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\begin{document}
\frontmatter
\npptitlepage{Largest Gaps between Eigenvalues\\ of the Gaussian Orthogonal Ensemble}{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 largest gaps between consecutive eigenvalues of the Gaussian orthogonal ensemble (GOE) in the bulk of the spectrum. After an explicit centering and scaling, the largest gaps in a fixed interval converge to a Poisson point process with an exponential intensity, so the largest gap has a shifted Gumbel limit. The analogous result for the Gaussian unitary ensemble was proved by Feng and Wei. Our proof expresses the squared probability that the GOE has no eigenvalues in a union of intervals as the corresponding probability for the Gaussian unitary ensemble times a determinant of fixed size, for every matrix dimension. We control this determinant on intervals whose length in units of the local mean spacing grows with the dimension, using an uncertainty principle of Nazarov and exact identities for Hermite functions.
\end{abstractpage}
\tableofcontents
\mainmatter

\input{sections/01-introduction}
\input{sections/02-hermite}
\input{sections/03-exact-formula}
\input{sections/04-bulk-estimates}
\input{sections/05-inverse-bound}
\input{sections/06-endpoint-matrix}
\input{sections/07-one-interval}
\input{sections/08-several-intervals}
\input{sections/09-first-moment}
\input{sections/10-factorial-moments}

\appendix

\input{sections/A-poisson}

\backmatter

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