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\begin{document}
\frontmatter
\npptitlepage{Smallest Gaps between Eigenvalues of Wigner\\ Matrices without Smoothness Assumptions}{Patrick Lopatto}
\begin{authorsnote}
With the exception of this note, the entire document is machine-written. I
have carefully supervised its preparation to ensure that the proofs are
readable and that previous literature is appropriately cited. I welcome
suggestions concerning mathematical corrections, omitted citations, and
attribution of ideas.

The author was partially supported by NSF grant DMS-2450004.
\end{authorsnote}
\begin{abstractpage}
We determine the asymptotic distribution of the smallest gaps between eigenvalues in the bulk of the spectrum of real symmetric Wigner matrices, including Bernoulli matrices. After multiplication by $N^{3/2}$, the gaps and their locations converge to a Poisson point process with the same intensity as for the Gaussian orthogonal ensemble. Since changing a single entry can move eigenvalues by an amount comparable to the smallest gaps, standard perturbative comparison estimates do not directly resolve this scale for arbitrary entry distributions. Instead, we use that resampling a few random entries does not change the law of the matrix, and we show that the resampling moves each pair of nearby eigenvalues by a random walk that satisfies a local limit theorem; this requires excluding arithmetic structure in the eigenvectors. We also prove a law of large numbers for the number of gaps at all scales between $N^{-3/2}$ and $N^{-1}$, and we sketch the proof of the analogous smallest-gap result for complex Hermitian Wigner matrices.
\end{abstractpage}
\tableofcontents
\mainmatter

\input{sections/01-introduction}
\input{sections/02-overview}
\input{sections/03-preliminaries}
\input{sections/04-reduction}
\input{sections/05-comparison}
\input{sections/06-pair-statistics}
\input{sections/07-linearization}
\input{sections/08-local-limit}
\input{sections/09-hermitian}

\appendix

\input{sections/A-goe}

\backmatter

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\end{document}