_J^{\prime }) \in \mathcal {F}$ . For this, let $w=w(X):=\frac {A^{-1}\widetilde {X}}{\mu _1}$ and assume $(J,X_J,X_J^{\prime }) \in \mathcal {F}$ . By Lemma 5.5, we have

(5.14) $$ \begin{align} \big|\mathbb{E}_{X_I} e^{ 4\pi i {\theta} \langle X_I, w\rangle +\langle X_I,2u \rangle }\big| \lesssim \exp\left(-c \min_{r \in [1,c^{-1}]} \|2r\theta w_I\|_{\mathbb{T}}^2\right). \end{align} $$

Note that $\|w_I\|_2\leqslant \|\widetilde {X}\|_2\leqslant c^{-1}\sqrt {n}$ , by the definition of $\mu _1 = \sigma _{\max }(A^{-1})$ and line (5.8) in the definition of $\mathcal {F}(A)$ .

Now, from property (5.10) in that definition and by the hypothesis $\delta> e^{-c_1 n}$ , we may choose $c_1> 0$ small enough so that

$$\begin{align*}D_{\alpha,\gamma}(w_I/\|w_I\|_2)\geqslant 2c^{-2}n^{1/2} /\delta \geqslant 2c^{-1}\|w_I\|_2/\delta .\end{align*}$$

By the definition of the least common denominator, for $|\theta | \leqslant 1/\delta $ , we have

(5.15) $$ \begin{align} \min_{r \in [1,c^{-1}]} \| 2r\theta w_I \|_{\mathbb{T}} = \min_{r \in [1,c^{-1}]} \left\|2r\theta\|w_I\|_2\cdot \frac{w_I}{\|w_I\|_2}\right\|_{\mathbb{T}} \geqslant \min\left\lbrace \gamma\theta\|w_I\|_2, \sqrt{\alpha |I|}\right\rbrace. \end{align} $$

So, for $|\theta |\leqslant 1/\delta $ , we use (5.15) in (5.14) to bound the right-hand side of (5.12) as

(5.16) $$ \begin{align} \mathbb{E}_{J,X_J,X_J^{\prime}}^{\mathcal{F}} e^{\langle (X + X')_J,u \rangle } \cdot \left|\mathbb{E}_{X_I}\, e^{4\pi i \theta\langle w,X_I\rangle+2\langle X_I,u\rangle} \right| \lesssim \mathbb{E}_{J,X_J,X_J^{\prime}}^{\mathcal{F}}\, e^{\langle (X + X')_J,u \rangle }e^{- c \min\{\gamma^2\theta^2\|w_I\|_2^2,\alpha |I|\}}. \end{align} $$

We now use that $(J,X_J,X_J^{\prime }) \in \mathcal {F}$ to see that $w \in \mathrm {Incomp\,}(\delta ,\rho )$ and that we chose $\mu $ to be sufficiently small, compared to $\rho ,\delta $ , to guarantee that

$$\begin{align*}\|w\|_2 \leqslant C\|w_I\|_2, \end{align*}$$

for some $C> 0$ (see (4.7)). Thus, the right-hand side of (5.16) is

$$\begin{align*}\lesssim \mathbb{E}_{J,X_J,X_J^{\prime}}^{\mathcal{F}} e^{\langle (X + X')_J,u \rangle } e^{-c' \theta^2 \|w \|_2^2} + e^{-\Omega( n)} \,. \end{align*}$$

Combining this with (5.16), (5.12) obtains the desired bound in the case $(J,X_J,X^{\prime }_J) \in \mathcal {F}$ . Combining this with (5.13) completes the proof of Lemma 5.2.

6 Preparation for the “base step” of the iteration

As we mentioned at (2.1), Vershynin [46] gave a natural way of bounding the least singular value of a random symmetric matrix:

$$\begin{align*}\mathbb{P}( \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} ) \lesssim \sup_{r \in \mathbb{R}} \mathbb{P}_{A_n,X}\big( |\langle A_n^{-1}X, X \rangle - r| \leqslant \varepsilon \|A_n^{-1}X\|_2 \big)\, , \end{align*}$$

where we recall that $A_n$ is obtained from $A_{n+1}$ by deleting its first row and column. The main goal of this section is to prove the following lemma which tells us that we may intersect with the event $\sigma _{\min }(A_{n}) \geqslant \varepsilon n^{-1/2}$ in the probability on the right-hand side, at a loss of $C\varepsilon $ . This will be crucial for the base step in our iteration, since the bound we obtain on $\mathbb {P}( \sigma _{\min }(A_{n+1}) \leqslant \varepsilon n^{-1/2} )$ deteriorates as $\sigma _{\min }(A_n)$ decreases.

Lemma 6.1. For $B> 0$ , $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta ) $ and let $X \sim \mathrm {Col\,}_n(\zeta )$ . Then

$$ \begin{align*}\mathbb{P}\left(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \right) \lesssim \varepsilon + \sup_{r \in \mathbb{R}}\, \mathbb{P}\left(\frac{|\langle A_n^{-1}X, X\rangle - r|}{ \|A_n^{-1} X \|_2} \leqslant C \varepsilon , \sigma_{\min}(A_{n}) \geqslant \varepsilon n^{-1/2} \right) + e^{-\Omega(n)} \,,\end{align*} $$

for all $\varepsilon>0$ . Here, $C> 0$ depends only on B.

We deduce Lemma 6.1 from a geometric form of the lemma, which we state here. Let $X_j$ denote the jth column of $A_{n+1}$ , and let

$$\begin{align*}H_j = \mathrm{Span}\{ X_1,\ldots,X_{j-1},X_{j+1},\ldots,X_{n+1}\} \text{ and }d_j(A_{n+1}) := \mathrm{dist}(X_j,H_j).\end{align*}$$

We shall prove the following “geometric” version of Lemma 6.1.

Lemma 6.2. For $B>0$ , $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ . Then for all $\varepsilon>0$ ,

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} ) \lesssim \varepsilon + \mathbb{P}\left(d_1(A_{n+1}) \leqslant C\varepsilon \text{ and } \sigma_{\min}(A_n) \geqslant \varepsilon n^{-1/2} \right) + e^{-\Omega(n)}\,, \end{align*}$$

where $C> 0$ depends only on B.

The deduction of Lemma 6.1 from Lemma 6.2 is straightforward given the ideas from [46]; so we turn to discuss the proof of Lemma 6.2.

For this, we want to intersect the event $\sigma _{\min }(A_{n+1}) \leqslant \varepsilon n^{-1/2}$ with the event $\sigma _{\min }(A_n) \geqslant \varepsilon n^{-1/2}$ , where we understand $A_n$ to be the principal minor $A_{n+1}^{(n+1)}$ of $A_{n+1}$ . To do this, we first consider the related “pathological” event

$$\begin{align*}\mathcal{P} := \left\lbrace \sigma_{\min}(A_{n+1}^{(i)})\leqslant \varepsilon n^{-1/2} \text{ for at least } cn \text{ values of } i \in [n+1] \right\rbrace \end{align*}$$

and then split our probability of interest into the sum

(6.1) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal{P} ) + \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal{P}^c) , \end{align} $$

and work with each term separately. Here, $c = c_{\rho ,\delta }/2$ , where $c_{\rho ,\delta }$ is the constant defined in Section 4.

We deal with the second term on the right-hand side by showing

(6.2) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal{P}^c) \leqslant \mathbb{P}( d_1(A_{n+1}) \lesssim \varepsilon \text{ and } \sigma_{\min}(A_n) \geqslant \varepsilon n^{-1/2} ) + e^{-\Omega(n)}\, ,\end{align} $$

by a straightforward argument in a manner similar to Rudelson and Vershynin in [31]. We then deal with the first term on the right-hand side of (6.1) by showing that

(6.3) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal{P} ) \lesssim \varepsilon + e^{-\Omega(n)}.\end{align} $$

Putting these two inequalities together then implies Lemma 6.2.

6.1 Proof of the inequality at (6.2)

Here, we prove (6.2) in the following form.

Lemma 6.3. For $B>0$ , $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ . Then, for all $\varepsilon>0$ , we have

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal{P}^c) \lesssim \mathbb{P}\big( d_1(A_{n+1}) \lesssim \varepsilon \text{ and } \sigma_{\min}(A_n) \geqslant \varepsilon n^{-1/2} \big) + e^{-\Omega(n)}.\end{align*}$$

For this, we use a basic but important fact which is at the heart of the geometric approach of Rudelson and Vershynin (see, e.g. [31, Lemma 3.5]).

Fact 6.4. Let M be an $n \times n$ matrix and v be a unit vector satisfying $\| M v \|_2 = \sigma _{\min }(M)$ . Then

$$\begin{align*}\sigma_{\min}(M) \geqslant |v_j| \cdot d_j(M) \quad \text{ for each } j \in [n]\,. \end{align*}$$

We are now ready to prove the inequality mentioned at (6.2).

Proof of Lemma 6.3.

We rule out another pathological event: Let v denote a unit eigenvector corresponding to the least singular value of $A_{n+1}$ , and let $\mathcal {C}$ denote the event that v is $(\rho ,\delta )$ -compressible. ⁷ By Lemma 3.2, $\mathbb {P}(\mathcal {C})\leqslant e^{-\Omega (n)}$ . Thus

(6.4) $$ \begin{align} \mathbb{P}( \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{P}^c) \leqslant \mathbb{P}( \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{C}^c \cap \mathcal{P}^c) + e^{-\Omega(n)}. \end{align} $$

We now look to bound this event in terms of the distance of the column $X_j$ to the subspace $H_j$ , in the style of [31]. For this, we define

$$\begin{align*}S := \{j : d_j(A_{n+1}) \leqslant \varepsilon/c_{\rho, \delta} \text{ and } \sigma_{\min}(A_{n+1}^{(j)}) \geqslant \varepsilon n^{-1/2} \}. \end{align*}$$

We now claim

(6.5) $$ \begin{align} \{ \sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \} \cap \mathcal{C}^c \cap \mathcal{P}^c \Longrightarrow |S| \geqslant c_{\rho,\delta} n/2. \end{align} $$

To see this, fix a matrix A satisfying the left-hand side of (6.5) and let v be an eigenvector corresponding to the least singular value. Now, since v is not compressible, there are $\geqslant c_{\rho ,\delta } n$ values of $j \in [n+1]$ for which $|v_j| \geqslant c_{\rho ,\delta }n^{-1/2}$ . Thus, Fact 6.4 immediately tells us there are $\geqslant c_{\rho ,\delta } n$ values of $j \in [n+1]$ for which $d_{j}(A) \leqslant \varepsilon /c_{\rho ,\delta }$ . Finally, by definition of $\mathcal {P}^c$ , at most $c_{\rho ,\delta }n/2$ of these values of j satisfy $\sigma _{n+1}(A^{(j)}) \leqslant \varepsilon n^{-1/2}$ , and so (6.5) is proved.

We now use (6.5) along with Markov’s inequality to bound

(6.6) $$ \begin{align} \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{C}^c \cap \mathcal{P}^c) \leqslant \mathbb{P}( |S| \geqslant c_{\rho, \delta}n/2 ) \leqslant \frac{2}{c_{\rho, \delta}n} \mathbb{E} |S|.\end{align} $$

Now, by definition of S and symmetry of the coordinates, we have

$$ \begin{align*} \mathbb{E} |S| &= \sum_j \mathbb{P}\big( d_j(A_{n+1}) \leqslant \varepsilon/c_{\rho, \delta}, ~\sigma_{\min}(A_{n+1}^{(j)}) \geqslant \varepsilon n^{-1/2} \big) \\ &= n\cdot \mathbb{P}\big(d_1(A_{n+1}) \leqslant \varepsilon/c_{\rho, \delta},~\sigma_{\min}(A_{n+1}^{(1)}) \geqslant \varepsilon n^{-1/2} \big)\,. \end{align*} $$

Putting this together with (6.6) and (6.5) finishes the proof.

6.2 Proof of the inequality at (6.3)

We now prove the inequality discussed at (6.3) in the following form.

Lemma 6.5. For $B>0$ , $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ . Then, for all $\varepsilon>0$ , we have

(6.7) $$ \begin{align} \mathbb{P}\left(\sigma_{\min}( A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{P} \right) \lesssim \varepsilon + e^{-\Omega(n)}\,.\end{align} $$

For the proof of this lemma, we will need a few results from the random matrix literature. The first such result is a more sophisticated version of Lemma 3.2, which tells us that the mass of the eigenvectors of A does not “localize” on a set of coordinates of size $o(n)$ . The theorem we need, due to Rudelson and Vershynin (Theorem 1.5 in [35]), tells us that the mass of the eigenvectors of our random matrix does not “localize” on a set of coordinates of size $(1-c)n$ , for any fixed $c>0$ . We state this result in a way to match our application.

Theorem 6.6. For $B>0$ , $\zeta \in \Gamma _B$ , let $A \sim \mathrm {Sym\,}_{n}(\zeta )$ and let v denote the unit eigenvector of A corresponding to the least singular value of A. Then there exists $c_2>0$ , such that for all sufficiently small $c_1>0$ , we have

$$ \begin{align*}\mathbb{P}\big(\, |v_j| \geqslant (c_2c_1)^6 n^{-1/2} \text{ for at least } (1-c_1)n \text{ values of } j \big) \geqslant 1- e^{-c_1 n}\, ,\end{align*} $$

for n sufficiently large.

We also require an elementary, but extremely useful, fact from linear algebra. This fact is a key step in the work of Nguyen et al. on eigenvalue repulsion in random matrices (see [24, Section 4]); we state it here in a form best suited for our application.

Fact 6.7. Let M be an $n\times n$ real symmetric matrix, and let ${\lambda }$ be an eigenvalue of M with corresponding unit eigenvector u. Let $j\in [n]$ , and let ${\lambda }'$ be an eigenvector of the minor $M^{(j)}$ with corresponding unit eigenvector v. Then

$$\begin{align*}|\langle v, X^{(j)} \rangle| \leqslant |\lambda - \lambda'|/ |u_j |,\end{align*}$$

where $X^{(j)}$ is the jth column of M with the jth entry removed.

Proof. Without loss of generality, take $j = n$ and express $u = (w,u_{n})$ , where $w \in \mathbb {R}^{n-1}$ . Then we have $(M^{(n)} - \lambda I )w + X^{(n)} u_{n} = 0$ . Multiplying on the left by $v^T$ yields

$$ \begin{align*}|u_{n} \langle v,X^{(n)} \rangle | = |\lambda -\lambda'| |\langle v,w\rangle | \leqslant |\lambda- \lambda'|\,.\\[-37pt] \end{align*} $$

We shall also need the inverse Littlewood-Offord theorem of Rudelson and Vershynin [31], which we have stated here in simplified form. Recall that $D_{\alpha ,\gamma }(v)$ is the least common denominator of the vector v, as defined at (1.11).

Theorem 6.8. For $n\in \mathbb {N}$ , $B>0$ , $\gamma ,\alpha \in (0,1)$ , and $\varepsilon> 0$ , let $v\in {\mathbb {S}}^{n-1}$ satisfy $D_{\alpha ,\gamma }(v)> c\varepsilon ^{-1}$ and let $X \sim \mathrm {Col\,}_n(\zeta )$ , where $\zeta \in \Gamma _B$ . Then

$$\begin{align*}\mathbb{P}(|\langle X, v \rangle|\leqslant \varepsilon)\lesssim \varepsilon + e^{-c\alpha n}\,. \end{align*}$$

Here, $c>0$ depends only on B and $\gamma $ .

We are now in a position to prove Lemma 6.5.

Proof of Lemma 6.5.

Let A be an instance of our random matrix, and let v be the unit eigenvector corresponding to the least singular value of A. Let $w_j = w(A^{(j)})$ denote a unit eigenvector of $A^{(j)}$ corresponding to the least singular value of $A^{(j)}$ .

We introduce two “quasi-randomness” events $\mathcal {Q}$ and $\mathcal {A}$ that will hold with probability $1-e^{\Omega (n)}$ . Indeed, define

$$\begin{align*}\mathcal{Q}_j = \{ D_{\alpha,\gamma}(w_j)\geqslant e^{c_3 n} \} \text{ for all } j \in [n+1] \text{ and set } \mathcal{Q} = \bigcap \mathcal{Q}_j. \end{align*}$$

Here, $\alpha , \gamma , c_3$ are chosen according to Lemma 4.1, which tells us that $\mathbb {P}(\mathcal {Q}^c) \leqslant e^{-\Omega (n)}$ . Define

$$\begin{align*}S_1 = \{j : \sigma_n(A_{n+1}^{(j)}) \leqslant \varepsilon n^{-1/2} \ \}\, \text{ and } \, S_2 = \{ j : |v_j| \geqslant (cc_2/2)^6n^{-1/2} \}.\end{align*}$$

Note that $\mathcal {P}$ holds exactly when $|S_1| \geqslant cn $ . Let $\mathcal {A}$ be the “non-localization” event that $|S_2| \geqslant (1-c/2)n$ . By Theorem 6.6, we have $\mathbb {P}(\mathcal {A}^c)\leqslant e^{-\Omega (n)}$ . Here, $c/2 = c_{\rho ,\delta }/4$ . Now, if we let $X^{(j)}$ denote the jth column of A with the jth entry removed, we define

$$\begin{align*}T = \{ j : |\langle w_j, X^{(j)}\rangle| \leqslant C\varepsilon \}, \end{align*}$$

where $C = 2^7/(c_2c)^6$ . We now claim

(6.8) $$ \begin{align} \{ \sigma_{\min}( A) \leqslant \varepsilon n^{-1/2} \} \cap \mathcal{P} \cap \mathcal{A} \Longrightarrow |T| \geqslant cn/2. \end{align} $$

To see this, first note that if $\mathcal {P} \cap \mathcal {A}$ holds, then $|S_1\cap S_2| \geqslant cn/2$ . Also, for each $j \in S_1 \cap S_2$ , we may apply Fact 6.7 to see that $|\langle w_j, X^{(j)}\rangle | \leqslant C \varepsilon $ since j is such that $\sigma _{\min }( A^{(j)}) \leqslant \varepsilon n^{-1/2}$ and $\sigma _{\min }(A) \leqslant \varepsilon n^{-1/2}$ . This proves (6.8).

To finish the proof of Lemma 6.5, we define the random variable

$$\begin{align*}R = n^{-1} \sum_j {\mathbf{1}}\left( |\langle w_j , X^{(j)} \rangle| \leqslant C\varepsilon \text{ and } \mathcal{Q}_j \right),\end{align*}$$

and observe that $ \mathbb {P}(\sigma _{\min }(A_{n+1}) \leqslant \varepsilon n^{-1/2} \cap \mathcal {P} ) $ is at most

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{A} \cap \mathcal{Q} \cap \mathcal{P} ) + e^{-\Omega(n)} \leqslant \mathbb{P}( R \geqslant c/4 ) + e^{-\Omega(n)}. \end{align*}$$

We now apply Markov and expand the definition of R to bound

$$\begin{align*}\mathbb{P}( R \geqslant c/4) \lesssim n^{-1} \sum_{j} \mathbb{E}_{A^{(j)}_{n+1}}\mathbb{P}_{X^{(j)}}\left( |\langle w_j , X^{(j)} \rangle| \leqslant C\varepsilon \cap \mathcal{Q}_j \right) \lesssim \varepsilon + e^{-\Omega(n)},\end{align*}$$

where the last inequality follows from the fact that $X^{(j)}$ is independent of the events $Q_j$ and $w_j$ , and therefore we may put the property $\mathcal {Q}_j$ to use by applying the inverse Littlewood-Offord theorem of Rudelson and Vershynin, Theorem 6.8.

6.3 Proofs of Lemmas 6.2 and 6.1

All that remains is to put the pieces together and prove Lemmas 6.2 and 6.1.

Proof of Lemma 6.2.

As we saw at (6.1), we simply express $\mathbb {P}( \sigma _{\min }(A_{n+1})\leqslant \varepsilon n^{-1/2} )$ as

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{P} ) + \mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} \text{ and } \mathcal{P}^c), \end{align*}$$

and then apply Lemma 6.5 to the first term and Lemma 6.3 to the second term.

Proof of Lemma 6.1.

If we set $a_{1,1}$ to be the first entry of $A = A_{n+1}$ , then, by [46, Proposition 5.1], we have that

$$ \begin{align*}d_1(A_{n+1}) = \frac{|\langle A^{-1} X, X \rangle - a_{1,1} |}{\sqrt{1 + \|A^{-1}X\|_2^2 } }\,.\end{align*} $$

Additionally, by [46, Proposition 8.2], we have $\|A^{-1}X\|_2> 1/15$ with probability at least $1 - e^{-\Omega (n)}$ . Replacing $a_{1,1}$ with r and taking a supremum completes the proof of Lemma 6.1.

7 Eigenvalue crowding (and the proofs of Theorems 1.2 and 1.3)

The main purpose of this section is to prove the following theorem, which gives an upper bound on the probability that $k \geqslant 2$ eigenvalues of a random matrix fall in an interval of length $\varepsilon $ . The case $\varepsilon = 0$ of this theorem tells us that the probability that a random symmetric matrix has simple spectrum (that is, has no repeated eigenvalue) is $1-e^{-\Omega (n)}$ , which is sharp and confirms a conjecture of Nguyen et al. [24].

Given an $n\times n$ real symmetric matrix M, we let $\lambda _1(M)\geqslant \ldots \geqslant \lambda _n(M)$ denote its eigenvalues.

Theorem 7.1. For $B>0$ , $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta ) $ . Then for each $j \leqslant cn$ and all $\varepsilon \geqslant 0$ , we have

$$ \begin{align*}\max_{k \leqslant n-j} \, \mathbb{P}( |\lambda_{k+j}(A_n) - \lambda_{k}(A_n)| \leqslant \varepsilon n^{-1/2} ) \leqslant \left(C\varepsilon \right)^j + 2e^{-cn} \, ,\end{align*} $$

where $C,c>0$ are constants depending on B.

We suspect that the bound in Lemma 1.3 is actually far from the truth, for $\varepsilon> e^{-cn}$ and $j \geqslant 1 $ . In fact, one expects quadratic dependence on j in the exponent of $\varepsilon $ . This type of dependence was recently confirmed by Nguyen [27] for $\varepsilon> e^{-n^{c}}$ .

For the proof of Lemma 1.3, we remind the reader that if $u \in \mathbb {R}^n \cap \mathrm {Incomp\,}(\rho ,\delta )$ , then at least $c_{\rho ,\delta }n$ coordinates of u have absolute value at least $c_{\rho ,\delta }n^{-1/2}$ .

In what follows, for an $n \times n$ symmetric matrix A, we use the notation $A^{(i_1,\ldots , i_r)}$ to refer to the minor of A for which the rows and columns indexed by $i_1,\ldots ,i_r$ have been deleted. We also use the notation $A_{S \times T}$ to refer to the $|S| \times |T|$ submatrix of A defined by $(A_{i,j})_{i \in S, j\in T}$ .

The following fact contains the key linear algebra required for the proof of Theorem 1.3.

Fact 7.2. For $1\leqslant k +j < n$ , let A be an $n \times n$ symmetric matrix for which

$$\begin{align*}| {\lambda}_{k+j}(A) - {\lambda}_k(A)| \leqslant \varepsilon n^{-1/2}. \end{align*}$$

Let $(i_1,\ldots ,i_j) \in [n]^j$ be such that $i_1,\ldots , i_j$ are distinct. Then there exist unit vectors $w^{(1)},\ldots ,w^{(k)}$ for which

$$\begin{align*}\langle w^{(r)}, X_r \rangle \leqslant (\varepsilon n^{-1/2} ) \cdot (1/|w_{i_r}^{(r-1)}|), \end{align*}$$

where $X_r \in \mathbb {R}^{n-r} $ is the $i_r$ th column of A with coordinates indexed by $i_1,\ldots ,i_r$ removed. That is, $X_r := A_{ [n] \setminus \{i_1,\ldots , i_r \} \times \{i_r\} }$ and $w^{(r)}$ is a unit eigenvector corresponding to ${\lambda }_{k}(A^{(i_1,\ldots , i_r)})$ .

Proof. For $(i_1,\ldots ,i_j)\in [n]^j$ , define the matrices $M_0,M_1,\ldots ,M_j$ by setting $M_r = A^{(i_1,\ldots ,i_r)}$ for $r = 1,\ldots , j$ and then $M_0 := A$ . Now if

$$\begin{align*}|\lambda_{k+j}(A) - \lambda_{k}(A)| \leqslant \varepsilon n^{-1/2},\end{align*}$$

then Cauchy’s interlacing theorem implies

$$\begin{align*}|\lambda_{k}(M_r) - \lambda_k(M_{r-1})| \leqslant \varepsilon n^{-1/2}, \end{align*}$$

for all $r = 1,\ldots ,j$ . So let $w^{(r)}$ denote a unit eigenvector of $M_r$ corresponding to eigenvalue $\lambda _k(M_r)$ . Thus, by Fact 6.7, we see that

$$\begin{align*}|\langle w^{(r)} , X_r \rangle| \leqslant (\varepsilon n^{-1/2} ) \cdot (1/|w^{(r-1)}_{i_r}| ),\end{align*}$$

for $r=1, \ldots , j$ , where $X_r \in \mathbb {R}^{n-r}$ is the $i_r$ th column of $M_{r-1}$ , with the diagonal entry removed. In other words, $X_r \in \mathbb {R}^{n-r} $ is the $i_r$ th column of A with coordinates indexed by $i_1,\ldots ,i_r$ removed. This completes the proof of Fact 7.2.

Proof of Theorem 1.3.

Note, we may assume that $\varepsilon> e^{-cn}$ ; the general case follows by taking c sufficiently small. Now, define $\mathcal {A}$ to be the event that all unit eigenvectors v of all $\binom {n}{j}$ of the minors $A^{(i_1,\ldots ,i_j)}_n$ lie in $\mathrm {Incomp\,}(\rho ,\delta )$ and satisfy $D_{\alpha , \gamma }(v)>e^{c_3 n}$ , where $\alpha , \gamma , c_3$ are chosen according to Lemma 4.1. Note that by Lemmas 4.1 and 3.2, we have

$$\begin{align*}\mathbb{P}(\mathcal{A}^c) \leqslant \binom{n}{j+1} e^{-\Omega(n)} \leqslant n\left(\frac{en}{j} \right)^{j} e^{-\Omega(n)} \lesssim e^{-cn},\end{align*}$$

by taking c small enough, so that $j\log (en/j) < cn$ is smaller than the $\Omega (n)$ term.

With Fact 7.2 in mind, we define the event, $ \mathcal {E}_{i_1,\ldots ,i_j}$ , for each $(i_1,\ldots ,i_j) \in [n]^j$ , $i_r$ distinct, to be the event that

$$ \begin{align*}|\langle w^{(r)}, X_r \rangle | \leqslant \varepsilon/c_{\rho,\delta} \quad \text{for all } r\in [j]\, ,\end{align*} $$

where $X_r \in \mathbb {R}^{n-r} $ is the $i_r$ th column of A with coordinates indexed by $i_1,\ldots ,i_r$ removed and $w^{(r)}$ is a unit eigenvector corresponding to ${\lambda }_{k}(A^{(i_1,\ldots , i_r)})$ .

If $\mathcal {A}$ holds, then each $w^{(r)}$ has at least $c_{\rho ,\delta }n$ coordinates with absolute value at least $c_{\rho ,\delta }n^{-1/2}$ . Thus, if additionally we have

$$\begin{align*}|\lambda_{k+j}(A_n) - \lambda_k(A_n)| \leqslant \varepsilon n^{-1/2} .\end{align*}$$

Fact 7.2 tells us that $\mathcal {E}_{i_1,\ldots ,i_j}$ occurs for at least $(c_{\rho ,\delta }n /2)^j$ tuples $(i_1,\ldots ,i_j)$ .

Define N to be the number of indices $(i_1,\ldots ,i_j)$ for which $\mathcal {E}_{i_1,\ldots ,i_j}$ occurs, and note

(7.1) $$ \begin{align} \mathbb{P}(\, |\lambda_{k+j}(A_n) - \lambda_k(A_n)| \leqslant \varepsilon n^{-1/2} ) &\leqslant \mathbb{P}\big( N \geqslant (c_{\rho,\delta} n/2)^j \text{ and } \mathcal{A} \big) + O(e^{-cn}) \end{align} $$

(7.2) $$ \begin{align} &\qquad\qquad\qquad\quad\qquad\qquad\qquad\leqslant \left(\frac{2}{c_{\rho,\delta}}\right)^j\mathbb{P}(\mathcal{E}_{1,\ldots,j} \cap \mathcal{A} ) + O(e^{-cn}) , \end{align} $$

where, for the second inequality, we applied Markov’s inequality and used the symmetry of the events $\mathcal {E}_{i_1,\ldots ,i_j}$ .

Thus, we need only show that there exists $C>0$ , such that $\mathbb {P}(\mathcal {E}_{1,\ldots ,j} \cap \mathcal {A} ) \leqslant (C\varepsilon )^j$ . To use independence, we replace each of $w^{(r)}$ with the worst case vector, under $\mathcal {A}$

(7.3) $$ \begin{align} \mathbb{P}(\mathcal{E}_{1,\ldots,j} \cap \mathcal{A} ) &\leqslant \max_{w_1,\ldots,w_j : D_{\alpha, \gamma}(w_i)> e^{c_3n}} \mathbb{P}_{X_1,\ldots,X_r}\big(\, |\langle w_r, X_r \rangle | \leqslant \varepsilon/c_{\rho,\delta} \text{ for all } r\in [j]\, \big) \end{align} $$

(7.4) $$ \begin{align} &\kern1pt\quad\ \,\leqslant \max_{w_1,\ldots,w_j : D_{\alpha, \gamma}(w_i)> e^{c_3n}} \prod_{r=1}^j \mathbb{P}_{X_r}\big(\, |\langle w_r, X_r \rangle | \leqslant \varepsilon/c_{\rho,\delta}\, \big), \leqslant (C\varepsilon)^j, \end{align} $$

where the first inequality follows from the independence of the vectors $\{X_r \}_{r\leqslant j}$ and the last inequality follows from the fact that $D_{\alpha ,\gamma }(w_r)> e^{c_3 n}\gtrsim 1/\varepsilon $ (by choosing $c>0$ small enough relative to $c_3$ ), and the Littlewood-Offord theorem of Rudelson and Vershynin, Lemma 6.8. Putting (7.2) and (7.4) together completes the proof of Theorem 1.3.

Of course, the proof of Theorem 1.2 follows immediately.

Proof of Theorem 1.2.

Simply take $\varepsilon =0$ in Theorem 1.3.

8 Properties of the spectrum

In this section, we describe and deduce Lemma 8.1 and Corollary 8.2, which are the tools we will use to control the “bulk” of the eigenvalues of $A^{-1}$ . Here, we understand “bulk” relative to the spectral measure of $A^{-1}$ : our interest in an eigenvalue ${\lambda }$ of $A^{-1}$ is proportional to its contribution to $\|A^{-1}\|_{\mathrm {HS}}$ . Thus, the behavior of smallest singular values of A are of the highest importance for us.

For this, we let $\sigma _n \leqslant \sigma _{n-1} \leqslant \cdots \leqslant \sigma _1$ be the singular values of A and let $\mu _1 \geqslant \ldots \geqslant \mu _n$ be the singular values of $A^{-1}$ . Of course, we have $\mu _k=1/\sigma _{n-k+1}$ for $1\leqslant k \leqslant n$ .

In short, these two lemmas, when taken together, tell us that

(8.1) $$ \begin{align} \sigma_{n - k+1} \approx k n^{-1/2}, \end{align} $$

for all $n \geqslant k \gg 1$ in some appropriate sense.

Lemma 8.1. For $p> 1$ , $B>0$ and $\zeta \in \Gamma _B$ , let $A \sim \mathrm {Sym\,}_n(\zeta )$ . There is a constant $C_p$ depending on $B,p$ so that

$$ \begin{align*}\mathbb{E}\, \left( \frac{\sqrt{n}}{\mu_k k}\right)^p \leqslant C_p\,,\end{align*} $$

for all k.

We shall deduce Lemma 8.1 from the “local semicircular law” of Erdős et al. [13], which gives us good control of the bulk of the spectrum at “scales” of size $\gg n^{-1/2}$ .

We also record a useful corollary of this lemma. For this, we define the function $\| \cdot \|_{\ast } $ for an $n \times n$ symmetric matrix M to be

(8.2) $$ \begin{align} \|M\|_\ast^2 = \sum_{k = 1}^n \sigma_k(M)^2 (\log(1 + k))^2. \end{align} $$

The point of this definition is to give some measure to how the spectrum of $A^{-1}$ is “distorted” from what it “should be,” according to the heuristic at (8.1). Indeed, if we have $\sigma _{n - k+1} = \Theta ( k/\sqrt {n})$ for all k, say, then we have that

$$\begin{align*}\|A^{-1}\|_{\ast} = \Theta( \mu_1 ). \end{align*}$$

Conversely, any deviation from this captures some macroscopic misbehavior on the part of the spectrum. In particular, the “weight function” $k \mapsto (\log (1+k))^2$ is designed to bias the smallest singular values, and thus we are primarily looking at this range for any poor behavior.

Corollary 8.2. For $p> 1$ , $B>0$ , and $\zeta \in \Gamma _B$ , let $A \sim \mathrm {Sym\,}_n(\zeta )$ . Then there exists constants $C_p, c_p>0$ depending on $B,p$ , such that

$$ \begin{align*}\mathbb{E} \left[\left(\frac{\|A^{-1}\|_\ast}{\mu_1} \right)^p \right] \leqslant C_p\,.\end{align*} $$

In the remainder of this section, we describe the results of Erdős et al. [13] and deduce Lemma 8.1. We then deduce Corollary 8.2.

8.1 The local semicircular law and Lemma 8.1

For $ a < b $ , we define $N_A(a,b)$ to be the number of eigenvalues of A in the interval $(a,b)$ . One of the most fundamental results in the theory of random symmetric matrices is the semicircular law, which says that

$$\begin{align*}\lim_{n \rightarrow \infty} \frac{N_A(a\sqrt{n},b\sqrt{n})}{n} = \frac{1}{2\pi}\int_{a}^b(4 - x^2)^{1/2}_+\,dx, \end{align*}$$

almost surely, where $A \sim \mathrm {Sym\,}_n(\zeta )$ .

We use a powerful “local” version of the semicircle law developed by Erdős et al. in a series of important papers [10, 11, 13]. Their results show that the spectrum of a random symmetric matrix actually adheres surprisingly closely to the semicircular law. In this paper, we need control on the number of eigenvalues in intervals of the form $[-t,t]$ , where $1/n^{1/2} \ll t \ll n^{1/2}$ . The semicircular law predicts that

$$\begin{align*}N_A(-t,t) \approx \frac{n}{2\pi} \int_{-t n^{-1/2}}^{tn^{-1/2}}(4 - x^2)^{1/2}_+\,dx = \frac{2t n^{1/2}}{\pi}(1+o(1)). \end{align*}$$

Theorem 1.11 of [12] makes this prediction rigorous. ⁸

Theorem 8.3. Let $B>0$ , $\zeta \in \Gamma _B$ , and let $A \sim \mathrm {Sym\,}_n(\zeta )$ . Then, for $t \in [C n^{-1/2}, n^{1/2}]$ ,

(8.3) $$ \begin{align} \mathbb{P}\left(\, \left| \frac{N_A(-t,t)}{n^{1/2}t} - 2\pi^{-1} \right|> \pi \right) \lesssim \exp\left(-c_1(t^2n)^{1/4} \right)\, , \end{align} $$

where $C,c_1>0$ are absolute constants.

Lemma 8.1 follows quickly from Theorem 8.3. In fact, we shall only use two corollaries.

Corollary 8.4. Let $B>0$ , $\zeta \in \Gamma _B$ , and let $A \sim \mathrm {Sym\,}_n(\zeta )$ . Then for all $s \geqslant C$ and $k \in \mathbb {N}$ satisfying $sk \leqslant n$ , we have

$$ \begin{align*}\mathbb{P}\left( \frac{\sqrt{n}}{\mu_k k} \geqslant s\right) \lesssim \exp\big(-c(sk)^{1/2}\big)\,,\end{align*} $$

where $C,c>0$ are absolute constants.

Proof. Let C be the maximum of the constant C from Lemma 8.3 and $\pi $ . If $\frac {\sqrt {n}}{\mu _k k} \geqslant s$ , then $N_A(-sk n^{-1/2},skn^{-1/2}) \leqslant k$ . We now apply Lemma 8.3 with $t = sk n^{-1/2} \geqslant sn^{-1/2} \geqslant Cn^{-1/2}$ to see that this event occurs with probability $\lesssim \exp (-c\sqrt {sk})$ .

An identical argument provides a similar bound in the other direction.

Corollary 8.5. Let $B>0$ , $\zeta \in \Gamma _B$ , and let $A \sim \mathrm {Sym\,}_n(\zeta )$ . Then for all $k \in \mathbb {N}$ , we have

$$ \begin{align*}\mathbb{P}\left( \mu_k \geqslant \frac{C \sqrt{n}}{k}\right) \lesssim \exp\big(-c k^{1/2}\big)\,, \end{align*} $$

where $C,c>0$ are absolute constants.

Proof of Lemma 8.1.

Let $ C $ be the constant from Corollary 8.4. From the standard tail estimates on $\|A\|_{op}$ , like (4.11) for example, we immediately see that for all $k \geqslant n/C$ , we have

$$\begin{align*}\mathbb{E}\, \left(\frac{\sqrt{n}}{\mu_kk} \right)^p \leqslant \mathbb{E}_A\left( \frac{\sigma_{1}(A)\sqrt{n}}{k} \right)^p = O_p((n/k)^p) = O_p(1).\end{align*}$$

Thus, we can restrict our attention to the case when $k \leqslant n/C$ . Define the events

$$ \begin{align*}E_1 = \left\{\frac{\sqrt{n}}{\mu_kk} \leqslant C\right\}, \quad E_2 = \left\{\frac{\sqrt{n}}{\mu_kk} \in [C, n/k ] \right\}, \quad E_3 = \left\{\frac{\sqrt{n}}{\mu_kk} \geqslant \frac{n}{k} \right\}.\end{align*} $$

We may bound

(8.4) $$ \begin{align} \mathbb{E}\, \left(\frac{\sqrt{n}}{\mu_kk} \right)^p \leqslant C^p+ \mathbb{E}\, \left(\frac{\sqrt{n}}{\mu_kk}\right)^p {\mathbf{1}}_{E_2} +\mathbb{E} \left(\frac{\sqrt{n}}{\mu_kk}\right)^p {\mathbf{1}}_{E_3} \,. \end{align} $$

To deal with the second term in (8.4), we use Corollary 8.4 to see that

$$\begin{align*}\mathbb{E}\, \left(\frac{\sqrt{n}}{\mu_kk}\right)^p {\mathbf{1}}_{E_2} \lesssim \int_{C}^{n/k} ps^{p-1}e^{-c\sqrt{sk}} ds = O_p(1).\end{align*}$$

To deal with the third term in (8.4), we note that since $n/k \geqslant C$ , we may apply Corollary 8.4, with $s=n/k$ , to conclude that $\mathbb {P}(E_3) \lesssim e^{-c\sqrt {n}}$ . Thus, by Cauchy-Schwarz, we have

$$\begin{align*}\mathbb{E} \left(\frac{\sqrt{n}}{\mu_kk}\right)^p {\mathbf{1}}_{E_3} \leqslant \left(\mathbb{E} \left(\frac{\sigma_{1}\sqrt{n}}{k}\right)^{2p}\right)^{1/2} \mathbb{P}(E_3)^{1/2} \leqslant O_p(1) \cdot n^{p} e^{-c\sqrt{n}} = O_p(1), \end{align*}$$

where we have used the upper tail estimate in $\sigma _1$ from (4.11) to see $\mathbb {E}\, \sigma _1^{2p} = O_p(n^{p})$ .

8.2 Deduction of Corollary 8.2

We now conclude this section by deducing Corollary 8.2 from Lemma 8.1 and Corollary 8.5.

Proof of Corollary 8.2.

Recall

$$\begin{align*}\|A^{-1}\|_{\ast}^2 = \sum_{k = 1}^n \mu_k^2 (\log(1 + k))^2.\end{align*}$$

By Hölder’s inequality, we may assume without loss of generality that $p \geqslant 2$ . Applying the triangle inequality for the $L^{p/2}$ norm gives

$$\begin{align*}\left[ \mathbb{E} \left( \sum_{k = 1}^n \frac{\mu_k^2 (\log (1 + k))^2}{\mu_1^2} \right)^{p/2}\right]^{2/p} \leqslant \sum_{k = 1}^n (\log(1 + k))^2 \mathbb{E} \left[\frac{\mu_k^{p}}{\mu_1^{p}}\right]^{2/p }\,. \end{align*}$$

Taking C to be the constant from Corollary 8.5 bound

$$ \begin{align*} \mathbb{E} \left[\frac{\mu_k^{p}}{\mu_1^{p}}\right]&\leqslant C^pk^{-p} \mathbb{E}\left[\left(\frac{\sqrt{n}}{\mu_1}\right)^p \right] + \mathbb{P}\left(\mu_k \geqslant C \frac{\sqrt{n}}{k} \right) \lesssim C^p k^{-p} ,\end{align*} $$

where we used Lemma 8.1 and Corollary 8.5 for the second inequality. Combining the previous two equations completes the proof.

9 Controlling small balls and large deviations

The goal of this section is to prove the following lemma, which will be a main ingredient in our iteration in Section 10. We shall then use it again in the final step and proof of Theorem 1.1, in Section 11.

Lemma 9.1. For $B>0$ and $\zeta \in \Gamma _B$ , let $A = A_n \sim \mathrm {Sym\,}_{n}(\zeta )$ and let $X \sim \mathrm {Col\,}_n(\zeta )$ . Let $u\in \mathbb {R}^{n-1}$ be a random vector with $\|u\|_2 \leqslant 1$ that depends only on A. Then, for $\delta , \varepsilon> e^{-cn}$ and $s\geqslant 0$ , we have

(9.1) $$ \begin{align} &\mathbb{E}_A \sup_r\mathbb{P}_X\left(\frac{|\langle A^{-1}X,X\rangle -r|}{\|A^{-1}\|_{\ast}} \leqslant \delta,~\langle X, u\rangle\geqslant s,~ \frac{\mu_1}{\sqrt{n}} \leqslant \varepsilon^{-1} \right) \nonumber \\ &\qquad \lesssim \delta e^{-s} \left[ \mathbb{E}_{A} \left(\frac{\mu_1}{\sqrt{n}}\right)^{7/9} {\mathbf{1}}\left\{\frac{\mu_1}{\sqrt{n}} \leqslant \varepsilon^{-1} \right\} \right]^{6/7} + e^{-cn}\,, \end{align} $$

where $c>0$ depends only on $B>0$ .

Note that with this lemma, we have eliminated all “fine-grained” information about the spectrum of $A^{-1}$ and all that remains is $\mu _1$ , which is the reciprocal of the least singular value of the matrix A. We also note that we will only need the full power of Lemma 9.1 in Section 11; until then, we will apply it with $s=0, u=0$ .

We now turn our attention to proving Lemma 9.1. We start with an application of Theorem 1.5, our negative correlation theorem, which we restate here in its full-fledged form.

Theorem 9.2. For $n \in \mathbb {N}$ , $\alpha ,\gamma \in (0,1), B> 0$ , and $\mu \in (0,2^{-15})$ , there are constants $c,R> 0$ depending only on $\alpha ,\gamma ,\mu ,B$ so that the following holds. Let $0\leqslant k \leqslant c \alpha n$ and $\varepsilon \geqslant \exp (-c\alpha n)$ , let $v \in {\mathbb {S}}^{n-1}$ , and let $w_1,\ldots ,w_k \in {\mathbb {S}}^{n-1}$ be orthogonal. For $\zeta \in \Gamma _B$ , let $\zeta '$ be an independent copy of $\zeta $ and $Z_\mu $ a Bernoulli variable with parameter $\mu $ ; let $\widetilde {X} \in \mathbb {R}^n$ be a random vector whose coordinates are i.i.d. copies of the random variable $(\zeta - \zeta ')Z_\mu $ .

If $D_{\alpha ,\gamma }(v)> 1/\varepsilon $ , then

(9.2) $$ \begin{align} \mathbb{P}_X\left( |\langle \widetilde{X}, v \rangle| \leqslant \varepsilon\, \text{ and }\, \sum_{j = 1}^k \langle w_j, \widetilde{X}\rangle^2 \leqslant c k \right) \leqslant R \varepsilon \cdot e^{-c k}\,. \end{align} $$

The proof of Theorem 9.2 is provided in the Appendix. We now prove Lemma 9.3.

Lemma 9.3. Let A be an $n \times n$ real symmetric matrix with $A \in \mathcal {E}$ , and set $ \mu _i := \sigma _{i}(A^{-1})$ , for all $i \in [n]$ . For $B>0$ , $\zeta \in \Gamma _B$ , let $X,X' \sim \mathrm {Col\,}_n(\zeta )$ be independent, let $J \subseteq [n]$ be a $\mu $ -random subset with $\mu \in (0, 2^{-15})$ , and set $\widetilde {X} := (X - X')_J$ . If $k \in [1,c n]$ is such that $s \in (e^{-c n} , \mu _k/\mu _1)$ , then

(9.3) $$ \begin{align} \mathbb{P}_{\widetilde{X}}\left( \|A^{-1} \widetilde{X}\|_2 \leqslant s\mu_1 \right) \lesssim s e^{-ck}\,, \end{align} $$

where $c> 0$ depends only on B.

Proof. For each $j\in [n]$ , we let $v_j$ denote a unit eigenvector of $A^{-1}$ corresponding to $\mu _j$ . Using the resulting singular value decomposition of $A^{-1}$ , we may express

$$\begin{align*}\|A^{-1} \widetilde{X}\|^2_2 = \langle A^{-1} \widetilde{X} , A^{-1}\widetilde{X} \rangle = \sum_{j=1}^n \mu_j^2 \langle \widetilde{X}, v_j \rangle^2, \end{align*}$$

and thus

(9.4) $$ \begin{align} \mathbb{P}_{\widetilde{X}}\left( \|A^{-1} \widetilde{X}\|_2 \mu_1^{-1} \leqslant s \right) \leqslant \mathbb{P}_{\widetilde{X}}\left( |\langle v_1 , \widetilde{X} \rangle| \leqslant s \text{ and } \sum_{j = 2}^k \frac{\mu_j^2}{\mu_1^2}\langle v_j, \widetilde{X} \rangle^2 \leqslant s^2 \right). \end{align} $$

We now use that $s \leqslant 1$ and $\mu _k/\mu _1 \leqslant 1$ in (9.4) to obtain

(9.5) $$ \begin{align} \mathbb{P}_{\widetilde{X}}\left( \|A^{-1} \widetilde{X}\|_2 \mu_1^{-1} \leqslant s \right) \leqslant \mathbb{P}_{\widetilde{X}}\left( |\langle v_1 , \widetilde{X} \rangle| \leqslant s \text{ and } \sum_{j = 2}^k \langle v_j, \widetilde{X} \rangle^2 \leqslant 1 \right) \,. \end{align} $$

We now carefully observe that we are in a position to apply Theorem 1.5 to the right-hand side of (9.5). The coordinates of $\widetilde {X}$ are of the form $(\zeta -\zeta ')Z_{\mu }$ , where $Z_{\mu }$ is a Bernoulli random variable taking $1$ with probability $\mu \in (0,2^{-15})$ and $0$ otherwise. Also, the $ v_2,\ldots ,v_k$ are orthogonal and, importantly, we use that $A \in \mathcal {E}$ to learn that ⁹ $D_{\alpha ,\gamma }(v_1)>1/s$ by property (4.3), provided we choose the constant $c>0$ (in the statement of Lemma 9.3) to be sufficiently small, depending on $\mu ,B$ . Thus, we may apply Theorem 1.5 and complete the proof of the Lemma 9.3.

With this lemma in hand, we establish the following corollary of Lemma 5.2.

Lemma 9.4. For $B>0$ and $\zeta \in \Gamma _B$ , let $X \sim \mathrm {Col\,}_n(\zeta )$ and let A be an $n\times n$ real symmetric matrix with $A \in \mathcal {E}$ . If $s>0$ , $\delta \in (e^{-c n},1)$ and $u \in {\mathbb {S}}^{n-1}$ , then

(9.6) $$ \begin{align} \sup_r\mathbb{P}_{X}\big( \left|\langle A^{-1} X, X \rangle -r \right| \leqslant \delta \mu_1 ,\langle X, u \rangle \geqslant s \big) \lesssim \delta e^{-s} \sum_{k = 2}^{cn} e^{-ck}\left(\frac{\mu_1}{\mu_k} \right)^{2/3} + e^{-cn}\, , \end{align} $$

where $c>0$ is a constant depending only on B.

Proof. We apply Lemma 5.2 to the left-hand side of (9.6) to get

(9.7) $$ \begin{align} \sup_r\mathbb{P}_{X}\big( \left|\langle A^{-1} X, X \rangle - r \right| \leqslant \delta \mu_1 ,\langle X, u \rangle \geqslant s \big) \lesssim \delta e^{-s} \int_{-1/\delta}^{1/\delta} I({\theta})^{1/2} \,d{\theta} + e^{-\Omega(n)} \, , \end{align} $$

where

$$\begin{align*}I({\theta}) := \mathbb{E}_{J,X_J,X_J^{\prime}} \exp\left( \langle (X + X')_J,u \rangle -c' \theta^2 \mu_1^{-2} \| A^{-1}(X - X')_J \|_2^2 \right) ,\end{align*}$$

and $c' = c'(B)>0 $ is a constant depending only on B and $J \subseteq [n]$ is a $\mu $ -random subset. Set

$$\begin{align*}\widetilde{X}=(X-X')_J \qquad \text{ and } \qquad v = A^{-1}\widetilde{X},\end{align*}$$

and apply Hölder’s inequality

(9.8) $$ \begin{align} I({\theta}) = \mathbb{E}_{J,X_J,X_J^{\prime}} \left[e^{\langle (X + X')_J,u \rangle} e^{-c' \theta^2 \|v \|_2^2/\mu_1^2 } \right] \lesssim \left(\mathbb{E}_{\widetilde{X}} e^{-c" \theta^2 \|v \|_2^2/\mu_1^2} \right)^{8/9}\left( \mathbb{E}_{J,X_J,X_J^{\prime}}\, e^{9\langle (X + X')_J,u \rangle} \right)^{1/9} .\end{align} $$

Thus, we apply (5.2) to see that the second term on the right-hand side of (9.8) is $O(1)$ . Thus, for each ${\theta }> 0$ , we have

$$ \begin{align*}I(\theta)^{9/8} \lesssim_{B} \mathbb{E}_{\widetilde{X}} e^{-c" \theta^2 \|v\|_2^2/\mu_1^2} \leqslant e^{-c" \theta^{1/5}} + \mathbb{P}_{\widetilde{X}}( \|v\|_2 \leqslant \mu_1\theta^{-9/10})\,.\end{align*} $$

As a result, we have

$$ \begin{align*}\int_{-1/\delta}^{1/\delta} I({\theta})^{1/2} \,d\theta \lesssim 1 + \int_{1}^{1/\delta} \mathbb{P}_{\widetilde{X}}(\|v\|_2 \leqslant \mu_1 \theta^{-9/10} )^{4/9}\,d\theta\, \lesssim 1 + \int_{\delta}^{1} s^{-19/9} \mathbb{P}_{\widetilde{X}}(\|v\|_2 \leqslant \mu_1 s )^{4/9}\, ds .\end{align*} $$

To bound this integral, we partition $ [\delta ,1] = [\delta , \mu _{c n}/\mu _1 ] \cup \bigcup _{k=2}^{c n} [\mu _{k}/\mu _1,\mu _{k-1}/\mu _1]$ and apply Lemma 9.3 to bound the integrand depending on which interval s lies in. Note, this lemma is applicable since $A \in \mathcal {E}$ . We obtain

$$\begin{align*}\int_{\mu_k/\mu_1}^{\mu_{k-1}/\mu_1} s^{-19/9} \mathbb{P}_{\widetilde{X}}(\|v\|_2 \leqslant \mu_1 s )^{4/9} \leqslant e^{-ck} \int_{\mu_k/\mu_1}^{\mu_{k-1}/\mu_1} s^{-15/9} \, ds \leqslant e^{-ck}(\mu_1/\mu_k)^{2/3}, \end{align*}$$

while

$$\begin{align*}\int_{\delta}^{\mu_{c n}/\mu_1} s^{-19/9} \mathbb{P}_{\widetilde{X}}(\|v\|_2 \leqslant \mu_1 s )^{4/9} \leqslant e^{-c n} \delta^{-3/2} \leqslant e^{-\Omega(n)}. \end{align*}$$

Summing over all k and plugging the result into (9.7) completes the lemma.

We may now prove Lemma 9.1 by using the previous Lemma 9.4 along with the properties of the spectrum of A established in Section 8.

Proof of Lemma 9.1.

Let $\mathcal {E}$ be our quasi-random event as defined in Section 4, and let

$$\begin{align*}\mathcal{E}_0=\mathcal{E}\cap \left\lbrace\frac{\mu_1}{\sqrt{n}} \leqslant \varepsilon^{-1}\right\rbrace.\end{align*}$$

For fixed $A \in \mathcal {E}_0$ and $u = u(A) \in \mathbb {R}^{n}$ with $\|u\|_2\leqslant 1$ , we may apply Lemma 9.4 with $\delta ' =\delta \frac {\|A^{-1}\|_{\ast }}{\mu _1}$ to see that

$$\begin{align*}\sup_{r \in \mathbb{R}} \mathbb{P}_{X}\big( \left|\langle A^{-1} X, X \rangle -r \right| \leqslant \delta \|A\|_{\ast} ,\langle X, u \rangle \geqslant s \big) \lesssim \delta e^{-s} \left(\frac{\|A^{-1}\|_{*}}{\mu_1}\right) \sum_{k = 2}^{cn} e^{-ck}\left(\frac{\mu_1}{\mu_k} \right)^{2/3} + e^{-cn}\, .\end{align*}$$

By Lemma 4.1, $\mathbb {P}_A(\mathcal {E}^c)\lesssim \exp (-\Omega (n))$ . Therefore, it is enough to show that

(9.9) $$ \begin{align} \mathbb{E}_A^{\mathcal{E}_0} \left(\frac{\|A^{-1}\|_{*}}{\mu_1}\right) \left(\frac{\mu_1}{\mu_k} \right)^{2/3} \lesssim k\cdot \mathbb{E}_{A}^{\mathcal{E}_0} \left[ \left(\frac{\mu_1}{\sqrt{n}}\right)^{7/9}\right]^{6/7},\end{align} $$

for each $k \in [2,c n]$ . For this, apply Hölder’s inequality to the left-hand side of (9.9) to get

$$\begin{align*}\mathbb{E}_A^{\mathcal{E}_0} \left(\frac{\|A^{-1}\|_{*}}{\mu_1}\right)\left(\frac{\mu_1}{\mu_k} \right)^{2/3}\leqslant \mathbb{E}_A^{\mathcal{E}_0} \left[\left(\frac{\|A^{-1}\|_{*}}{\mu_1}\right)^{14}\right]^{1/14}\mathbb{E}_A^{\mathcal{E}_0} \left[\left(\frac{\sqrt{n}}{\mu_k}\right)^{28/3}\right]^{1/14}\mathbb{E}_A^{\mathcal{E}_0} \left[\left(\frac{\mu_1}{\sqrt{n}} \right)^{7/9}\right]^{6/7}.\end{align*}$$

We now apply Corollary 8.2 to see the first term is $O(1)$ and Lemma 8.1 to see that the second term is $O(k)$ . This establishes (9.9) and thus Lemma 9.1.

10 Intermediate bounds: Bootstrapping the lower tail

In this short section, we will use the tools developed so far to prove an “up-to-logarithms” version of Theorem 1.1. In the next section, Section 11, we will bootstrap this result (once again) to prove Theorem 1.1.

Lemma 10.1. For $B>0 $ , let $\zeta \in \Gamma _B$ , and let $A_n \sim \mathrm {Sym\,}_{n}(\zeta )$ . Then for all $\varepsilon>0$

$$ \begin{align*}\mathbb{P}(\sigma_{\min}(A_{n}) \leqslant \varepsilon n^{-1/2} ) \lesssim \varepsilon \cdot (\log \varepsilon^{-1})^{1/2} + e^{-\Omega(n)}\,.\end{align*} $$

To prove Lemma 10.1, we first prove the following “base step” (Lemma 10.3), which we then improve upon in three increments, ultimately arriving at Lemma 10.1.

The “base step” is an easy consequence of Lemmas 6.2 and 9.1 and actually already improves upon the best known bounds on the least-singular value problem for random symmetric matrices. For this, we will need the well-known theorem due to Hanson and Wright [18, 51]. See [47, Theorem 6.2.1]) for a modern exposition.

Theorem 10.2 (Hanson-Wright).

For $B>0$ , let $\zeta \in \Gamma _B$ , let $X \sim \mathrm {Col\,}_n(\zeta )$ , and let M be an $m\times n$ matrix. Then for any $t\geqslant 0$ , we have

$$\begin{align*}\mathbb{P}_X\big( \left| \|MX\|_2 - \|M\|_{\mathrm{HS}} \right|>t \big) \leqslant 2 \exp\left(- \frac{ct^2}{B^4\|M\|^2} \right)\, , \end{align*}$$

where $c>0$ is absolute constant.

We now prove the base step of our iteration.

Lemma 10.3 (Base step).

For $B>0$ , let $\zeta \in \Gamma _B$ and let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ . Then

$$ \begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1}) \leqslant \varepsilon n^{-1/2} ) \lesssim \varepsilon^{1/4} + e^{-\Omega(n)}\,,\end{align*} $$

for all $\varepsilon>0$ .

Proof. As usual, we let $A := A_{n}$ . By Lemma 6.1, it will be sufficient to show that for $r\in \mathbb {R}$ ,

(10.1) $$ \begin{align} \mathbb{P}_{A,X}\left(\, \frac{|\langle A^{-1} X, X \rangle - r|}{\|A^{-1}X\|_2} \leqslant C\varepsilon,\, \sigma_n(A)\geqslant \varepsilon n^{-1/2} \right) \lesssim \varepsilon^{1/4} + e^{-\Omega(n)}\,. \end{align} $$

By the Hanson-Wright inequality (Theorem 10.2), there exists $C'>0$ so that

(10.2) $$ \begin{align} \mathbb{P}_X\big(\, \|A^{-1} X\|_2 \geqslant C'(\log \varepsilon^{-1} )^{1/2} \cdot \| A^{-1} \|_{\mathrm{HS}}\, \big) \leqslant \varepsilon\,\end{align} $$

and so the left-hand side of (10.1) is bounded above by

$$\begin{align*}\varepsilon + \mathbb{P}_{A,X}\left(\, \frac{|\langle A^{-1} X, X \rangle - r|}{\|A^{-1}\|_{\mathrm{HS}}} \leqslant \delta,\, \sigma_n(A)\geqslant \varepsilon n^{-1/2} \right) \, , \end{align*}$$

where $\delta :=C" \varepsilon \cdot ( \log \varepsilon ^{-1} )^{1/2}$ . Now, by Lemma 9.1 with the choice of $u=0, s=0$ , we have

(10.3) $$ \begin{align} \mathbb{P}_{A,X}\left(\, \frac{|\langle A^{-1} X, X \rangle - r|}{\|A^{-1}\|_{\mathrm{HS}}} \leqslant \delta,\, \sigma_n(A)\geqslant \varepsilon n^{-1/2} \right) \lesssim \delta \varepsilon^{-2/3} + e^{-\Omega(n)} \lesssim \varepsilon^{1/4} + e^{-\Omega(n)}\, , \end{align} $$

where we have used that $\|A^{-1}\|_{\ast }\geqslant \|A^{-1}\|_{\mathrm {HS}}$ . We also note that Lemma 9.1 actually gives an upper bound on $\mathbb {E}_A \sup _r \mathbb {P}_X( \mathcal {A})$ , where $\mathcal {A}$ is the event on the left-hand side of (10.7). Since $\sup _r \mathbb {P}_{A,X}(\mathcal {A}) \leqslant \mathbb {E}_A \sup _r \mathbb {P}_X( \mathcal {A}) $ , the bound (10.3), and thus Lemma 10.3, follows.

The next lemma is our “bootstrapping step”: Given bounds of the form

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_n)\leqslant \varepsilon n^{-1/2} ) \lesssim \varepsilon^{\kappa}+ e^{-cn}, \end{align*}$$

this lemma will produce better bounds for the same problem with $A_{n+1}$ in place of $A_n$ .

Lemma 10.4 (Bootstrapping step).

For $B>0$ , let $\zeta \in \Gamma _B$ , let $A_{n+1} \sim \mathrm {Sym\,}_{n+1}(\zeta )$ , and let ${\kappa } \in (0,1) \setminus \{7/10\}$ . If for all $\varepsilon>0$ , and all n, we have

(10.4) $$ \begin{align} \mathbb{P}\big(\sigma_{\min}(A_n)\leqslant \varepsilon n^{-1/2} \big )\lesssim \varepsilon^{\kappa}+ e^{-\Omega(n)}\, , \end{align} $$

then for all $\varepsilon>0$ and all n, we have

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n+1})\leqslant \varepsilon n^{-1/2} )\lesssim (\log \varepsilon^{-1} )^{1/2} \cdot \varepsilon^{\min\left\{1, 6\kappa/7+1/3\right\}}+ e^{-\Omega(n)}\,. \end{align*}$$

Proof. Let $c>0$ denote the implicit constant in the exponent on the right-hand side of (10.4). Note that if $0< \varepsilon <e^{-cn}$ , by the assumption of the lemma, then we have

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A_{n})\leqslant \varepsilon n^{-1/2} )\lesssim e^{-\Omega(n)},\end{align*}$$

for all n, in which case, we are done. So we may assume $\varepsilon> e^{-cn}$ .

As in the proof of the “base step,” Lemma 10.3, we look to apply Lemmas 6.2 and 9.1 in sequence. For this, we write $A = A_n$ and bound (9.1) as in the conclusion of Lemma 9.1

(10.5) $$ \begin{align} \mathbb{E}_{A}\, \left(\frac{\mu_1}{\sqrt{n}}\right)^{7/9} {\mathbf{1}}\left\{\frac{\mu_1}{\sqrt{n}} \leqslant \varepsilon^{-1} \right\} \leqslant \int_{0}^{\varepsilon^{-7/9}}\mathbb{P}\left(\sigma_{\min}(A)\leqslant x^{-9/7} n^{-1/2} \right)\, dx ,\end{align} $$

where we used that $\sigma _{\min }(A)=1/\mu _1(A)$ . Now use assumption (10.4) to see the right-hand side of (10.5) is

(10.6) $$ \begin{align} \lesssim 1 + \int_{1}^{\varepsilon^{-7/9}} (x^{-9\kappa/7}+ e^{-cn})\,dx \lesssim \max\left\{1, \varepsilon^{\kappa-7/9}\right\}\,. \end{align} $$

Now, we apply Lemma 9.1 with $\delta = C\varepsilon \cdot (\log \varepsilon ^{-1} )^{1/2}$ , $s=0$ , and $u=0$ to see that

(10.7) $$ \begin{align} \mathbb{P}_{A,X}\left(\frac{|\langle A^{-1}X,X\rangle - r |}{\|A^{-1}\|_{\mathrm{HS}}} \leqslant \delta,\, \frac{\mu_1}{\sqrt{n}} \leqslant \varepsilon^{-1} \right) &\lesssim \max\left\{\varepsilon, \varepsilon^{6\kappa/7+1/3}\right\} \cdot (\log \varepsilon^{-1} )^{1/2} + e^{-\Omega(n)}\, , \end{align} $$

for all r. Here, we used that $\|A^{-1}\|_{\mathrm {HS}} \leqslant \|A^{-1}\|_{\ast }$ .

Now, by Hanson-Wright (Theorem 10.2), there exists $C'>0$ , such that

$$\begin{align*}\mathbb{P}_X\big(\| A^{-1} X \|_2 \geqslant C' \|A^{-1}\|_{\mathrm{HS}}\cdot (\log \varepsilon^{-1} )^{1/2} \big) \leqslant \varepsilon.\end{align*}$$

Thus, we choose $C"$ to be large enough, so that

$$ \begin{align*}\mathbb{P}_{A,X}\left(\, \frac{|\langle A^{-1}X,X\rangle -r |}{\|A^{-1} X\|_2} \leqslant C"\varepsilon , \sigma_{n}(A) \geqslant \varepsilon n^{-1/2}\, \right) \lesssim \max\left\{\varepsilon, \varepsilon^{6\kappa/7+1/3}\right\} \cdot (\log \varepsilon^{-1} )^{1/2} + e^{-\Omega(n)} \, ,\end{align*} $$

for all r. Lemma 6.1 now completes the proof of Lemma 10.4.

Lemma 10.1 now follows by iterating Lemma 10.4 three times.

Proof of Lemma 10.1.

By Lemmas 10.3 and 10.4, we have

$$\begin{align*}\mathbb{P}(\sigma_{\min}(A)\leqslant \varepsilon n^{-1/2} )\lesssim \varepsilon^{13/21} \cdot (\log \varepsilon^{-1} )^{1/2}+ e^{-\Omega(n)} \lesssim \varepsilon^{13/21-\eta}+ e^{-\Omega(n)}\, , \end{align*}$$

for some small $\eta>0$ . Applying Lemma 10.4 twice more gives an exponent of $\frac {127}{147}-\frac {6}{7}\eta $ and then $1$ , for $\eta $ small, thus completing the proof.

11 Proof of Theorem 1.1

We are now ready to prove our main result, Theorem 1.1. We use Lemma 6.1 (as in the proof of Lemma 10.1) and the inequality at (4.5) to see that it is enough to prove

(11.1) $$ \begin{align} \mathbb{P}^{\mathcal{E}}\left(\, \frac{|\langle A^{-1}X,X\rangle-r|}{\|A^{-1}X\|_2}\leqslant C\varepsilon, \text{ and } \sigma_n(A) \geqslant \varepsilon n^{-1/2} \right) \lesssim \varepsilon + e^{-\Omega(n)} \,, \end{align} $$

where C is as in Lemma 6.1 and the implied constants do not depend on r. Recall that $\mathcal {E}$ is the quasi-random event defined in Section 4.

To prepare ourselves for what follows, we put $\mathcal {E}_0 := \mathcal {E} \cap \{\sigma _{\min }(A) \geqslant \varepsilon n^{-1/2} \}$ and

$$\begin{align*}Q(A, X):=\frac{|\langle A^{-1}X,X\rangle-r|}{\|A^{-1}X\|_2}\, \, \text{ and } \, Q_{\ast}(A, X):=\frac{|\langle A^{-1}X,X\rangle-r|}{\|A^{-1}\|_{\ast}} ,\end{align*}$$

where

$$\begin{align*}\|A^{-1}\|_{*}^2 =\sum_{k=1}^n \mu_k^{2}(\log (1 + k) )^2 \, ,\end{align*}$$

as defined in Section 8. We now split the left-hand side of (11.1) as

(11.2) $$ \begin{align} \mathbb{P}^{\mathcal{E}_0}\left( Q(A, X)\leqslant C\varepsilon \right) &\leqslant \mathbb{P}^{\mathcal{E}_0}\left(Q_{\ast}(A, X)\leqslant 2C\varepsilon \right) + \mathbb{P}^{\mathcal{E}_0}\left(Q(A, X)\leqslant C\varepsilon, \frac{\|A^{-1}X\|_2}{\|A^{-1}\|_{\ast}}\geqslant 2 \right)\,. \end{align} $$

We can take care of the first term easily by combining Lemmas 9.1 and 10.1.

Lemma 11.1. For $\varepsilon>0$ ,

$$ \begin{align*}\mathbb{P}^{\mathcal{E}_0}(Q_{\ast}(A, X) \leqslant 2 C\varepsilon ) \lesssim \varepsilon + e^{-\Omega(n)}\,.\end{align*} $$

Proof. Apply Lemma 9.1, with $\delta =2C\varepsilon $ , $u=0$ , and $s=0$ to obtain

$$ \begin{align*}\mathbb{P}^{\mathcal{E}_0}(Q_{\ast}(A, X)\leqslant 2 C \varepsilon ) \lesssim \varepsilon \left( \mathbb{E}_{A} \left(\frac{\mu_1}{\sqrt{n}} \right)^{7/9} {\mathbf{1}}\left\lbrace\frac{\mu_1}{\sqrt{n}}\leqslant \varepsilon^{-1} \right\rbrace \right)^{6/7} + e^{-\Omega(n)}\,.\end{align*} $$

By Lemma 10.1 and the calculation at (10.6), the expectation on the right is bounded by a constant.

We now focus on the latter term on the right-hand side of (11.2). By considering the dyadic partition $ 2^j \leqslant \|A^{-1}X\|_2 / \|A^{-1}\|_{*} \leqslant 2^{j+1}$ , we see the second term on the right hand side (RHS) of (11.2) is

(11.3) $$ \begin{align} \lesssim \sum_{j=1}^{\log n}\mathbb{P}^{\mathcal{E}_0}\left(Q_{\ast}(A, X) \leqslant 2^{j+1}C \varepsilon\,, \frac{\|A^{-1}X\|_2}{ \|A^{-1}\|_{*}}\geqslant 2^{j}\right)\ + e^{-\Omega(n)}\,. \end{align} $$

Here, we have dealt with the terms for which $j \geqslant \log n$ by using the fact that

$$ \begin{align*} \mathbb{P}_X\big(\, \|A^{-1} X\|_2 \geqslant \sqrt{n} \|A^{-1}\|_{\ast} \big) \lesssim e^{-\Omega(n)}\,, \end{align*} $$

which follows from Hanson-Wright and the inequality $\|A^{-1}\|_{\ast }\geqslant \|A^{-1}\|_{\mathrm {HS}}$ .

We now show that the event $\|A^{-1}X\|_2 \geqslant t \|A^{-1}\|_\ast $ implies that X must correlate with one of the eigenvectors of A.

Lemma 11.2. For $t>0$ , we have

$$ \begin{align*} \mathbb{P}_{X}\left(Q_{\ast}(A,X)\leqslant 2Ct \varepsilon, \frac{ \|A^{-1}X\|_2}{ \|A^{-1}\|_{*}}\geqslant t\right) \leqslant 2\sum_{k=1}^n\mathbb{P}_X\left(Q_{\ast}(A,X)\leqslant 2Ct \varepsilon , \langle X,v_k\rangle\geqslant t \log (1 + k)\right) ,\end{align*} $$

where $\{v_k\}$ is an orthonormal basis of eigenvectors of A.

Proof. Assume that $\|A^{-1}X\|_2 \geqslant t \|A^{-1}\|_{*}$ , and use the singular value decomposition associated with $\{v_k\}_k$ to write

$$\begin{align*}t^2\sum_{k} \mu_i^2(\log(k+1))^2 = t^2\|A\|^2_{\ast} \leqslant \| A^{-1} X\|_2^2 = \sum_{k} \mu_k^{2} \langle v_k,X\rangle^2. \end{align*}$$

Thus

$$ \begin{align*}\{\|A^{-1} X \|_2 \geqslant t \| A^{-1}\|_\ast \} \subset \bigcup_{k} \big\lbrace |\langle X, v_k \rangle| \geqslant t \log(k+1) \big\rbrace \,.\end{align*} $$

To finish the proof of Lemma 11.2, we union bound and treat the case of $-X$ the same as X (by possibly changing the sign of $v_k$ ) at the cost of a factor of $2$ .

Proof of Theorem 1.1.

Recall that it suffices to establish (11.1). Combining (11.2) with Lemma 11.2 and Lemma 11.1 tells us that

(11.4) $$ \begin{align} \mathbb{P}^{\mathcal{E}_0}\left(Q(A,X) \leqslant C\varepsilon \right) \lesssim \varepsilon + 2\sum_{j=1}^{\log n}\sum_{k = 1}^n \mathbb{P}^{\mathcal{E}_0}\left(Q_{\ast}(A,X)\leqslant 2^{j+1}C \varepsilon , \langle X,v_k\rangle\geqslant 2^j \log(1 + k)\right) + e^{-\Omega(n)} \,. \end{align} $$

We now apply Lemma 9.1 for all $t>0$ , with $\delta = 2Ct\varepsilon $ , $s=t \log (k+1)$ and $u=v_k$ to see that,

(11.5) $$ \begin{align} \mathbb{P}^{\mathcal{E}_0}\big( Q_{\ast}(A,X)\leqslant 2Ct \varepsilon, \langle X,v_k\rangle\geqslant t \log(1 + k) \big) \lesssim \varepsilon t (k+1)^{-t} \cdot I^{6/7}+ e^{-\Omega(n)}\, , \end{align} $$

where

$$\begin{align*}I := \mathbb{E}_{A} \left(\frac{\mu_1(A)}{\sqrt{n}}\right)^{7/9} {\mathbf{1}}\left\{\frac{\mu_1(A)}{\sqrt{n}} \leqslant \varepsilon^{-1} \right\}. \end{align*}$$

Using (11.5) in (11.4) yields

$$\begin{align*}\mathbb{P}^{\mathcal{E}_0}(Q(A,X) \leqslant C\varepsilon ) \lesssim \varepsilon I^{6/7} \sum_{j=1}^{\log n}\sum_{k = 1}^{n} 2^j(k+1)^{-2^j} + e^{-\Omega(n)} \lesssim \varepsilon \cdot I^{6/7} + e^{-\Omega(n)}, \end{align*}$$

since $\sum _{j=1}^{\infty }\sum _{k = 1}^{\infty } 2^j(k+1)^{-2^j} = O(1)$ . Now we write

$$\begin{align*}I = \mathbb{E}_{A}\, \left(\frac{\mu_1(A)}{\sqrt{n}}\right)^{7/9} {\mathbf{1}}\left\{\frac{\mu_1(A)}{\sqrt{n}} \leqslant \varepsilon^{-1} \right\} \leqslant \int_{0}^{\varepsilon^{-7/9}}\mathbb{P}\left(\sigma_{\min}(A)\leqslant x^{-9/7} n^{-1/2} \right)\, dx\end{align*}$$

and apply Lemma 10.1 to see

$$\begin{align*}\int_{0}^{\varepsilon^{-7/9}}\mathbb{P}\left(\sigma_{\min}(A)\leqslant x^{-9/7} n^{-1/2} \right)\, dx \lesssim \int_{1}^{\infty} s^{-9/7} \, ds + 1 \lesssim 1. \end{align*}$$

Thus, Lemma 6.1 completes the proof of Theorem 1.1.