\chapter{Introduction}\label{s:intro}

\section{Background}\label{s:intro-background}

The statistics of spacings between consecutive energy levels have been studied since Wigner proposed to model the spectra of heavy nuclei by the eigenvalues of large random matrices \cite{wigner1951statistical,wigner1957results,wigner1967random}. A basic feature of such spectra is the rarity of very close neighboring levels, which is known as \emph{level repulsion}. The origin of level repulsion goes back to an observation of von Neumann and Wigner \cite{vonneumann1929verhalten}. Real symmetric matrices with a repeated eigenvalue form a set of codimension two, and complex Hermitian matrices with a repeated eigenvalue form a set of codimension three. Heuristically, the probability that a given spacing of a random matrix is smaller than $s$ therefore behaves like $s^{\beta + 1}$, where $\beta = 1$ in the real symmetric case and $\beta = 2$ in the complex Hermitian case, and the density of spacings vanishes like $s^\beta$ near zero \cite{guhr1998random,haake2018quantum}. The same exponent appears in Wigner's surmise for the spacing distribution \cite{wigner1957results} and in the factor $\prod_{i < j} |\lambda_i - \lambda_j|^\beta$ in the eigenvalue densities of Dyson's ensembles \cite{dyson1962statistical,dyson1962threefold,mehta2004random}.

The real symmetric case corresponds to systems with time-reversal symmetry. For such systems, Bohigas, Giannoni, and Schmit conjectured that quantum spectra whose classical dynamics is chaotic share the local statistics of the Gaussian orthogonal ensemble (GOE) \cite{bohigas1984characterization}, in contrast with the Poisson statistics, without repulsion, expected for integrable systems \cite{berry1977level}. This picture is supported by experimental data (see, for example, \cite{haq1982fluctuation,stockmann1990quantum} for nuclear spectra and microwave billiards).

This paper concerns the smallest spacings in a spectrum of $N$ levels. If the spacings were independent, with a density proportional to $s^\beta$ near zero, then the smallest spacing would be of order $N^{-1/(\beta + 1)}$ times the mean spacing, and the smallest spacings, suitably rescaled, would form a Poisson process with an intensity proportional to $s^\beta \dd s$. This heuristic is described in \cite{forrester2010log,smaczynski2013extremal}, and numerical simulations agree with it \cite{lecaer2007nearest,smaczynski2013extremal}. (We emphasize that this Poisson process describes the rare small spacings. Its intensity vanishes at zero because of level repulsion, and it should not be confused with the Poisson level statistics of integrable systems.) Smallest gaps also appear in numerical linear algebra. For example, the rate at which the Toda flow, which is closely related to the QR algorithm, diagonalizes a symmetric matrix is governed by its smallest eigenvalue gap \cite{deift1983ordinary,deift2008some,benarous2013extreme}, and numerical experiments suggest that the resulting running times are universal for Wigner matrices, including matrices with discrete entries \cite{pfrang2014how}.

We call real symmetric or complex Hermitian random matrices with independent entries on and above the diagonal \emph{Wigner matrices}. The class considered in this paper is specified in \zcref{d:wigner} below. In mathematics, the local eigenvalue statistics of Wigner matrices are now known to be universal, in the sense that under mild conditions on the entry distributions they agree asymptotically with those of the Gaussian ensembles. This was established through the work of many authors. We mention \cite{johansson2001universality,erdos2010bulk,erdos2011universality,tao2011random,erdos2011universalitybernoulli,erdos2012bulk} for the averaged bulk statistics, \cite{tao2013asymptotic,erdos2015gap} for the distribution of a single gap, and \cite{bourgade2016fixed,landon2017convergence,landon2019fixed} for statistics at a fixed energy, and we refer to the book \cite{erdos2017dynamical} for an account of the dynamical approach. Matrices with discrete entries are natural in this context. Wigner's first paper on the semicircle law concerns real symmetric matrices whose off-diagonal entries are $\pm 1$ with random signs \cite{wigner1955characteristic}, and the centered and rescaled adjacency matrix of a dense Erd\H{o}s--R\'enyi graph has independent Bernoulli entries above the diagonal.

Universality of a single gap does not determine the smallest gap among $N$ gaps, since the latter depends on rare events. For the classical ensembles, the smallest gaps have been computed from explicit formulas for the correlation functions. When these functions are determinants, as for the circular unitary ensemble (CUE) and the Gaussian unitary ensemble (GUE), Vinson found the limiting law of the smallest gap \cite{vinson2001closest}. Building on Soshnikov's analysis of extreme spacings in determinantal point processes \cite{soshnikov2005statistics}, Ben Arous and Bourgade \cite{benarous2013extreme} obtained the joint limit of the smallest gaps of the CUE and the GUE together with their locations, and Figalli and Guionnet carried these results over to several-matrix models \cite{figalli2016universality}. Determinantal structure is also behind the results of Shi and Jiang for complex Ginibre matrices \cite{shi2012smallest} and those of Charlier for the two-dimensional Coulomb gas at $\beta = 2$ with a general potential, which describes the eigenvalues of random normal matrices \cite{charlier2025smallest}.

For the Gaussian orthogonal ensemble (GOE), the correlation functions are Pfaffians, and the smallest gaps were determined by Feng, Tian, and Wei \cite{feng2019small}. Feng, Li, and Yao \cite{feng2024small} gave a detailed treatment of the Gaussian symplectic ensemble, together with a sketch of a GOE version that also records the locations of the gaps, and the smallest gaps of real Gaussian matrices were studied in \cite{lopatto2026smallest}. Feng and Wei \cite{feng2021small} treated the circular $\beta$-ensembles for every positive integer $\beta$. For the largest gaps of the CUE and the GUE, Ben Arous and Bourgade \cite{benarous2013extreme} determined the leading order, and Feng and Wei \cite{feng2025large} determined the fluctuations.

For Wigner matrices, Bourgade \cite{bourgade2022extreme} proved universality of the smallest gaps for both real symmetric and complex Hermitian generalized Wigner matrices under smoothness assumptions on the entry distributions. Bourgade also proved universality of the largest gaps in the complex Hermitian case under a smoothness assumption. Landon, Lopatto, and Marcinek \cite{landon2020comparison} proved a comparison theorem that removes this smoothness assumption. In both works, the explicit largest-gap limit laws were stated for complex Hermitian matrices because the corresponding limiting distribution for the GOE was not known at the time; their comparison arguments also apply to real symmetric matrices. Whether the smallest gaps are universal without smoothness assumptions was raised in \cite{bourgade2022extreme}. Recently, Zhang \cite{zhang2026quantitative} proved universality of the smallest gaps for complex Hermitian Wigner matrices whose entries have real and imaginary parts with sufficiently many well-separated atoms (at least ten suffice), through a comparison argument that matches many moments of the entries. In \cite[Conjecture 1.6]{zhang2026quantitative}, he conjectured the analogous result for real symmetric matrices, noting that it does not seem accessible by his methods, and he further conjectured that two atoms suffice.

For real symmetric matrices with discrete entries, the strongest results available were lower bounds. Nguyen, Tao, and Vu \cite{nguyen2017random} proved tail bounds for gaps that imply, in the normalization of this paper, that the smallest gap is at least $N^{-2 - o(1)}$ with probability tending to one, and related bounds were obtained in \cite{lopatto2021tail,campos2024least,song2025eigenvalue}. The absence of repeated eigenvalues was proved in \cite{tao2017random,luh2020sparse,campos2024least}. These lower bounds are smaller than the true order $N^{-3/2}$ of the smallest gaps by a factor $N^{1/2}$.

\section{Main results}\label{s:intro-results}

We now state our results. We begin with the definition of the matrices we consider.

\begin{definition}\label{d:wigner}
Let $X$ and $Z$ be real random variables such that $\E X = \E Z = 0$, $\E X^2 = 1$, and $\E Z^2 > 0$, and such that there exist constants $c_0, C_0 > 0$ with
\begin{equation}\label{e:subgaussian}
\Prob(|X| > y) + \Prob(|Z| > y) \le C_0 e^{-c_0 y^2} \qquad \text{for all } y \ge 0.
\end{equation}
For every integer $N \ge 2$, let $(X_{ab})_{1 \le a < b \le N}$ be independent copies of $X$, and let $(Z_a)_{1 \le a \le N}$ be independent copies of $Z$, independent of the $X_{ab}$. We call the $N \times N$ real symmetric matrix $H = H_N$ with entries
\begin{equation}\label{e:Hentries}
H_{ab} = H_{ba} = N^{-1/2} X_{ab} \quad (1 \le a < b \le N), \qquad H_{aa} = N^{-1/2} Z_a \quad (1 \le a \le N),
\end{equation}
the \emph{real symmetric Wigner matrix} associated with $X$ and $Z$.
\end{definition}

No assumption on the symmetry, smoothness, or support of the laws of $X$ and $Z$ is made beyond \eqref{e:subgaussian}. In particular, $X$ may take only two values, and $\E X^3$ may be nonzero; for example, $X$ may be a centered and normalized Bernoulli variable with any parameter. The law of $Z$ may equal that of $X$, in which case all entries on and above the diagonal are identically distributed.

For a matrix $H$ as in \zcref{d:wigner}, let $\lambda_1 \le \dots \le \lambda_N$ denote its eigenvalues. For every positive integer $n$, we set $[n] = \{ 1, \dots, n \}$. For $i \in [N - 1]$, we let $g_i = \lambda_{i + 1} - \lambda_i$ denote the $i$-th gap. We let
\begin{equation}\label{e:cE}
\rhosc(E) = \frac{1}{2\pi} \sqrt{\max\{ 4 - E^2, 0 \}}, \qquad c(E) = \frac{\pi^2}{6} \rhosc(E)^3 \qquad (E \in \R)
\end{equation}
denote the semicircle density and the function that determines the intensity in our main result. We identify a locally finite point configuration on $[0, \infty) \times (-2, 2)$ with its counting measure. We equip the set of such measures with the \emph{vague topology}, which we define as the coarsest topology for which the maps $\mu \mapsto \int f \dd \mu$ are continuous for all continuous functions $f$ with compact support in $[0, \infty) \times (-2, 2)$. We use the convention that the minimum over the empty set is $+\infty$. Our main result is the following theorem.

\begin{theorem}\label{t:main}
Let $H$ be as in \zcref{d:wigner}. Then the point processes
\begin{equation}\label{e:XiN}
\Xi_N = \sum_{i \in [N - 1] : \lambda_i \in (-2, 2)} \delta_{(N^{3/2} g_i, \, \lambda_i)}
\end{equation}
on $[0, \infty) \times (-2, 2)$ converge in distribution, with respect to the vague topology, to a Poisson point process with intensity
\begin{equation}\label{e:nu}
\nu(\dd x \dd E) = c(E) \, x \dd x \dd E.
\end{equation}
Consequently, for every nonempty interval $I$ whose closure is contained in $(-2, 2)$ and every $x \ge 0$,
\begin{equation}\label{e:minimum}
\lim_{N \to \infty} \Prob\Big( N^{3/2} \min_{i \in [N - 1] : \lambda_i \in I} g_i > x \Big) = \exp\Big( - \frac{\pi^2 x^2}{12} \int_I \rhosc(E)^3 \dd E \Big).
\end{equation}
\end{theorem}

\zcref[S]{t:main} identifies the smallest gaps in the bulk of the spectrum for every matrix as in \zcref{d:wigner}, including Bernoulli matrices. It settles \cite[Conjecture 1.6]{zhang2026quantitative}, together with the stronger version with two atoms, for the matrices of \zcref{d:wigner}. The conjecture concerns real generalized Wigner matrices whose entries have tails at most $c^{-1} e^{-x^c}$ and laws supported on sufficiently many well-separated atoms (two atoms in the stronger version), whereas \zcref{t:main} requires sub-Gaussian entries, $\E Z^2 > 0$, and identically distributed entries above the diagonal. The limit agrees with the one obtained for the GOE in \cite{feng2019small,feng2024small} and with the one obtained under smoothness assumptions in \cite{bourgade2022extreme}. For every interval $I$ of positive length whose closure is contained in $(-2, 2)$, the Poisson limit determines the joint limiting law of the $k$ smallest gaps in $I$ for every $k$. We assign $+\infty$ to the $k$-th smallest gap when fewer than $k$ gaps have left endpoints in $I$. For example, as in \cite[Corollary 1.3]{bourgade2022extreme}, which gives the limiting law of the $k$-th smallest gap, the $k$-th smallest value of $N^{3/2} (2^{-1} \int_I c(E) \dd E)^{1/2} g_i$ over $i$ with $\lambda_i \in I$ converges in law to the distribution with density $2 x^{2k - 1} e^{-x^2}/(k - 1)!$ on $(0, \infty)$.

We have not attempted to optimize the assumptions. The sub-Gaussian condition \eqref{e:subgaussian} is used in several places, for instance through the results of \cite{nguyen2017random}. The condition $\E Z^2 > 0$ is used through results on generalized Wigner matrices \cite{erdos2012rigidity,bourgade2022extreme}, which assume that all entry variances are bounded below by a multiple of $N^{-1}$; in particular, \zcref{t:main} does not apply directly to centered adjacency matrices of random graphs, whose diagonal vanishes. We assume that the entries above the diagonal are identically distributed because the resampling argument of this paper and the results of \cite{nguyen2017random} use this property.

Our second result is a law of large numbers for the number of gaps at scales between the smallest gaps and the typical spacing. It is the main input to the proof of \zcref{t:main} and may be of independent interest. We let $E_i = (\lambda_i + \lambda_{i + 1})/2$ denote the midpoint of the $i$-th gap. We extend every function on $(-2, 2)$ by zero to a function on $\R$, and every function on $[0, \infty) \times (-2, 2)$ by zero to a function on $[0, \infty) \times \R$.

\begin{theorem}\label{t:pairs}
Let $H$ be as in \zcref{d:wigner}, and let $\alpha \in (0, 1/2)$. Then for every continuous function $\varphi \colon (-2, 2) \to \R$ with compact support and every continuous function $f \colon [0, \infty) \to \R$ with compact support,
\begin{equation}\label{e:pairs}
N^{2\alpha - 1} \sum_{i = 1}^{N - 1} \varphi(E_i) \, f\big( N^{1 + \alpha} g_i \big) \longrightarrow \int_{-2}^{2} \varphi(E) c(E) \dd E \cdot \int_0^\infty y f(y) \dd y
\end{equation}
in probability as $N \to \infty$.
\end{theorem}

We call a compact interval contained in $(-2, 2)$ a \emph{bulk interval}. In words, \zcref{t:pairs} states that for every bulk interval $I$ and every $y > 0$, the number of gaps at most $y N^{-1 - \alpha}$ with midpoints in $I$ is asymptotic to $N^{1 - 2\alpha} (y^2/2) \int_I c(E) \dd E$ in probability. By \zcref{t:main}, this number tends to zero in probability for $\alpha > 1/2$, and it converges in law to a Poisson random variable for $\alpha = 1/2$, so a law of large numbers cannot hold for $\alpha \ge 1/2$. For the GOE, the corresponding law of large numbers for pair statistics with $\alpha < 1/4$ (\zcref{p:goepairs}) follows from the Pfaffian formulas for its correlation functions. We prove it in \zcref{s:goe}, where it serves as the main input to the base case of our proof.

Our methods also apply to complex Hermitian matrices, as follows.

\begin{definition}\label{d:hermitian}
Let $Z$ be a real random variable such that $\E Z = 0$ and $\E Z^2 > 0$, and let $X_{\C}$ be a complex random variable such that $\E X_{\C} = 0$, $\E |X_{\C}|^2 = 1$, and $\E X_{\C}^2 = 0$. Suppose that there exist constants $c_0, C_0 > 0$ such that $\Prob(|X_{\C}| > y) + \Prob(|Z| > y) \le C_0 e^{-c_0 y^2}$ for all $y \ge 0$. For every integer $N \ge 2$, let $(X_{\C, ab})_{1 \le a < b \le N}$ be independent copies of $X_{\C}$, and let $(Z_a)_{1 \le a \le N}$ be independent copies of $Z$, independent of the $X_{\C, ab}$. We call the $N \times N$ complex Hermitian matrix $H$ with entries $H_{ab} = N^{-1/2} X_{\C, ab}$ and $H_{ba} = \overline{H_{ab}}$ for $a < b$, and $H_{aa} = N^{-1/2} Z_a$, the \emph{complex Hermitian Wigner matrix} associated with $X_{\C}$ and $Z$.
\end{definition}

The condition $\E X_{\C}^2 = 0$ states that the real and imaginary parts of $X_{\C}$ are uncorrelated and have the same variance. The third moments of $X_{\C}$ are arbitrary. For example, the real and imaginary parts of $X_{\C}$ may be independent copies of $2^{-1/2} X$ for every $X$ as in \zcref{d:wigner}.

\begin{theorem}\label{t:hermitian}
Let $H$ be as in \zcref{d:hermitian}, with eigenvalues $\lambda_1 \le \dots \le \lambda_N$ and gaps $g_i = \lambda_{i + 1} - \lambda_i$, and let $c_2(E) = (\pi^2/3) \rhosc(E)^4$. Then the point processes
\begin{equation}\label{e:XiNherm}
\sum_{i \in [N - 1] : \lambda_i \in (-2, 2)} \delta_{(N^{4/3} g_i, \, \lambda_i)}
\end{equation}
on $[0, \infty) \times (-2, 2)$ converge in distribution, with respect to the vague topology, to a Poisson point process with intensity $c_2(E) \, x^2 \dd x \dd E$. Consequently, for every nonempty interval $I$ whose closure is contained in $(-2, 2)$ and every $x \ge 0$,
\begin{equation}\label{e:minimumherm}
\lim_{N \to \infty} \Prob\Big( N^{4/3} \min_{i \in [N - 1] : \lambda_i \in I} g_i > x \Big) = \exp\Big( - \frac{\pi^2 x^3}{9} \int_I \rhosc(E)^4 \dd E \Big).
\end{equation}
\end{theorem}

The limit agrees with the one for the GUE \cite{benarous2013extreme}. \zcref[S]{t:hermitian} removes the smoothness assumption of \cite{bourgade2022extreme} and the assumption on the number of atoms of \cite{zhang2026quantitative} for complex Hermitian Wigner matrices with identically distributed entries above the diagonal. We give the proof of \zcref{t:hermitian} in less detail than that of \zcref{t:main}. In \zcref{s:hermitian}, we sketch how the arguments of \zcref{s:comparison,s:pairs} and \zcref{s:goe} adapt to complex Hermitian matrices; the resampling argument is not needed in this case. We give the details of the two estimates that are new. The first bounds, for $\kappa \in (0, 1/8)$, the number of triples of eigenvalues in a bulk interval within distance $N^{-1 - \kappa}$ of each other by $N^{1 - 8\kappa + o(1)}$, with probability tending to one. This is small compared with the number $N^{1 - 3\alpha'}$ of gaps at a smaller scale $N^{-1 - \alpha'}$ when $\alpha' < 8\kappa/3$, which is the comparison that the induction needs. The second bounds the probability that a Gaussian perturbation of a $3 \times 3$ diagonal matrix has a small gap.

\section{Methods and relation to previous work}\label{s:intro-methods}

We give an overview of the proofs in \zcref{s:overview}. Here we describe how their ingredients relate to the literature.

In \cite{bourgade2022extreme}, the smoothness of the entry distributions is used through the reverse heat flow \cite{erdos2010bulk}, which represents the matrix, up to a small error in total variation, as a matrix with a Gaussian component. Dyson Brownian motion \cite{dyson1962brownian} must run for a time at least $N^{-1/2 + 2\eps}$, for some $\eps > 0$, for its relaxation to resolve the gaps at the scale $N^{-3/2}$. This time corresponds to a Gaussian component of standard deviation $N^{-1/4 + \eps}$ in the variables $X_{ab}$, and this is where smoothness on the scale $N^{-1/4 + \eps}$ is needed.

The alternative is a comparison argument, which replaces the entries one at a time. As explained in \zcref{s:ov-difficulty}, for real symmetric matrices a single entry moves a gap by an amount comparable to the smallest gaps, so the standard perturbative comparison estimate does not give a small error at this scale. Zhang gives a different reason for restricting his result to the complex Hermitian case, which is consistent with ours. The error in his Green function comparison, which matches a fixed number of moments, is small only for spectral parameters at distance much larger than $N^{-3/2}$ from the real axis (see \cite[(11)]{zhang2026quantitative} and the paragraph after it), while the smallest gaps of real symmetric matrices are of order $N^{-3/2}$. We avoid comparison at the scale $N^{-3/2}$.

Instead, we use the fact that resampling a small random set of entries does not change the law of the matrix. Resampling a few random entries of a Wigner matrix, and the heuristic that each resampled entry moves an eigenvalue by one step of a random walk, were used by Bordenave, Lugosi, and Zhivotovskiy to study the noise sensitivity of the top eigenvector \cite{bordenave2020noise}; see also \cite{chatterjee2014superconcentration} for a related dynamics. We use resampling instead to smooth the law of the gaps while preserving the law of the matrix. The relevant random walk acts on the traceless part of the $2 \times 2$ compression of the matrix to the span of two consecutive eigenvectors. This traceless part can be identified with a vector in $\R^2$ whose norm is half of the gap, which reflects the codimension-two phenomenon of \cite{vonneumann1929verhalten}.

The analysis of this random walk requires a local limit theorem that resolves the law of the walk at the scale of a single step, which for discrete entries requires excluding arithmetic structure in the eigenvectors. For this, we use net and decoupling arguments developed for the invertibility of random matrices and the structure of their eigenvectors by Rudelson and Vershynin \cite{rudelson2008littlewood}, Vershynin \cite{vershynin2014invertibility}, Nguyen, Tao, and Vu \cite{nguyen2017random}, and Tao and Vu \cite{tao2017random}. The local limit theorem itself is proved by Fourier analysis with smooth test functions, as in the classical theorems of Shepp and Stone \cite{shepp1964local,stone1965local}, and the passage from local limits to Poisson convergence uses the Bonferroni inequalities, as in the factorial moment method used for extreme gaps in \cite{soshnikov2005statistics,benarous2013extreme}. We also use the tail bound of Nguyen, Tao, and Vu for three consecutive eigenvalues \cite[Corollary 2.5]{nguyen2017random} at several points.

The resampling argument requires the number of gaps at a scale slightly larger than $N^{-3/2}$, which is the content of \zcref{t:pairs}. Its proof uses a comparison argument along the Ornstein--Uhlenbeck flow. Our comparison estimate follows the continuity estimate of Bourgade and Yau \cite[Lemma A.1]{bourgade2017eigenvector} (see also \cite[Chapter 15]{erdos2017dynamical}), which descends from the Lindeberg method of Chatterjee \cite{chatterjee2006generalization}, the four moment theorem of Tao and Vu \cite{tao2011random}, and the Green function comparison of Erd\H{o}s, Yau, and Yin \cite{erdos2012bulk}; the expansion in a single entry is the cumulant expansion of Khorunzhy, Khoruzhenko, and Pastur \cite{khorunzhy1996asymptotic}, in the form of \cite[Proposition 3.1]{lytova2009central}.

Comparison arguments for statistics that involve all eigenvalues appear in \cite{landon2020comparison}, which uses regularizations of eigenvalues through the Helffer--Sj\"ostrand formula following Knowles and Yin \cite{knowles2013eigenvector}, and in \cite{zhang2026quantitative}, which uses resolvent expansions with spectral parameters at distance less than $N^{-1}$ from the real axis, together with many matched moments. In these works, derivatives of the observables are controlled through resolvent or perturbation formulas. Our observables are instead bounded by one in a complex neighborhood of the real entries, which follows from a trace representation of the pair statistic, the argument principle, and the local semicircle law \cite{erdos2009local,erdos2012rigidity,benaychgeorges2016lectures}, and we bound their derivatives by Cauchy's estimates. We are not aware of an earlier use of this device in comparison arguments.

The proof of \zcref{t:pairs} is an induction on $\alpha$. Its base case $\alpha < 1/8$ uses the coupling of Bourgade \cite[Corollary 3.3]{bourgade2022extreme} between Dyson Brownian motion started from $H$ and from a GOE matrix, together with the GOE result of \zcref{s:goe}, which relies on the Pfaffian formulas for the GOE correlation functions \cite{mehta2004random,tracy1998correlation,adler2000classical} in the form given in \cite{feng2024small}.

To go beyond $\alpha = 1/8$, we analyze matrices with a small Gaussian component conditionally on the eigenvectors of $H$, as in the work of Johansson \cite{johansson2001universality} and Erd\H{o}s, P\'ech\'e, Ram\'irez, Schlein, and Yau \cite{erdos2010bulk} (see also \cite{erdos2011universality} and \cite[Chapter 12]{erdos2017dynamical}); the regime of Dyson Brownian motion run for times much smaller than $N^{-1}$, which is the one relevant here, was studied by von Soosten and Warzel \cite{vonsoosten2018phase}. Our perturbation lemma for entrywise small perturbations of diagonal matrices is a variant of the quadratic residual bounds of Mathias \cite{mathias1998quadratic} and Li and Li \cite{li2005note}, with entrywise bounds in place of operator norms. The computation of gap counts at a smaller scale through independent $2 \times 2$ Gaussian blocks makes rigorous, in this regime, the $2 \times 2$ heuristic behind Wigner's surmise. The induction on the scale is in the spirit of the multiscale arguments for local laws (see \cite[Section 5]{benaychgeorges2016lectures}) and of the induction on the size of index sets in \cite{benigni2022fluctuations}.

\section{Notation}\label{s:intro-notation}

Constants $c, C > 0$ may depend on the laws of $X$ and $Z$ in \zcref{d:wigner} (or of $X_{\C}$ and $Z$ in \zcref{d:hermitian}) and on fixed parameters such as bulk intervals, test functions, integers, and small exponents, but never on $N$; their values may change from line to line. The function $c(E)$ of \eqref{e:cE} always appears with its argument, which distinguishes it from these constants. The letter $\eps$ denotes a small positive exponent. Its value is fixed within each argument and may change between arguments. When the dependence on a parameter matters, we indicate it by a subscript, as in $C_k$.

We say that an event holds \emph{with overwhelming probability} if for every $D > 0$ there exists $C_D > 0$ such that its complement has probability at most $C_D N^{-D}$ for all $N$. We let $o(1)$ denote a deterministic quantity that tends to zero as $N \to \infty$, $o_{\Prob}(1)$ a random quantity that tends to zero in probability, and $O_{\Prob}(1)$ a family of random quantities that is tight.

For $x, y \in \R^n$, we let $|x|$ denote the Euclidean norm of $x$ and $x \cdot y$ the Euclidean inner product. For $v \in \R^N$, we let $\|v\|_2$ and $\|v\|_\infty$ denote its Euclidean and supremum norms. For a matrix $A$, we let $\|A\|$ denote its operator norm, and $A^T$ and $A^*$ its transpose and conjugate transpose. For an $N \times N$ matrix $A$ and a set $\mathcal K \subset [N]$, we let $A_{\mathcal K \mathcal K} = (A_{jk})_{j, k \in \mathcal K}$ denote the corresponding principal submatrix. We let $\Id_n$ denote the $n \times n$ identity matrix and $e_a$ the $a$-th standard basis vector of $\R^N$.

For a real symmetric or complex Hermitian $N \times N$ matrix $A$, we let $\lambda_1(A) \le \dots \le \lambda_N(A)$ denote its eigenvalues, counted with multiplicity. For the matrix $H$ of \zcref{d:wigner}, we write $\lambda_i = \lambda_i(H)$, and we fix a measurable choice of orthonormal eigenvectors $u_1, \dots, u_N$ with $H u_i = \lambda_i u_i$; we let $u_i(a) = e_a^T u_i$ denote the $a$-th coordinate of $u_i$. Probabilities and expectations conditional on $H$ are conditional on $H$ and on this choice of eigenvectors.