\chapter{The hole probability of one interval}\label{s:oneinterval}

In this section, we prove that the GOE hole probability of an interval $(x_0, x_0 + s/(2 \varrho_N(x_0)))$ in the bulk, with $s \le \gamma \sqrt{\log N}$, is asymptotic to its limit $q_1$, with a relative error that is polynomially small in $N$. We first recall the limiting objects.

\section{Sine-kernel determinants}

Recall from \eqref{e:Dpmintro} that $D_\pm(t) = \det( \Id - S_{\pm, t} )$ for $t > 0$, where $S_{\pm, t}$ is the integral operator on $L^2(0, t)$ with kernel $S(u - v) \pm S(u + v)$, and $S$ is the sine kernel \eqref{e:sinedef}. We set $D_\pm(0) = 1$. The operators $S_{\pm, t}$ are the restrictions of the integral operator with kernel $S(u - v)$ on $L^2(-t, t)$ to even and odd functions, respectively, so their eigenvalues lie in $[0, 1)$, and
\begin{equation}\label{e:Dproduct}
D_+(t) D_-(t) = \det \big( \Id - S_{(-t, t)} \big) = \det \big( \Id - S_{(0, 2t)} \big) ,
\end{equation}
where $S_A$ denotes the operator on $L^2(A)$ with kernel $S(u - v)$, and the second equality follows from translation invariance; see \cite[Section~IV]{TracyWidom1996} or \cite[(4)]{Ehrhardt2007}. The functions $D_\pm$ are continuous and positive on $[0, \infty)$, and they are nonincreasing. To see the monotonicity, note that $S_{\pm, t}$ is a compression of $S_{\pm, t'}$ for $t < t'$, so its eigenvalues increase with $t$ by the Courant--Fischer min-max principle. By definition, $q_1(s) = D_+(s/4)$. We define
\begin{equation}\label{e:q1mudef}
\mu(s) = \frac{D_+(s/4)}{D_-(s/4)} , \qquad s \ge 0 ,
\end{equation}
and we set $q_1(0) = 1$. (The interpretation of $q_1$ as a gap probability, recalled in \zcref{s:mainresults}, is not used in our proofs.) Recall from \eqref{e:ehrhardtintro} the asymptotics of $D_\pm(t)$ as $t \to \infty$, which were proved by Ehrhardt \cite[Theorem~4.8]{Ehrhardt2007}.

\begin{lemma}\label{l:q1}
Recall the function $F(s) = s^2/64 + s/8$ and the constant $\kappa_1$ from \eqref{e:kappa1def}. Then the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item We have $q_1(s) = \kappa_1 s^{-1/8} e^{-F(s)} (1 + o(1))$ as $s \to \infty$, and $q_1$ is nonincreasing.
\item We have $\mu(s) = \sqrt 2 \, e^{-s/4} (1 + o(1))$ as $s \to \infty$, and there exists $c > 0$ such that $\mu(s) \ge c \, e^{- s/4}$ for all $s \ge 0$.
\end{enumerate}
\end{lemma}

\begin{proof}
Setting $t = s/4$ in \eqref{e:ehrhardtintro} with the upper sign, we find
\begin{equation}
\log q_1(s) = - \frac{s^2}{64} - \frac s8 - \frac18 \log s + \frac14 \log 2 + \frac{7}{24} \log 2 + \frac32 \zeta'(-1) + o(1) ,
\end{equation}
which is the asymptotic formula in (a), since $1/4 + 7/24 = 13/24$. The monotonicity in (a) was shown above. Subtracting the two cases of \eqref{e:ehrhardtintro}, we find
\begin{equation}
\log \mu(s) = - s/4 + (\log 2)/2 + o(1) .
\end{equation}
Since $\mu$ is continuous and positive on $[0, \infty)$, this implies the lower bound in (b).
\end{proof}

Next, we identify the limit of the endpoint matrix of one interval. For $\ell > 0$, we define
\begin{equation}\label{e:Asinedef}
\Amat_\sine(\ell) = 1 - \Big( \big( (\Id - S_{(0, \ell)})^{-1} g_\sine \big)(\ell^-) - \big( (\Id - S_{(0, \ell)})^{-1} g_\sine \big)(0^+) \Big) ,
\end{equation}
where $g_\sine$ was defined in \eqref{e:gsinedef}, and the values at the endpoints are defined through the analogue of \eqref{e:resolventidentity} for $S$, with $g_\sine(0^+) = -1/2$. The following identity is due to Tracy and Widom.

\begin{lemma}\label{l:TW}
For all $\ell > 0$, we have $\Amat_\sine(\ell) = D_+(\ell / 2) / D_-(\ell / 2)$. In particular, $\Amat_\sine(s/2) = \mu(s)$ for all $s > 0$.
\end{lemma}

\begin{proof}
Set $t = \ell / 2$ and $J = (-t, t)$, and define
\begin{equation}
\Delta G = G(t^-) - G((-t)^+) .
\end{equation}
Since the sine kernel is translation invariant and $g_\sine$ is evaluated relative to the left endpoint, we have
\begin{equation}
\Amat_\sine(\ell) = 1 - \Delta \big[ (\Id - S_J)^{-1} \big( \1_J \, g_\sine(\cdot + t) \big) \big] .
\end{equation}
Let $R(x, y)$, $x \in \R$, $y \in \overline J$, denote the kernel of $(\Id - S \1_J)^{-1} S \1_J$. Then
\begin{equation}
R(x, y) = S(x - y) + \int_J S(x - z) R(z, y) \dd z ,
\end{equation}
and $R$ is symmetric on $J \times J$.

Set $w = \1_{J^c} \eps_{-t}$. Tracy and Widom showed \cite[(19), (25), (30)]{TracyWidom1996} that
\begin{equation}\label{e:TW}
1 - \int_{J^c} w(x) \big( R(x, t) - R(x, -t) \big) \dd x = \frac{D_+(t)}{D_-(t)} ,
\end{equation}
where the integral over $J^c$ is the limit as $M \to \infty$ of the integral over $J^c \cap [-M, M]$ (we show below that this limit exists). To deduce this identity from the equations of Tracy and Widom, set
\begin{equation}
a_\pm = \int_{J^c} w(x) R(x, \pm t) \dd x .
\end{equation}
Since $\1_{J^c} \eps_{-t} = \1_{J^c} \eps_t$ almost everywhere, the determinant in \cite[(19)]{TracyWidom1996} with $m = 1$ is
\begin{equation}
\det \begin{pmatrix} 1 + a_- & - a_+ \\ a_- & 1 - a_+ \end{pmatrix} = 1 - (a_+ - a_-) ,
\end{equation}
which is the left side of \eqref{e:TW}. This determinant is computed in \cite[(25)]{TracyWidom1996} and is equal to $D_+(t)/D_-(t)$ by \cite[(30)]{TracyWidom1996}.

It remains to identify the left side of \eqref{e:TW} with $\Amat_\sine(\ell)$. To that end, for $z \in \overline J$, we set
\begin{equation}
\Phi(z) = \lim_{M \to \infty} \int_{J^c \cap [-M, M]} w(x) S(x - z) \dd x .
\end{equation}
A direct computation using $\int_0^\infty S = 1/2$ gives
\begin{equation}
\lim_{M \to \infty} \int_{-M}^M \eps_{-t}(x) S(x - z) \dd x = \int_0^{z + t} S(v) \dd v = \eps_{-t}(z) + g_\sine(z + t) ,
\end{equation}
and the convergence is uniform in $z \in \overline J$, since $|\int_M^\infty S| \le 2/M$. Therefore
\begin{equation}
\Phi = (\Id - S_J)(\1_J \eps_{-t}) + \1_J \, g_\sine(\cdot + t) .
\end{equation}
Inserting the equation for $R$ into the left side of \eqref{e:TW} and exchanging the order of integration over $[-M, M]$ and $J$, we find
\begin{equation}
\int_{J^c} w(x) R(x, y) \dd x = \Phi(y) + \int_J \Phi(z) R(z, y) \dd z = \big( (\Id - S_J)^{-1} \Phi \big) (y) , \qquad y = \pm t .
\end{equation}
Since $\eps_{-t}$ is equal to $1/2$ on $J$, we have
\begin{equation}
\Delta[(\Id - S_J)^{-1} \Phi] = \Delta [ (\Id - S_J)^{-1} (\1_J \, g_\sine(\cdot + t)) ] ,
\end{equation}
which completes the proof.
\end{proof}

\section{The comparison}

\begin{proposition}\label{p:onehole}
Let $\delta \in (0, 1/2]$ and $\gamma > 0$. There exist $C, N_0 > 0$ depending only on $\delta$ and $\gamma$ such that the following holds for $N \ge N_0$. Let $x_0 \in \bulk_N(\delta)$, set $\varrho = \varrho_N(x_0)$, and let $0 < s \le \gamma \sqrt{\log N}$. Then the interval defined by $J = (x_0, x_0 + s/(2\varrho))$ satisfies
\begin{equation}\label{e:onehole}
\Big| \frac{\Pgoe_N(J)}{q_1(s)} - 1 \Big| \le C e^{C s} N^{-1/2} , \qquad \Big| \frac{\Dgue_N(J)}{D_+(s/4) D_-(s/4)} - 1 \Big| \le C e^{C s} N^{-1/2} .
\end{equation}
\end{proposition}

Before giving the proof, we record a standard bound for determinants. We let $\| Z \|_1$ and $\| Z \|_2$ denote the trace norm and the Hilbert--Schmidt norm of a trace class operator $Z$.

\begin{lemma}\label{l:detbound}
Let $Z$ be a trace class operator on a Hilbert space with $\| Z \|_2 \le 1/2$, and let $(y_k)$ denote its eigenvalues, repeated according to algebraic multiplicity. We define $\log \det(\Id - Z) = \sum_k \log(1 - y_k)$, where $\log$ denotes the principal branch of the logarithm. Then
\begin{equation}\label{e:detbound}
\big| \log \det(\Id - Z) + \Tr Z \big| \le \| Z \|_2^2 .
\end{equation}
\end{lemma}

\begin{proof}
Recall that, by Lidskii's theorem,
\begin{equation}
\Tr Z = \sum_k y_k
\end{equation}
and
\begin{equation}
\det(\Id - Z) = \prod_k (1 - y_k) ,
\end{equation}
and that, by Weyl's inequality (see \cite{Simon2005}),
\begin{equation}
\sum_k |y_k|^2 \le \| Z \|_2^2 .
\end{equation}
In particular, $|y_k| \le 1/2$ for all $k$. The claim now follows from the elementary inequality $|\log(1 - y) + y| \le |y|^2$ for complex $y$ with $|y| \le 1/2$.
\end{proof}

\begin{proof}[Proof of \zcref{p:onehole}]
Set $\ell = s/2$. Then $J = (x_0, x_0 + \ell/\varrho)$ and
\begin{equation}
L(J) = \sqrt{2N} \ell / \varrho \le C s .
\end{equation}
Since $s \le \gamma \sqrt{\log N}$, we may assume that $\overline J \subset \bulk_N(\delta/2)$, $L(J) \le N^{1/8}$, and $C e^{C s} N^{-1/2} \le N^{-1/4}$ for every fixed $C$, by taking $N_0$ large. We use the results of \zcref{s:bulk,s:inverse,s:endpoint} with $\delta / 2$ in place of $\delta$.

We define a map from $L^2(J)$ to $L^2(0, \ell)$ by
\begin{equation}
(U f)(u) = \varrho^{-1/2} f(x_0 + u/\varrho) .
\end{equation}
This map is unitary. Let $\mathcal K_\ell$ denote the integral operator on $L^2(0, \ell)$ with kernel
\begin{equation}
\mathcal K(u, v) = \varrho^{-1} K_N(x_0 + u/\varrho, x_0 + v/\varrho) .
\end{equation}
Note that $U K_J U^{-1}$ coincides with this operator. We abbreviate $S_\ell = S_{(0, \ell)}$, and we set $\Xi = \mathcal K_\ell - S_\ell$. By \zcref{l:localsine}, we have $|\Xi(u, v)| \le C N^{-1/2}$ for $u, v \in [0, \ell]$. Further, \zcref{p:inverse} implies that $\| (\Id - \mathcal K_\ell)^{-1} \|$ and $\| (\Id - S_\ell)^{-1} \|$ are bounded by $C e^{C s}$.

We first prove the second bound in \eqref{e:onehole}. By \eqref{e:Dproduct}, we have $D_+(s/4) D_-(s/4) = \det(\Id - S_\ell)$, and $\Dgue_N(J) = \det(\Id - \mathcal K_\ell)$. Set $Z = (\Id - S_\ell)^{-1} \Xi$. Then
\begin{equation}
\det(\Id - \mathcal K_\ell) = \det(\Id - S_\ell) \det(\Id - Z) .
\end{equation}

We have $\| \Xi \|_2 \le C \ell N^{-1/2}$ and $\| Z \|_2 \le C e^{C s} N^{-1/2}$. The operators $\mathcal K_\ell$ and $S_\ell$ are positive and have continuous kernels, so their traces are the integrals of their kernels over the diagonal. Therefore, $|\Tr \Xi| \le C \ell N^{-1/2}$. Writing $(\Id - S_\ell)^{-1} = \Id + S_\ell (\Id - S_\ell)^{-1}$, we also have
\begin{equation}
\begin{split}
|\Tr Z| & \le |\Tr \Xi| + |\Tr( S_\ell (\Id - S_\ell)^{-1} \Xi )| \\
& \le |\Tr \Xi| + \| S_\ell \|_2 \| (\Id - S_\ell)^{-1} \| \| \Xi \|_2 \le C e^{C s} N^{-1/2} ,
\end{split}
\end{equation}
where we used $\| S_\ell \|_2 \le \ell / \pi$. Then \zcref{l:detbound} implies that $|\log \det(\Id - Z)| \le C e^{C s} N^{-1/2}$, which gives the second bound in \eqref{e:onehole}.

Next, we compare the endpoint matrices. We define
\begin{equation}
G = (\Id - K_J)^{-1} (\1_J g_{x_0}) .
\end{equation}
The endpoint matrix of $J$ is then the number $\Amat_N(J) = 1 - \Delta_1 G$. Set
\begin{equation}
\widetilde g(u) = g_{x_0}(x_0 + u/\varrho)
\end{equation}
and
\begin{equation}
\widetilde G(u) = G(x_0 + u/\varrho)
\end{equation}
for $u \in (0, \ell)$. Using \eqref{e:resolventidentity} and the change of variables $z = x_0 + v/\varrho$, we find that
\begin{equation}
\widetilde G = \widetilde g + \mathcal K_\ell \widetilde G ,
\end{equation}
and this identity extends to $u \in \{0, \ell\}$.

Let $G_\sine = (\Id - S_\ell)^{-1} g_\sine$. By \zcref{c:gt}(c), we have $|\widetilde g(u) - g_\sine(u)| \le C (1 + \ell) N^{-1/2}$ for $u \in (0, \ell)$. Note that
\begin{equation}
\widetilde G - G_\sine = (\Id - \mathcal K_\ell)^{-1} \big( (\widetilde g - g_\sine) + \Xi G_\sine \big)
\end{equation}
and that $\| G_\sine \|_2 \le C e^{C s} \ell^{1/2}$. Therefore,
\begin{equation}
\| \widetilde G - G_\sine \|_2 \le C e^{C s} N^{-1/2} .
\end{equation}
For $u \in \{0, \ell\}$, we then have
\begin{equation}
\begin{split}
\big| \widetilde G(u) - G_\sine(u) \big| & \le |\widetilde g(u) - g_\sine(u)| + \int_0^\ell |\Xi(u, v)| |\widetilde G(v)| \dd v \\
& \quad + \int_0^\ell |S(u - v)| |\widetilde G(v) - G_\sine(v)| \dd v \\
& \le C e^{C s} N^{-1/2} ,
\end{split}
\end{equation}
by the Cauchy--Schwarz inequality. Therefore, by \zcref{l:TW},
\begin{equation}\label{e:Acompare}
\big| \Amat_N(J) - \mu(s) \big| = \big| \Amat_N(J) - \Amat_\sine(\ell) \big| \le C e^{C s} N^{-1/2} .
\end{equation}

We now conclude. By \zcref{c:exact}, \zcref{p:endpoint}, \eqref{e:Acompare}, and \zcref{l:q1}(b), we have
\begin{equation}
\frac{\Pgoe_N(J)^2}{\Dgue_N(J)} = \det \Mmat_N^\sharp(J) = \mu(s) + O\big( e^{C s} N^{-1/2} \big) = \mu(s) \big( 1 + O(e^{C s} N^{-1/2}) \big) .
\end{equation}
Combining this with the second bound in \eqref{e:onehole} and $D_+(s/4) D_-(s/4) \mu(s) = q_1(s)^2$, we find that $\Pgoe_N(J)^2 = q_1(s)^2 (1 + O(e^{C s} N^{-1/2}))$. Taking square roots, we obtain the first bound in \eqref{e:onehole}.
\end{proof}

We now translate \zcref{p:onehole} to eigenvalue coordinates. Recall from the introduction that $E_{1, N}(A)$ denotes the probability that $H_N$ has no eigenvalue in $A$, and that $\sigma(y) = \sqrt{4 - y^2}$.

\begin{corollary}\label{c:onehole}
Let $\delta \in (0, 1)$ and $\gamma > 0$. There exist $C, N_0 > 0$ depending only on $\delta$ and $\gamma$ such that for $N \ge N_0$, all $y \in [-2 + \delta, 2 - \delta]$, and all $0 < s \le \gamma \sqrt{\log N}$,
\begin{equation}\label{e:oneholelambda}
\Big| E_{1, N} \Big( \Big( y, y + \frac{s}{N \sigma(y)} \Big) \Big) - q_1(s) \Big| \le C e^{C s} N^{-1/2} q_1(s) .
\end{equation}
\end{corollary}

\begin{proof}
Set $x_0 = \sqrt{N/2} \, y$. Then $x_0 \in \bulk_N(\delta/2)$, $\varrho_N(x_0) = \sqrt{N/2} \, \sigma(y)$, and the image of the interval $(y, y + s/(N \sigma(y)))$ under $\lambda \mapsto \sqrt{N/2} \, \lambda$ is $(x_0, x_0 + s/(2 \varrho_N(x_0)))$. The claim therefore follows from \zcref{p:onehole}.
\end{proof}