\chapter{Several intervals}\label{s:severalintervals}

In this section, we prove two estimates for unions of intervals, which are used in the proof of \zcref{t:main}. The first states that the hole probability of a union of well-separated intervals factorizes. The second states that the hole probability of an arbitrary union of disjoint intervals is bounded by $e^{C L}$ times the product of the individual hole probabilities. For the GUE, the second estimate holds without the factor $e^{C L}$, by the negative association of determinantal point processes \cite[Lemma~3.8]{BenArousBourgade2013}. For the GOE, the factor $e^{C L}$ comes from the endpoint matrix. This factor is harmless, since $L$ is of order $\sqrt{\log N}$ in our application.

Throughout this section, we fix $\delta \in (0, 1/2]$, an integer $r_0 \ge 1$, and $\gamma > 0$, and we consider sets $J = \bigcup_{j = 1}^r J_j$ as in \eqref{e:Jdef} with
\begin{equation}\label{e:severalassumption}
r \le r_0 , \qquad \overline J \subset \bulk_N(\delta) , \qquad L(J) \le \gamma \sqrt{\log N} .
\end{equation}
The constants $C$, $c$, and $N_0$ depend only on $\delta$, $r_0$, and $\gamma$, and all statements hold for $N \ge N_0$.

\section{The determinantal part}

\begin{lemma}\label{l:fischer}
Let $J', J'' \subset \R$ be disjoint bounded measurable sets. Then
\begin{equation}\label{e:fischer}
\Dgue_N(J' \cup J'') \le \Dgue_N(J') \, \Dgue_N(J'') .
\end{equation}
Moreover, if $J'$ and $J''$ satisfy the assumptions of \zcref{p:inverse} and $\| K_{J' J''} \|_2 \le c e^{-C (L(J') + L(J''))}$, where $K_{J' J''}$ denotes the operator from $L^2(J'')$ to $L^2(J')$ with kernel $K_N(x, y)$, then
\begin{equation}\label{e:Dsplit}
\Big| \log \frac{\Dgue_N(J' \cup J'')}{\Dgue_N(J') \, \Dgue_N(J'')} \Big| \le C e^{C (L(J') + L(J''))} \| K_{J' J''} \|_2^2 .
\end{equation}
\end{lemma}

\begin{proof}
With respect to the decomposition $L^2(J' \cup J'') = L^2(J') \oplus L^2(J'')$, we write
\begin{equation}
\Id - K_{J' \cup J''} = \begin{pmatrix} \Id - K_{J'} & - K_{J' J''} \\ - K_{J'' J'} & \Id - K_{J''} \end{pmatrix} .
\end{equation}
Factoring this operator into block triangular operators whose diagonal blocks are $\Id - K_{J'}$ and the Schur complement, and using the multiplicativity of Fredholm determinants, we have
\begin{equation}\label{e:schur}
\frac{\Dgue_N(J' \cup J'')}{\Dgue_N(J') \, \Dgue_N(J'')} = \det(\Id - Z) ,
\end{equation}
where we define
\begin{equation}
Z = (\Id - K_{J''})^{-1/2} K_{J'' J'} (\Id - K_{J'})^{-1} K_{J' J''} (\Id - K_{J''})^{-1/2} .
\end{equation}

Note that the operator $Z$ is trace class and nonnegative. Since $\Id - K_{J' \cup J''}$ is nonnegative, so is its Schur complement $\Id - K_{J''} - K_{J'' J'} (\Id - K_{J'})^{-1} K_{J' J''}$, and therefore $Z \le \Id$. Hence the eigenvalues of $Z$ lie in $[0, 1]$, and $\det(\Id - Z) \le 1$, which proves \eqref{e:fischer}.

We now prove the second claim. Under the additional assumptions, \zcref{p:inverse} gives
\begin{equation}
\| Z \|_1 \le C e^{C (L(J') + L(J''))} \| K_{J' J''} \|_2^2 \le 1/2 .
\end{equation}
Therefore, \eqref{e:Dsplit} follows from
\begin{equation}
|\log \det(\Id - Z)| \le \sum_k |\log (1 - z_k)| \le 2 \| Z \|_1 ,
\end{equation}
where we let $z_k \in [0, 1/2]$ denote the eigenvalues of $Z$.
\end{proof}

\section{Factorization and the quasi-product bound}

We first record a lower bound for the endpoint matrix of a single interval.

\begin{lemma}\label{l:singlelower}
Let $J'$ be an interval satisfying \eqref{e:severalassumption}. Then $\det \Mmat^\sharp_N(J') \ge c \, e^{- C L(J')}$ and $\Amat_N(J') \ge c \, e^{- C L(J')}$.
\end{lemma}

\begin{proof}
We write
\begin{equation}
J' = (x_0, x_0 + s/(2\varrho))
\end{equation}
with $\varrho = \varrho_N(x_0)$, so that $s \le 2 L(J')$. By \zcref{c:exact} and \zcref{p:onehole}, we have
\begin{equation}
\det \Mmat^\sharp_N(J') = \Pgoe_N(J')^2 / \Dgue_N(J') = \mu(s) (1 + O(e^{C s} N^{-1/2}))
\end{equation}
and \zcref{l:q1}(b) gives $\mu(s) \ge c e^{-s/4}$. This yields the first claim for $N \ge N_0$, since $e^{C s} N^{-1/2} \le N^{-1/4}$. The second claim follows from the first and \zcref{p:endpoint}.
\end{proof}

\begin{proposition}\label{p:factorization}
Assume \eqref{e:severalassumption}, and suppose that $\operatorname{sep}(J) \ge \sqrt N$. Then
\begin{equation}\label{e:factorization}
\Big| \frac{\Pgoe_N(J)}{\prod_{j = 1}^r \Pgoe_N(J_j)} - 1 \Big| \le C e^{C L(J)} N^{-1/4} .
\end{equation}
\end{proposition}

\begin{proof}
We write $L = L(J)$ and $\mathsf d = \operatorname{sep}(J)$. We first consider the determinantal part. Set
\begin{equation}
J'_k = J_1 \cup \dots \cup J_{k - 1}
\end{equation}
for $2 \le k \le r$. By \zcref{l:pointwise}(c), we have $|K_N(x, y)| \le C \sqrt N \mathsf d^{-1}$ for $x \in J'_k$ and $y \in J_k$, which implies that
\begin{equation}
\| K_{J'_k J_k} \|_2^2 \le C N \mathsf d^{-2} |J|^2 \le C L^2 \mathsf d^{-2} \le C L^2 N^{-1} .
\end{equation}
Applying \eqref{e:Dsplit} for $k = r, r - 1, \dots, 2$, we find
\begin{equation}\label{e:Dfactor}
\Dgue_N(J) = \Big( \prod_{j = 1}^r \Dgue_N(J_j) \Big) \big( 1 + O(e^{C L} N^{-1}) \big) .
\end{equation}

Next, we consider the endpoint matrices. Let $\Amat_0$ denote the diagonal matrix with entries $\Amat_N(J_j)$, and write $\Amat_N(J) = \Amat_0 + E$. It follows from \zcref{l:offdiagonal} and $\mathsf d \ge \sqrt N$ that the entries of $E$ are bounded by $C e^{C L} N^{-1/4}$ in absolute value. By \zcref{l:singlelower}, the diagonal entries of $\Amat_0$ are at least $c e^{-C L}$. Therefore, we have
\begin{equation}
\det \Amat_N(J) = \det \Amat_0 \det(\Id + \Amat_0^{-1} E) = \Big( \prod_{j = 1}^r \Amat_N(J_j) \Big) \big( 1 + O(e^{C L} N^{-1/4}) \big) .
\end{equation}
In particular, $\det \Amat_N(J) \ge c e^{-C L}$.

We now apply \zcref{p:endpoint} to $J$ and to each $J_j$. Together with \zcref{l:singlelower}, this gives
\begin{equation}
\det \Mmat^\sharp_N(J) = \det \Amat_N(J) (1 + O(e^{C L} N^{-1/2}))
\end{equation}
and
\begin{equation}
\det \Mmat^\sharp_N(J_j) = \Amat_N(J_j)(1 + O(e^{C L} N^{-1/2})) .
\end{equation}
Hence
\begin{equation}\label{e:Mfactor}
\det \Mmat^\sharp_N(J) = \Big( \prod_{j = 1}^r \det \Mmat^\sharp_N(J_j) \Big) \big( 1 + O(e^{C L} N^{-1/4}) \big) .
\end{equation}
Combining \eqref{e:Dfactor} and \eqref{e:Mfactor} with \zcref{c:exact}, we find
\begin{equation}
\Pgoe_N(J)^2 = \prod_j \Pgoe_N(J_j)^2 \, (1 + O(e^{C L} N^{-1/4})) .
\end{equation}
Taking square roots completes the proof.
\end{proof}

\begin{proposition}\label{p:quasi}
Assume \eqref{e:severalassumption}. Then
\begin{equation}\label{e:quasi}
\Pgoe_N(J) \le C e^{C L(J)} \prod_{j = 1}^r \Pgoe_N(J_j) .
\end{equation}
\end{proposition}

\begin{proof}
Combining \zcref{p:endpoint}, \zcref{l:fischer}, \zcref{c:exact}, and \zcref{l:singlelower}, we have
\begin{equation}
\begin{split}
\Pgoe_N(J)^2 & \le C e^{C L(J)} \Dgue_N(J) \le C e^{C L(J)} \prod_{j = 1}^r \Dgue_N(J_j) \\
& = C e^{C L(J)} \prod_{j = 1}^r \frac{\Pgoe_N(J_j)^2}{\det \Mmat^\sharp_N(J_j)} \le C e^{C L(J)} \prod_{j = 1}^r \Pgoe_N(J_j)^2 .
\end{split}
\end{equation}
Taking square roots, we obtain \eqref{e:quasi}.
\end{proof}

We now translate these results to eigenvalue coordinates.

\begin{corollary}\label{c:several}
Let $\delta \in (0, 1)$, $r_0 \ge 1$, and $\gamma > 0$. There exist $C, N_0 > 0$ depending only on $\delta$, $r_0$, and $\gamma$ such that the following holds for $N \ge N_0$. Let $A_1, \dots, A_r \subset [-2 + \delta, 2 - \delta]$ be open intervals with disjoint closures, where $r \le r_0$, and suppose that $L \le \gamma \sqrt{\log N}$, where we set $L = N \sum_j |A_j|$. Then
\begin{equation}\label{e:quasilambda}
E_{1, N} \Big( \bigcup_{j = 1}^r A_j \Big) \le C e^{C L} \prod_{j = 1}^r E_{1, N}(A_j) .
\end{equation}
If, in addition, $\dist(A_i, A_j) \ge N^{-1/2}$ for all $i \neq j$, then
\begin{equation}\label{e:factorlambda}
\Big| E_{1, N} \Big( \bigcup_{j = 1}^r A_j \Big) - \prod_{j = 1}^r E_{1, N}(A_j) \Big| \le C e^{C L} N^{-1/4} \prod_{j = 1}^r E_{1, N}(A_j) .
\end{equation}
\end{corollary}

\begin{proof}
Let $J_j$ denote the image of $A_j$ under $\lambda \mapsto \sqrt{N/2} \, \lambda$, and set $J = \bigcup_j J_j$. These images are intervals, and their union satisfies \eqref{e:severalassumption} with $\delta / 2$ in place of $\delta$. Moreover, we have $L(J) = L$ and
\begin{equation}
\operatorname{sep}(J) = N \min_{i \neq j} \dist(A_i, A_j) .
\end{equation}
So \eqref{e:quasilambda} follows from \zcref{p:quasi}. Under the additional assumption that $\dist(A_i, A_j) \ge N^{-1/2}$ for all $i \neq j$, we have $\operatorname{sep}(J) \ge \sqrt N$, and \eqref{e:factorlambda} follows from \zcref{p:factorization}.
\end{proof}