\chapter{Overview of the proof}\label{s:overview}

This section explains the proofs of \zcref{t:main} and \zcref{t:pairs}, and, in \zcref{s:ov-hermitian}, the changes needed for \zcref{t:hermitian}. It is written to be read on its own. The precise statements and proofs occupy \zcref{s:prelim,s:reduction,s:comparison,s:pairs,s:linearization,s:llt} and \zcref{s:goe}; a roadmap is given in \zcref{s:ov-organization}.

Throughout this section, except in \zcref{s:ov-hermitian}, $H$ denotes a matrix as in \zcref{d:wigner}. It is real symmetric, its entries above the diagonal are $N^{-1/2} X_{ab}$, where the $X_{ab}$ are independent copies of a centered, sub-Gaussian random variable $X$ of variance one, and its diagonal entries are independent, centered, and sub-Gaussian, with positive variance. Let $\lambda_1 \le \dots \le \lambda_N$ denote the eigenvalues of $H$, and let $u_1, \dots, u_N$ denote corresponding orthonormal eigenvectors. For $i \in [N - 1]$, let $g_i = \lambda_{i + 1} - \lambda_i$ denote the $i$-th gap and $E_i = (\lambda_i + \lambda_{i + 1})/2$ its midpoint. Recall that $c(E) = (\pi^2/6) \rhosc(E)^3$, where $\rhosc$ is the semicircle density. Recall also that we call a compact interval contained in $(-2, 2)$ a bulk interval, and that an event holds with overwhelming probability if for every $D > 0$ its complement has probability at most $C_D N^{-D}$. In this section, statements involving the exponent $\eps > 0$ hold for all sufficiently small $\eps$. Approximate statements about random quantities hold with probability tending to one, and in such statements $o(1)$ denotes a random error that is bounded by a deterministic sequence tending to zero, with probability tending to one.

\section{The main difficulty}\label{s:ov-difficulty}

We set $d = N^{-3/2}$, and we call $d$ the \emph{critical scale}. \zcref[S]{t:main} states that the smallest gaps of $H$ in the bulk of the spectrum are of order $d$, and that for every bulk interval $I$ and every $x > 0$, the number of indices $i$ with $g_i \le x d$ and $\lambda_i \in I$ converges to a Poisson random variable with mean
\begin{equation}\label{e:ov-mean}
\int_I \int_0^x c(E) y \dd y \dd E = \frac{x^2}{2} \int_I c(E) \dd E.
\end{equation}

The usual route to universality of local eigenvalue statistics compares $H$ with a matrix that has a Gaussian component, by replacing the entries one at a time and showing that each replacement changes the observable of interest by a small amount \cite{chatterjee2006generalization,tao2011random,erdos2012bulk}. For the smallest gaps of real symmetric matrices, the standard perturbative comparison estimate does not give a small error at this scale. First-order perturbation theory suggests that changing the entries $H_{ab}$ and $H_{ba}$ by an amount $h$ moves $\lambda_i$ by approximately $2 h u_i(a) u_i(b)$. The entries are of order $N^{-1/2}$, and the eigenvectors are delocalized, in the sense that $\max_{i, a} |u_i(a)| \le N^{-1/2 + \eps}$ with overwhelming probability for every $\eps > 0$ \cite{erdos2012rigidity,benaychgeorges2016lectures}. So a single entry moves an eigenvalue, and therefore a gap, by about $N^{-3/2} = d$, which is the scale of the smallest gaps. Observables that resolve gaps at the scale $d$ are therefore not stable under the replacement of a single entry.

In \cite{bourgade2022extreme}, this obstacle was avoided by assuming that the entry distributions are smooth, which allows one to replace $H$ by a matrix with a Gaussian component without a comparison at the scale $d$. For complex Hermitian matrices, the smallest gaps are of order $N^{-4/3}$, which is much larger than $N^{-3/2}$, and the obstacle does not arise; we return to this case in \zcref{s:ov-hermitian}.

Our argument has two parts. At the scale $d$, we do not compare $H$ with another ensemble. Instead, we use the sensitivity of the gaps to single entries through a resampling argument, which we describe in \zcref{s:ov-resampling,s:ov-linearization,s:ov-arithmetic}. This argument requires the number of gaps at a scale slightly larger than $d$. At the scales $N^{-1 - \alpha}$ with $\alpha \in (0, 1/2)$, which lie between $d$ and the typical spacing $N^{-1}$, the typical gap displacement caused by changing a single entry, of order $N^{-3/2}$, is small compared to this scale. In \zcref{s:ov-pairs}, we prove a law of large numbers for the number of gaps at such scales, using a comparison argument together with an induction on $\alpha$.

Four steps of the proof are new: the resampling argument, together with the reduction of its local limit theorem to the exclusion of eigenvectors whose coordinates take few values, which replaces the use of smooth entry distributions in \cite{bourgade2022extreme}; the holomorphic bound on derivatives in the comparison argument, which replaces bounds obtained from eigenvalue perturbation formulas or from resolvent regularizations of eigenvalues; the induction on $\alpha$, which computes gap counts through independent $2 \times 2$ Gaussian blocks and replaces the direct combination of coupling and comparison, which reaches only $\alpha < 1/8$ when $\E X^3 \neq 0$; and, for complex Hermitian matrices, a bound on triples of nearby eigenvalues relative to the number of gaps, together with an estimate for Gaussian perturbations of $3 \times 3$ diagonal matrices, which replaces the exclusion of such triples.

\section{Resampling}\label{s:ov-resampling}

Let $\tau = N^{1/100}$ and $m = 2 \lfloor \tau/8 \rfloor$. Then $m = \tau/4 + O(1)$. We choose positions $(a_1, b_1), \dots, (a_m, b_m)$ independently and uniformly from the set of pairs $(a, b)$ with $1 \le a < b \le N$, and we choose independent copies $Y_1, \dots, Y_m$ of $X$, all independent of $H$. We call $(a_1, b_1), \dots, (a_m, b_m)$ the \emph{selected positions}. Let $\widetilde H$ denote the matrix obtained from $H$ by replacing, for $l = 1, \dots, m$ in turn, the entries in the positions $(a_l, b_l)$ and $(b_l, a_l)$ by $N^{-1/2} Y_l$. Each replacement preserves the joint law of the entries, so $\widetilde H$ has the same law as $H$. It therefore suffices to prove \zcref{t:main} for $\widetilde H$, and we may perform all computations conditionally on $H$.

Heuristically, one replacement moves a gap by an amount of order $d$, so the $m$ replacements should move it by a random walk with $m \approx \tau/4$ steps of size of order $d$. To make this precise, we use the following elementary fact. The eigenvalues of a real symmetric $2 \times 2$ matrix with diagonal entries $h_1, h_2$ and off-diagonal entry $h_{12}$ differ by
\begin{equation}\label{e:ov-twobytwo}
2 \left| \left( \frac{h_1 - h_2}{2}, h_{12} \right) \right|.
\end{equation}

Fix $i \in [N - 1]$, and let $\tilde g_i$ denote the gap between the $i$-th and $(i + 1)$-th eigenvalues of $\widetilde H$. For $a < b$, we define the vector $D^{(i)}_{ab} \in \R^2$ by
\begin{equation}\label{e:ov-D}
D^{(i)}_{ab} = N \big( u_i(a) u_i(b) - u_{i + 1}(a) u_{i + 1}(b), \, u_i(a) u_{i + 1}(b) + u_i(b) u_{i + 1}(a) \big),
\end{equation}
and we set
\begin{equation}\label{e:ov-zS}
z_i = \left( - \frac{g_i}{2 d}, 0 \right), \qquad S_i = \sum_{l = 1}^m (Y_l - X_{a_l b_l}) D^{(i)}_{a_l b_l}.
\end{equation}
We apply \eqref{e:ov-twobytwo} to the compression of $\widetilde H$ to the span of $u_i$ and $u_{i + 1}$, which has diagonal entries $u_i^T \widetilde H u_i$ and $u_{i + 1}^T \widetilde H u_{i + 1}$ and off-diagonal entry $u_i^T \widetilde H u_{i + 1}$. When the selected positions are distinct, which holds with probability $1 - O(\tau^2 N^{-2})$, this compression has eigenvalue gap exactly $2 d |z_i + S_i|$. If we neglect the interaction of $u_i$ and $u_{i + 1}$ with the other eigenvectors, which we justify in \zcref{s:ov-linearization}, we therefore obtain the approximation $\tilde g_i \approx 2 d |z_i + S_i|$. We call $S_i$ the \emph{linearized increment} and $2 d |z_i + S_i|$ the \emph{linearized gap} of the pair $(i, i + 1)$.

Conditionally on $H$, the vector $S_i$ is a sum of $m$ independent copies of the random vector $(Y - X_{ab}) D^{(i)}_{ab}$, where $(a, b)$ is a uniformly chosen position with $a < b$ and $Y$ is an independent copy of $X$. By the delocalization of the eigenvectors, and since all entries $X_{ab}$ are bounded by $N^{\eps}$ with overwhelming probability, we have $|(Y - X_{ab}) D^{(i)}_{ab}| \le 2 N^{2\eps} (|Y| + N^{\eps})$. The conditional mean of this vector is small by the eigenvalue equation, and its conditional covariance is $4 \Id_2 + o(1)$. Heuristically, the factor $(Y - X_{ab})^2$ has average close to $2$ over the uniformly chosen position, and the orthonormality of $u_i$ and $u_{i + 1}$ makes the average of $D^{(i)}_{ab} (D^{(i)}_{ab})^T$ over $a < b$ close to $2 \Id_2$. The entries $X_{ab}$ are not independent of the eigenvectors, so these two averages cannot simply be multiplied. The two are decoupled by the bound $\|\mathring X\| \le N^{1/2 + \eps}$, which holds with overwhelming probability, for the matrix $\mathring X$ with zero diagonal and off-diagonal entries $X_{ab}^2 - 1$; \zcref{l:moments} contains the computation.

So $S_i$ is approximately a centered Gaussian vector with covariance $4 m \Id_2 \approx \tau \Id_2$, whose density at $y \in \R^2$ is $(2 \pi \tau)^{-1} e^{-|y|^2/(2\tau)}$. Suppose for the moment that $S_i$ satisfies a local limit theorem that resolves its law at the scale of a single step, in the sense that for every smooth function $\psi \colon \R^2 \to \R$ with compact support,
\begin{equation}\label{e:ov-llt}
\tau \, \E \big[ \psi(z_i + S_i) \bigm| H \big] = \frac{1}{2\pi} \exp\Big( - \frac{|z_i|^2}{2\tau} \Big) \int_{\R^2} \psi(y) \dd y + o(1).
\end{equation}
We set $\ell = d \tau^{1/2} = N^{-299/200}$. Approximating the indicator function of a disk by smooth functions, we find for every $x > 0$ that
\begin{equation}\label{e:ov-heuristic}
\Prob \big( \tilde g_i \le x d \, \big| \, H \big) \approx \Prob \Big( |z_i + S_i| \le \frac{x}{2} \, \Big| \, H \Big) \approx \frac{\pi x^2}{4} \cdot \frac{1}{2 \pi \tau} \exp\Big( - \frac{|z_i|^2}{2 \tau} \Big) = \frac{x^2}{8 \tau} \exp\Big( - \frac{g_i^2}{8 \ell^2} \Big).
\end{equation}
This mechanism is illustrated in \zcref{f:ov-planar}. A gap $g_i$ of order $\ell$ has a probability of order $\tau^{-1}$ to produce a gap of $\widetilde H$ of order $d$, and gaps $g_i$ much larger than $\ell$ contribute negligibly.

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.85]
  \fill[black!8] (-7.6,0) circle (2.6);
  \fill[black!4] (-7.6,0) circle (1.3);
  \draw[densely dashed, black!55] (-7.6,0) circle (1.3);
  \draw[densely dashed, black!55] (-7.6,0) circle (2.6);
  \draw[->, black!70] (-11.2,0) -- (2.2,0);
  \draw[->, black!70] (0,-3.0) -- (0,3.0);
  \fill[black!45] (0,0) circle (0.45);
  \draw (0,0) circle (0.45);
  \fill (-7.6,0) circle (0.06);
  \node[below] at (-7.6,-0.05) {$z_i$};
  \node[above right] at (0.3,0.35) {disk of radius $x/2$};
  \draw[thick, ->] (-7.6,0) -- (-7.1,0.35) -- (-6.7,-0.1) -- (-6.15,0.25) -- (-5.8,0.75) -- (-5.25,0.5) -- (-4.9,0.95) -- (-4.3,0.7);
  \fill (-4.3,0.7) circle (0.05);
  \node[above] at (-4.2,0.75) {$z_i + S_i$};
  \draw[decorate, decoration={brace, amplitude=5pt, mirror}] (-7.6,-2.85) -- (0,-2.85);
  \node[below] at (-3.8,-3.05) {$g_i/(2d)$};
  \draw[<->, black!60] (-7.6,0.08) -- (-7.6+1.3*0.0,1.3);
  \node[left] at (-7.65,0.75) {$\tau^{1/2}$};
\end{tikzpicture}
\caption{The linearized picture of a pair $(i, i + 1)$. The point $z_i = (- g_i/(2d), 0)$ encodes the gap $g_i$ of $H$. The resampling moves it by the linearized increment $S_i$, a sum of $m \approx \tau/4$ conditionally independent steps of order one, each with covariance close to $4 \Id_2$, whose law is close to a centered Gaussian law with covariance $\tau \Id_2$ (dashed circles of radius $\tau^{1/2}$ and $2 \tau^{1/2}$). Up to an error $o(d)$, the gap of $\widetilde H$ is at most $x d$ exactly when $z_i + S_i$ lies in the disk of radius $x/2$ around the origin; by \eqref{e:ov-heuristic}, this has probability close to $(x^2/(8\tau)) \exp(- g_i^2/(8 \ell^2))$.}
\label{f:ov-planar}
\end{figure}

To sum \eqref{e:ov-heuristic} over $i$, we need the number of gaps of $H$ at the scale $\ell$. For $\alpha = 99/200$, we have $N^{-1 - \alpha} = \ell$ and $N^{1 - 2\alpha} = \tau$, so \zcref{t:pairs} with this value of $\alpha$ counts the gaps of $H$ at the scale $\ell$. \zcref[S]{t:pairs} concerns continuous test functions with compact support. To apply it to the indicator function of a bulk interval $I$ and to the Gaussian function $y \mapsto e^{-y^2/8}$, we approximate these functions by continuous functions with compact support. We control the tail of the Gaussian function by an upper bound that we prove together with \zcref{t:pairs} (see \zcref{p:pairsl}). It states that for every bulk interval $I$, there exists $C_I > 0$ such that with probability tending to one,
\begin{equation}\label{e:ov-upper}
\sum_{1 \le i < j \le N : (\lambda_i + \lambda_j)/2 \in I} \Big( 1 + \frac{(\lambda_j - \lambda_i)^2}{\ell^2} \Big)^{-61} \le C_I \tau.
\end{equation}
We obtain
\begin{equation}\label{e:ov-pairs}
\frac{1}{\tau} \sum_{i : E_i \in I} \exp\Big( - \frac{g_i^2}{8 \ell^2} \Big) \longrightarrow 4 \int_I c(E) \dd E
\end{equation}
in probability, where we used that $\int_0^\infty y e^{-y^2/8} \dd y = 4$. Multiplying by $x^2/8$, we recover the mean \eqref{e:ov-mean}. We note that the dependence on $\tau$ cancels, as it must.

Summing \eqref{e:ov-heuristic} over all pairs requires care. The local limit theorem \eqref{e:ov-llt} has an error $o(\tau^{-1})$ for each pair, while the sums run over all pairs whose gap is small enough for the linearization to apply. These are the pairs with $g_i \le N^{-7/5}$ (see \zcref{s:ov-linearization}), and there are about $N^{1/5}$ of them, many more than $\tau$. We therefore restrict the sums to the pairs with $g_i \le R \ell$ for a large constant $R$, of which there are $O_R(\tau)$ by \eqref{e:ov-upper}. We treat the remaining pairs with a large-deviation bound. For $g_i \ge \ell$, the conditional probability that $|z_i + S_i| \le x/2$ is at most $C \tau^{-1} e^{-c g_i/\ell}$ (\zcref{p:llt}), and these bounds are summable against \eqref{e:ov-upper}, with a sum that tends to zero as $R \to \infty$.

To obtain a Poisson limit, we need the joint behavior of $S_{i_1}, \dots, S_{i_k}$ for indices $i_1 < \dots < i_k$ such that $i_{j + 1} \ge i_j + 2$ for all $j$, so that the pairs $(i_j, i_j + 1)$ are disjoint. This restriction suffices, since we show in \zcref{s:ov-linearization} that, with probability tending to one, the pairs with small gaps are disjoint. The vectors $S_{i_1}, \dots, S_{i_k}$ are built from the same selected positions and the same variables $Y_1, \dots, Y_m$, so they are not conditionally independent. However, their joint conditional covariance is close to $4 m \Id_{2k}$, again by the orthonormality of the eigenvectors, and we prove a joint local limit theorem for $(S_{i_1}, \dots, S_{i_k})$. Joint local limits, rather than independence, suffice to compute, through the Bonferroni inequalities, the conditional Laplace functional of the point process of the rescaled linearized gaps $2 |z_i + S_i|$, which approximate the rescaled gaps of $\widetilde H$. Finally, since $\widetilde H$ and $H$ have the same law, the Poisson limit for $\widetilde H$ is also the Poisson limit for $H$.

The choice of $\tau$ is a compromise. The local limit theorem needs many steps, so $\tau$ must tend to infinity. On the other hand, the error in the linearization grows with $\tau$, as we explain next. The choice $\tau = N^{1/100}$ is convenient, but not essential.

\section{Comparing actual and linearized gaps}\label{s:ov-linearization}

We now explain why the approximation $\tilde g_i \approx 2 d |z_i + S_i|$ holds up to an error $o(d)$, simultaneously for all relevant $i$. Let $\Delta = \widetilde H - H$, let $U$ denote the orthogonal matrix with columns $u_1, \dots, u_N$, and let $W = U^T \Delta U$ denote the perturbation in the eigenbasis of $H$. With probability tending to one, the matrix $\Delta$ has at most $2m$ nonzero entries, each of absolute value at most $2 N^{-1/2 + \eps}$, so the delocalization of the eigenvectors gives
\begin{equation}\label{e:ov-W}
\begin{gathered}
\max_{i, j} |W_{ij}| \le \sum_{a < b} |\Delta_{ab}| \, |u_i(a) u_j(b) + u_i(b) u_j(a)| \le 4 m N^{-3/2 + 3 \eps} \le w_N, \\
w_N = \tau N^{-3/2 + 3 \eps},
\end{gathered}
\end{equation}
where we used that $4m \le \tau$. This bound is much smaller than the typical spacing $N^{-1}$, but it is larger than $d$. So the first-order approximation is not enough, and we must control the interaction of each pair with the rest of the spectrum.

Let $s = N^{-7/5}$. We call a maximal set $\mathcal K \subset [N]$ of consecutive integers such that $\lambda_{j + 1} - \lambda_j \le s$ whenever $j, j + 1 \in \mathcal K$ a \emph{cluster}. By a result of Nguyen, Tao, and Vu \cite{nguyen2017random}, the expected number of indices $i \in [N - 2]$ with $\lambda_{i + 2} - \lambda_i \le r$ is $O(N (N r)^3)$, uniformly in $r \ge N^{-2}$. Applying this with $r = 4s$, we see that $\lambda_{i + 2} - \lambda_i > 4s$ for all $i$, and in particular that no cluster has three or more elements, with probability $1 - O(N (N s)^3) = 1 - O(N^{-1/5})$.

For each cluster $\mathcal K$, a Schur complement argument (\zcref{l:cluster}) shows that $H + \Delta$ has exactly $|\mathcal K|$ eigenvalues near the eigenvalues $\lambda_j$ with $j \in \mathcal K$, and that their ordered values differ from the ordered eigenvalues of the block $(\Lambda + W)_{\mathcal K \mathcal K}$, where $\Lambda = \diag(\lambda_1, \dots, \lambda_N)$, by at most
\begin{equation}\label{e:ov-schur}
\begin{gathered}
2 |\mathcal K| \Big( \max_{i, j} |W_{ij}| \Big)^2 \sup_{w} \sum_{j \notin \mathcal K} \frac{1}{|\lambda_j - w|} \le 4 w_N^2 \mathscr L_N = \frac{\delta_N}{2}, \\
\mathscr L_N = 2 s^{-1} \log N, \qquad \delta_N = 8 w_N^2 \mathscr L_N.
\end{gathered}
\end{equation}
Here the supremum is over the complex numbers $w$ in the rectangle obtained by enlarging the interval $[\lambda_{\min \mathcal K}, \lambda_{\max \mathcal K}]$ by $s/4$ in each direction. We show in \zcref{s:linearization} that the sum is bounded by $\mathscr L_N$, because every eigenvalue $\lambda_j$ with $j \notin \mathcal K$ is at distance at least $3s/4$ from such $w$, and at most two eigenvalues lie in every interval of length $4s$, since $\lambda_{j + 2} - \lambda_j > 4s$ for all $j$. Inserting the values of $\tau$, $s$, and $d$, we find
\begin{equation}\label{e:ov-deltaN}
\frac{\delta_N}{d} = 16 N^{-2/25 + 6\eps} \log N,
\end{equation}
which tends to zero when $\eps < 1/75$. In \zcref{s:linearization}, we take $\eps = 1/100$.

For a two-element cluster $\mathcal K = \{ i, i + 1 \}$, the block $(\Lambda + W)_{\mathcal K \mathcal K}$ is exactly the compression considered in \zcref{s:ov-resampling}, and by \eqref{e:ov-twobytwo} its eigenvalue gap equals $2 d |z_i + S_i|$ when the selected positions are distinct. Hence $\tilde g_i = 2 d |z_i + S_i| + o(d)$ for all two-element clusters at once, and all other gaps of $\widetilde H$ exceed $s/2$. The same estimate, applied with a Gaussian perturbation in place of $\Delta$, is used in \zcref{s:ov-pairs}.

\section{A local limit theorem at the scale of a single step}\label{s:ov-arithmetic}

The local limit theorem \eqref{e:ov-llt} is needed at unit scale, which is the size of a single increment $(Y - X_{ab}) D^{(i)}_{ab}$. At this resolution, the lack of smoothness of $X$ matters again. For example, if $X$ takes the values $\pm 1$, then $Y - X_{ab} \in \{ -2, 0, 2 \}$. If the linearized increments were supported on a fixed lattice with mesh bounded away from zero at unit resolution, the local limit theorem could fail. The fixed-lattice example at the beginning of \zcref{s:llt} illustrates this obstruction. The central limit theorem holds regardless, but the local limit theorem requires a quantitative upper bound on the modulus of the conditional characteristic function
\begin{equation}\label{e:ov-char}
\Phi_H(\xi) = \E \big[ \exp\big( \iu \xi \cdot (Y - X_{ab}) D^{(i)}_{ab} \big) \, \big| \, H \big], \qquad \xi \in \R^2,
\end{equation}
for $\xi$ in a range of intermediate frequencies, where $(a, b)$ is a uniformly chosen position and $Y$ is an independent copy of $X$. Small frequencies are handled by the Gaussian approximation, and large frequencies by the decay of the Fourier transform of $\psi$ in \eqref{e:ov-llt}.

The first step is a symmetrization, which removes the old entries $X_{ab}$ (whose values are fixed once we condition on $H$) and avoids any symmetry assumption on $X$. Let $Y'$ denote an independent copy of $Y$. Averaging over the uniformly chosen position $(a, b)$ and using the convexity of $z \mapsto |z|^2$, we find
\begin{equation}\label{e:ov-sym}
1 - |\Phi_H(\xi)|^2 \ge \binom{N}{2}^{-1} \sum_{a < b} \E \big[ 1 - \cos\big( (Y - Y') \, \xi \cdot D^{(i)}_{ab} \big) \big].
\end{equation}

For $b \in [N]$, let $r_b = N^{1/2} (u_i(b), u_{i + 1}(b)) \in \R^2$. For $\xi = (\xi(1), \xi(2)) \in \R^2$, let $\Gamma(\xi)$ denote the symmetric $2 \times 2$ matrix with rows $(\xi(1), \xi(2))$ and $(\xi(2), -\xi(1))$. A direct computation from \eqref{e:ov-D} gives $\xi \cdot D^{(i)}_{ab} = r_a^T \Gamma(\xi) r_b$. Suppose that the right side of \eqref{e:ov-sym} is small. Since $\Var X = 1$, the variable $|Y - Y'|$ lies in a fixed compact subset of $(0, \infty)$ with positive probability, and for such values of $Y - Y'$ the phases $(Y - Y') \, r_a^T \Gamma(\xi) r_b$ are close to $2 \pi \Z$ for most pairs $(a, b)$. We choose two indices $a', a''$ such that the $2 \times 2$ matrix $Q$ with rows $r_{a'}^T$ and $r_{a''}^T$ is well conditioned and such that most phases in these two rows are close to $2 \pi \Z$. Then for most $b$, there exists $\mathbf n_b \in \Z^2$ such that
\begin{equation}\label{e:ov-reconstruct}
(Y - Y') \, Q \Gamma(\xi) r_b = 2 \pi \mathbf n_b + (\text{small error}), \qquad \text{so that} \qquad r_b \approx \frac{2 \pi}{Y - Y'} \, \Gamma(\xi)^{-1} Q^{-1} \mathbf n_b .
\end{equation}
When $N^{-c} \le |\xi| \le N^{c}$ for a small constant $c > 0$, the vectors $\mathbf n_b$ range over a set of size at most $N^{C c}$. Consequently, $N^{1/2} u_i$ is close, in the norm $v \mapsto (N^{-1} \sum_{b} v(b)^2)^{1/2}$, to a vector whose coordinates take at most $N^{C c}$ distinct values. For $k$ pairs at once, the same argument applies with $2k$ indices in place of $a', a''$.

The second step excludes this. We show that with overwhelming probability, no delocalized unit eigenvector of $H$ is close, in this norm, to a vector whose coordinates take few distinct values (\zcref{l:fewvalues}). The proof replaces the eigenvector and its eigenvalue by nearby elements of a deterministic net of approximate eigenvectors and approximate eigenvalues, and it combines a union bound over this net with anticoncentration.

For a given vector $v$ and a given energy $E$ in the net, we split $[N]$ into a set of about $N/2$ indices on which $|v|$ is largest and its complement. For an index $a$ in the complement, the coordinate $N^{1/2} ((H - E) v)(a)$ equals the sum of $X_{ab} v(b)$ over the indices $b$ in the first set, plus a term that does not involve the entries $X_{\tilde a b}$ with $\tilde a$ in the complement and $b$ in the first set. These coordinates are independent over $a$ once the other entries are fixed, and each lies in a given short interval with small probability, by the Berry--Esseen theorem. If $(H - E) v$ is small, then at least $N/4$ of these coordinates lie in such intervals. The product of the corresponding probabilities is much smaller than the inverse of the size of the net, which is exponential in $N$ only through the number of possible assignments of the few distinct values. This part follows the net and decoupling arguments for structured approximate eigenvectors in \cite{rudelson2008littlewood,vershynin2014invertibility,nguyen2017random}.

\section{Gap counts at the scales \texorpdfstring{$N^{-1 - \alpha}$}{N\textasciicircum(-1-alpha)}}\label{s:ov-pairs}

It remains to explain \zcref{t:pairs}, which supplies \eqref{e:ov-pairs}. For $\alpha \in (0, 1/2)$, let $\ell_\alpha = N^{-1 - \alpha}$ and $M_\alpha = N^{1 - 2\alpha}$. \zcref[S]{t:pairs} states that for every $y > 0$ and every bulk interval $I$, the number of gaps $g_i \le y \ell_\alpha$ with $E_i \in I$ is asymptotic to $M_\alpha (y^2/2) \int_I c(E) \dd E$. At the scale $\ell_\alpha$, a single entry moves a gap by about $N^{-3/2} = o(\ell_\alpha)$, so comparison arguments are available in principle. We use the Ornstein--Uhlenbeck interpolation
\begin{equation}\label{e:ov-ou}
H_t = e^{-t/2} H + (1 - e^{-t})^{1/2} G, \qquad t \in [0, 1],
\end{equation}
where $G$ is a matrix from the Gaussian orthogonal ensemble (GOE), independent of $H$ and normalized so that its off-diagonal entries have variance $1/N$ and its diagonal entries have variance $2/N$.

We work with smoothed pair statistics. Let $p \ge 2$ be an integer. We define $f_p(w) = (1 + w^2)^{-p}$ for $w \in \C \setminus \{ \iu, -\iu \}$. Let $\varphi$ be a smooth nonnegative function with compact support in $(-2, 2)$, and let $\varphi_N$ denote the convolution of $\varphi$ with the centered Gaussian density of standard deviation $N^{-1/10}$. The function $\varphi_N$ extends to an entire function, which we also denote by $\varphi_N$. Let $\omega_p > 0$ be a small constant depending on $p$. For an $N \times N$ complex matrix $A$ whose eigenvalues $w_1, \dots, w_N$, listed with algebraic multiplicity, satisfy $|\Im w_j| < \omega_p \ell_\alpha$ for all $j$, and for $\theta > 0$, we set
\begin{equation}\label{e:ov-F}
\mathcal X_\alpha(A) = \sum_{1 \le i < j \le N} \varphi_N \Big( \frac{w_i + w_j}{2} \Big) f_p \Big( \frac{w_j - w_i}{\ell_\alpha} \Big), \qquad F(A) = \exp\Big( - \frac{\theta \mathcal X_\alpha(A)}{M_\alpha} \Big).
\end{equation}
The sum does not depend on the order of the eigenvalues, since $f_p$ is even. We call $\mathcal X_\alpha(A)$ the \emph{pair statistic} of $A$. (In \zcref{s:comparison}, it is denoted $\mathcal X_\alpha(A; \varphi_N, f_p)$.) For a real symmetric matrix $A$, it is a weighted count of the pairs of eigenvalues at distance of order $\ell_\alpha$. We will show that
\begin{equation}\label{e:ov-lln}
\frac{\mathcal X_\alpha(H)}{M_\alpha} \longrightarrow \int_0^\infty y f_p(y) \dd y \cdot \int_{-2}^2 \varphi(E) c(E) \dd E
\end{equation}
in probability, for all sufficiently large $p$ and all $\varphi$ as above.

For $p \ge 2$, the pairs $(i, j)$ with $j \ge i + 2$ and the pairs at distance much larger than $\ell_\alpha$ contribute $o(M_\alpha)$ to $\mathcal X_\alpha(H)$. Further, linear combinations of the functions $f_p$ approximate every continuous function on $[0, \infty)$ with compact support, and the smoothing of $\varphi$ can be removed. Hence \eqref{e:ov-lln} implies \zcref{t:pairs} at the scale $\ell_\alpha$. We carry out this deduction in the proof of \zcref{t:pairs} in \zcref{s:pairs-iteration}.

\paragraph{Base case.} For $\alpha < 1/8$, we use a coupling of Bourgade \cite[Corollary~3.3]{bourgade2022extreme}. Fix $c_* \in (0, 1/2)$. For $N^{-1 + \eps} \le t < 1$, there exists a coupling of $H_t$ with a GOE matrix $G'$ such that, with overwhelming probability, the gaps $\lambda_{i + 1}(H_t) - \lambda_i(H_t)$ and $\lambda_{i + 1}(G') - \lambda_i(G')$ differ by at most $N^{\eps}/(N^2 t)$ for all $i$ with $c_* N \le i \le (1 - c_*) N$ (\zcref{p:coupling}). We take $t = T$, where $T = N^{-3/4 - \alpha}$. Then this error is $N^{-5/4 + \alpha + \eps}$, which is $o(\ell_\alpha)$ when $\alpha < 1/8$, so \eqref{e:ov-lln} with $H_T$ in place of $H$ follows from the corresponding statement for the GOE. We prove the latter in \zcref{s:goe} from the Pfaffian formulas for the GOE correlation functions. It then remains to compare $H_T$ with $H$.

\paragraph{Comparison.} We compare the Laplace transforms $\E F(H_t)$ and $\E F(H)$. A comparison requires bounds on derivatives of $F$ with respect to the entries of the matrix. Derivatives of eigenvalues of order two and higher contain denominators $(\lambda_i(A) - \lambda_j(A))^{-1}$, so bounds obtained from perturbation formulas would require control of small gaps, which is the information we are trying to obtain.

We avoid this through holomorphy. For $a < b$, let $V^{ab} = e_a e_b^T + e_b e_a^T$. Let $A$ be a real symmetric matrix such that $\|A\| \le 3$, every unit eigenvector of $A$ has all coordinates bounded by $N^{-1/2 + \eps}$, and every interval of length $N^{-1}$ contains at most $N^{\eps}$ eigenvalues of $A$. By the local semicircle law \cite{erdos2012rigidity,benaychgeorges2016lectures}, these conditions hold for $H_t$ with overwhelming probability (\zcref{p:regularity}). We show that there exists $c = c_p > 0$ such that, for all sufficiently large $N$ and every complex $\zeta$ with $|\zeta| \le c N^{-\alpha - 4\eps}$, all eigenvalues $w$ of $A + \zeta V^{ab}$ satisfy $|\Im w| < \omega_p \ell_\alpha$ (\zcref{l:strip}). Hence $F(A + \zeta V^{ab})$ is defined. Since $\mathcal X_\alpha$ is a symmetric function of the eigenvalues, $F(A + \zeta V^{ab})$ is a holomorphic function of $\zeta$ (\zcref{l:holomorphic}). Since the imaginary parts of the eigenvalues are much smaller than $N^{-1/5}$, the square of the smoothing scale $N^{-1/10}$ of $\varphi_N$, and $\omega_p$ is small, every term of the sum in \eqref{e:ov-F} has nonnegative real part (\zcref{l:smoothing,l:holomorphic}). Therefore $|F(A + \zeta V^{ab})| \le 1$, and Cauchy's estimates give, for all real $\zeta$ with $|\zeta| \le c N^{-\alpha - 4\eps}/2$ and all integers $n \ge 1$,
\begin{equation}\label{e:ov-cauchy}
\Big| \frac{\dd^n}{\dd \zeta^n} F(A + \zeta V^{ab}) \Big| \le n! \, \Big( \frac{c N^{-\alpha - 4 \eps}}{2} \Big)^{-n}.
\end{equation}

No information about gaps enters this bound. With \eqref{e:ov-cauchy}, a Taylor expansion in one entry at a time along the interpolation \eqref{e:ov-ou}, in the style of the continuity estimate of Bourgade and Yau \cite{bourgade2017eigenvector}, gives for every $D > 0$
\begin{equation}\label{e:ov-comparison}
\big| \E F(H_t) - \E F(H) \big| \le C t N^{1/2 + 3\alpha + 12 \eps} + C_D N^{-D}.
\end{equation}
The factor $N^{1/2 + 3\alpha + 12\eps}$ comes from summing third-order Taylor terms, each of size $N^{-3/2} \cdot N^{3(\alpha + 4\eps)}$, over the $N^2$ entries. We do not assume that $\E X^3 = 0$. For $t = T$, the right side of \eqref{e:ov-comparison} is $C N^{-1/4 + 2\alpha + 12\eps} + C_D N^{-D}$, which tends to zero when $\alpha < 1/8$. Since this holds for all $\theta > 0$, the statement \eqref{e:ov-lln} for $H_T$ implies \eqref{e:ov-lln} for $H$. This proves \zcref{t:pairs} for $\alpha < 1/8$.

\paragraph{Induction on $\alpha$.} Let $\alpha$ and $\alpha'$ satisfy $1/10 \le \alpha < \alpha' < 1/2$, and suppose that \eqref{e:ov-lln} holds. To prove \eqref{e:ov-lln} with $\alpha'$ in place of $\alpha$, we use the Gaussian component of $H_t$ to transfer information from the scale $\ell_\alpha$ to the smaller scale $\ell_{\alpha'}$, and then compare back. Let $t_\alpha = N \ell_\alpha^2 = N^{-1 - 2\alpha}$, let $U$ and $\Lambda$ be as in \zcref{s:ov-linearization}, and let $W' = (1 - e^{-t_\alpha})^{1/2} U^T G U$. Then
\begin{equation}\label{e:ov-Ht}
U^T H_{t_\alpha} U = e^{-t_\alpha/2} \Lambda + W',
\end{equation}
so $H_{t_\alpha}$ has the same eigenvalues as $e^{-t_\alpha/2} \Lambda + W'$. (In \zcref{s:pairs}, the matrices $W'$ and $e^{-t_\alpha/2} \Lambda$ are denoted $W$ and $D_y$.) By the orthogonal invariance of $G$, conditionally on $H$, the matrix $W'$ is a GOE matrix multiplied by $(1 - e^{-t_\alpha})^{1/2}$, and its entries are of order $\ell_\alpha$.

We group the eigenvalues of $H$ into clusters as in \zcref{s:ov-linearization}, with the threshold $q = N^{-1 - \kappa}$ in place of $s$, where $\kappa = \alpha - \alpha'/6$. We apply the estimate \eqref{e:ov-schur} with $e^{-t_\alpha/2} \Lambda$, $W'$, and $q$ in place of $\Lambda$, $W$, and $s$. Since the entries of $W'$ are bounded by $N^{\eps} \ell_\alpha$ with overwhelming probability, it shows that for each two-element cluster $\mathcal K = \{ i, i + 1 \}$, the gap of $H_{t_\alpha}$ equals the eigenvalue gap of the block $(e^{-t_\alpha/2} \Lambda + W')_{\mathcal K \mathcal K}$ up to an error of order $N^{C \eps} \ell_\alpha^2/q$. By \eqref{e:ov-twobytwo}, the gap of this block equals
\begin{equation}\label{e:ov-block}
\big| \big( e^{-t_\alpha/2} g_i + W'_{i + 1, i + 1} - W'_{ii}, \, 2 W'_{i, i + 1} \big) \big| .
\end{equation}

Here $W'_{i + 1, i + 1} - W'_{ii}$ and $2 W'_{i, i + 1}$ are independent centered Gaussian variables with variance $4 (1 - e^{-t_\alpha})/N = 4 \ell_\alpha^2 (1 + o(1))$, and the blocks of different clusters are conditionally independent. Therefore, conditionally on $H$, the number of gaps of the blocks at the scale $\ell_{\alpha'}$, which approximates the corresponding number for $H_{t_\alpha}$, is a sum of independent bounded variables. Its conditional mean is a Gaussian smoothing, at the scale $\ell_\alpha$, of the gap counts of $H$ at the scale $\ell_\alpha$. These counts are known by the induction hypothesis, so we obtain \eqref{e:ov-lln} with $\alpha'$ in place of $\alpha$ for $H_{t_\alpha}$. The comparison \eqref{e:ov-comparison}, with $\alpha'$ in place of $\alpha$ and $t = t_\alpha$, transfers it to $H$ provided that $t_\alpha N^{1/2 + 3\alpha'} = N^{-1/2 - 2\alpha + 3\alpha'}$ tends to zero.

Two further conditions are needed. We have
\begin{equation}\label{e:ov-induction-errors}
\frac{\ell_\alpha^2}{q} = N^{-1 - 2\alpha + \kappa}, \qquad N (N q)^3 = N^{1 - 3\kappa}.
\end{equation}
The error in the Schur complement estimate, which is of order $\ell_\alpha^2/q$ up to factors $N^{C \eps}$, must be $o(\ell_{\alpha'})$, which holds when $\kappa < 2\alpha - \alpha'$. Clusters with three or more elements must be negligible. By the bound of Nguyen, Tao, and Vu, their expected number is $O(N (N q)^3)$, which must be $o(M_{\alpha'})$, and this holds when $3\kappa > 2\alpha'$. For $\kappa = \alpha - \alpha'/6$, both conditions are equivalent to $\alpha' < 6\alpha/5$. All conditions therefore hold when
\begin{equation}\label{e:ov-constraints}
\frac{1}{10} \le \alpha < \alpha' < \min\Big\{ \frac{6\alpha}{5}, \, \frac{1}{6} + \frac{2\alpha}{3} \Big\}.
\end{equation}
The bound $p \ge 61$, for which \eqref{e:ov-lln} is proved, and the lower bound $\alpha \ge 1/10$ are used only in the bound \eqref{e:pairs-cross-bound} in the proof of \zcref{p:induction}, where $2p(\alpha' - \kappa) > 1 + 2\alpha'$ is needed and holds because $\alpha' - \kappa > \alpha/6 \ge 1/60$. Starting from $\alpha = 1/10$, which is covered by the base case, finitely many steps reach every $\alpha < 1/2$, as shown in \zcref{f:ov-induction}.

\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{axis}[
  width=0.62\textwidth, height=0.5\textwidth,
  xmin=0.08, xmax=0.52, ymin=0.08, ymax=0.52,
  xlabel={$\alpha$}, ylabel={$\alpha'$},
  xtick={0.1,0.125,0.3125,0.5}, xticklabels={$\tfrac{1}{10}$,$\tfrac18$,$\tfrac{5}{16}$,$\tfrac12$},
  ytick={0.1,0.2,0.3,0.4,0.5},
  axis lines=left, tick label style={font=\small}, label style={font=\small},
  clip=true]
  \addplot[name path=diag, black!60, domain=0.08:0.52] {x};
  \addplot[name path=upper, black, thick, domain=0.08:0.52, samples=200] {min(6*x/5, 1/6 + 2*x/3)};
  \addplot[black!12] fill between[of=diag and upper, soft clip={domain=0.1:0.5}];
  \addplot[densely dashed, black!50, domain=0.08:0.52] {6*x/5};
  \addplot[densely dashed, black!50, domain=0.08:0.52] {1/6 + 2*x/3};
  \addplot[black, mark=*, mark size=1.2pt, const plot] coordinates {
    (0.100000,0.110000) (0.110000,0.121000) (0.121000,0.133100) (0.133100,0.146410) (0.146410,0.161051) (0.161051,0.177156) (0.177156,0.194872) (0.194872,0.214359) (0.214359,0.235795) (0.235795,0.259374) (0.259374,0.283437) (0.283437,0.305093) (0.305093,0.324584) (0.324584,0.342125) (0.342125,0.357913) (0.357913,0.372122) (0.372122,0.384909) (0.384909,0.396419) (0.396419,0.406777) (0.406777,0.416099)};
  \node[font=\small, anchor=west] at (axis cs:0.33,0.47) {$\alpha' = \tfrac16 + \tfrac{2\alpha}{3}$};
  \node[font=\small, anchor=west] at (axis cs:0.16,0.29) {$\alpha' = \tfrac{6\alpha}{5}$};
  \node[font=\small, anchor=west] at (axis cs:0.40,0.355) {$\alpha' = \alpha$};
\end{axis}
\end{tikzpicture}
\caption{The induction on $\alpha$. The shaded region is the set of pairs $(\alpha, \alpha')$ allowed by \eqref{e:ov-constraints}. The staircase shows the first steps of the sequence $\alpha^{(n + 1)} = \alpha^{(n)} + \min\{ \alpha^{(n)}, 1/2 - \alpha^{(n)} \}/10$ starting from $\alpha^{(0)} = 1/10$, which increases to $1/2$; the base case covers all $\alpha < 1/8$.}
\label{f:ov-induction}
\end{figure}

\section{Hermitian matrices}\label{s:ov-hermitian}

In this section only, $H$ denotes a complex Hermitian Wigner matrix as in \zcref{d:hermitian}, with eigenvalues $\lambda_1 \le \dots \le \lambda_N$ and gaps $g_i = \lambda_{i + 1} - \lambda_i$, and the matrix $G$ in \eqref{e:ov-ou} is taken from the Gaussian unitary ensemble (GUE), normalized so that $\E |G_{ab}|^2 = 1/N$ and $\E G_{ab}^2 = 0$ for $a < b$ and its diagonal entries have variance $1/N$. The smallest gaps are of order $N^{-4/3}$, which is the critical scale in this case, while a single entry moves a gap by only about $N^{-3/2}$. Comparison at the critical scale is therefore possible, and no resampling is needed. The scale $N^{-4/3}$ corresponds to $\alpha = 1/3$, and the analogue of \eqref{e:ov-comparison} at this scale has the right side
\begin{equation}\label{e:ov-herm-comparison}
C t N^{1/2} \big( N \cdot N^{-4/3} \big)^{-3} N^{C\eps} = C t N^{3/2 + C\eps}
\end{equation}
when the third moments of the entries do not vanish. So the Gaussian component must be taken at a time $t = N^{-1 - 2\alpha_1}$ with $\alpha_1 > 1/4$. We also take $\alpha_1 < 1/3$, so that the entries of the Gaussian part of $H_t$, which are of order $N^{-1 - \alpha_1}$, are larger than the critical scale.

An induction on $\alpha$ as in \zcref{s:ov-pairs}, with Gaussian vectors in $\R^3$ in place of $\R^2$ in \eqref{e:ov-block}, gives the Hermitian analogue of \eqref{e:ov-lln}. With $c_2(E)$ as in \zcref{t:hermitian}, it states that $\mathcal X_\alpha(H)/N^{1 - 3\alpha}$ converges in probability to $\int_0^\infty y^2 f_p(y) \dd y \cdot \int_{-2}^2 \varphi(E) c_2(E) \dd E$. We prove this along a finite chain of exponents that starts below $1/8$ and ends at $\alpha_1 = 8/25$, which is all that the argument needs; the induction step differs from the real case in its treatment of triples of eigenvalues, described below. At the time $t = N^{-1 - 2\alpha_1}$, the $2 \times 2$ block computation then gives the statistics of the gaps of $H_t$ at the scale $N^{-4/3}$, provided that clusters of three eigenvalues are under control and clusters of four or more eigenvalues are absent.

This is where nonzero third moments cause difficulty. We again group the eigenvalues into clusters, at a threshold $q = N^{-1 - \kappa}$ with $\kappa > 0$ to be chosen. For the GUE, the three-point correlation function vanishes quadratically in each of the three distances between its arguments, so the expected number of indices $i$ with $\lambda_{i + 2} - \lambda_i \le q$ and $\lambda_i$ in a bulk interval is of order $N (N q)^{2 + 6} = N^{1 - 8\kappa}$. Excluding such triples therefore requires $\kappa > 1/8$. However, transferring information at the scale $N^{-1 - \kappa}$ from the GUE to $H$ through Bourgade's coupling and the comparison estimate requires $\kappa < 1/8$.

Instead of excluding triples, we show that they rarely produce small gaps. For $\kappa < 1/8$, the number of indices $i$ with $\lambda_{i + 2} - \lambda_i \le N^{-1 - \kappa}$ and $\lambda_i$ in a bulk interval is at most $N^{1 - 8\kappa + C\eps}$ with probability tending to one. Further, a Gaussian perturbation at time $t = N^{-1 - 2\alpha_1}$ turns a given triple into a gap of order $N^{-4/3}$ with probability at most $N^{C \eps} (N^{-4/3}/N^{-1 - \alpha_1})^3 = N^{-1 + 3\alpha_1 + C\eps}$. The expected number of small gaps produced by triples is therefore bounded by
\begin{equation}\label{e:ov-herm-triples}
N^{1 - 8\kappa + C\eps} \cdot N^{-1 + 3\alpha_1 + C\eps} = N^{3\alpha_1 - 8\kappa + 2C\eps},
\end{equation}
which tends to zero when $\kappa > 3\alpha_1/8$. This is compatible with $\kappa < 1/8$ because $\alpha_1 < 1/3$. Clusters of four or more eigenvalues within distance $N^{-1 - \kappa}$ are absent with probability tending to one when $\kappa > 1/15$, by the analogous count of quadruples. A similar bound on triples is used in the steps of the induction. We sketch the details in \zcref{s:hermitian}, where we take $\alpha_1 = 8/25$.

\section{Organization of the paper}\label{s:ov-organization}

\zcref[S]{s:prelim} collects estimates on eigenvalues and eigenvectors from the literature, a perturbation lemma for matrices that are entrywise small perturbations of diagonal matrices, and an estimate for Gaussian kernels. In \zcref{s:reduction}, we state three propositions (the gap counts at the scale $\ell$, the local limit theorem for linearized increments, and the comparison of actual and linearized gaps) and deduce \zcref{t:main} from them. \zcref[S]{s:comparison} contains the holomorphic comparison estimate, which is the main input to \zcref{s:pairs}, where we prove \zcref{t:pairs} and \zcref{p:pairsl} through the base case and the induction on $\alpha$. In \zcref{s:linearization}, we compare actual and linearized gaps, and in \zcref{s:llt}, we prove the local limit theorem. \zcref[S]{s:hermitian} sketches the proof of \zcref{t:hermitian}. \zcref[S]{s:goe} computes the gap counts of the GOE needed for the base case.