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 [Reference Rudelson and Vershynin31] that allows for a geometric perspective on the least singular value problemFootnote 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 inequalityFootnote ⁴ due to Vershynin [Reference Vershynin46],

(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 [Reference Vershynin46] along with Corollary 8.4, which is based on the work of [Reference Erdős, Schlein and Yau13].

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 [Reference Campos, Jenssen, Michelen and Sahasrabudhe4]. 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 [Reference Costello6, Reference Meka, Nguyen and Vu23]. 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. [Reference Erdős, Schlein and Yau13].

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. [Reference Erdős, Schlein and Yau13].

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.

3.1 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 [Reference Rudelson and Vershynin31].

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 [Reference Rudelson and Vershynin31] will be useful throughout:

Fact 3.1 assures us that for each incompressible vector, we can find a large subvector that is “flat.” Using the work of Vershynin [Reference Vershynin46], we will safely be able to ignore compressible vectors. In particular, [Reference Vershynin46, 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.

Lemma 3.2 follows easily from [Reference Vershynin46, 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.

4.1 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 [Reference Vershynin46].

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 [Reference Campos, Jenssen, Michelen and Sahasrabudhe4], 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 [Reference Feldheim and Sodin15], we may deduceFootnote 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 holdsFootnote 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 [Reference Vershynin47], 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.

5.1 Proofs

We now dive in and prove our Esseen-type inequality. For this, we shall appeal to the classical Esseen inequality [Reference Esseen14]: 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, [Reference Vershynin47, 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 [Reference Kwan and Sauermann20].

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 [Reference Vershynin46].

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 [Reference Vershynin47], 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.

6.1 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. [Reference Rudelson and Vershynin31, 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.Footnote 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 [Reference Rudelson and Vershynin31]. 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 [Reference Rudelson and Vershynin35]), 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 [Reference Nguyen, Tao and Vu24, 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 [Reference Rudelson and Vershynin31], 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 [Reference Vershynin46, 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 [Reference Vershynin46, 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.

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 [Reference Erdős, Schlein and Yau10, Reference Erdős, Schlein and Yau11, Reference Erdős, Schlein and Yau13]. 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 [Reference Erdős12] makes this prediction rigorous.Footnote 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.

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.