\chapter{An exact formula for the hole probability}\label{s:exact}

In this section, we express $\Pgoe_N(J)^2$ as $\Dgue_N(J)$ times the determinant of a matrix whose size depends only on the number of intervals in $J$. The identity holds for every $N \ge 2$ and every finite union of bounded intervals $J$. For even $N$, an identity of this type was obtained by Tracy and Widom in \cite[(33)]{TracyWidom1996}, who manipulated Fredholm determinants of operators with matrix kernels. They gave a detailed justification of these manipulations for the sine kernel in \cite[Section~VIII]{TracyWidom1996}. Our derivation is different. It starts from de Bruijn's integration formula and uses only finite-dimensional linear algebra, so no regularization of the sign kernel is required. It treats $N$ of either parity and pairs the two endpoints of each interval from the start, which produces a matrix whose size is the number of intervals plus two, instead of twice the number of intervals plus one.

\section{Statement}

Throughout this section, we fix $N \ge 2$, an integer $r \ge 1$, and real numbers $a_1 < b_1 < a_2 < b_2 < \dots < a_r < b_r$. We set
\begin{equation}\label{e:Jdef}
J_j = (a_j, b_j) , \qquad J = \bigcup_{j = 1}^r J_j .
\end{equation}
For $1 \le j \le r$, we define the functions
\begin{equation}\label{e:kwdef}
k_j = K_N(\cdot, b_j) - K_N(\cdot, a_j) , \qquad w_j = \1_{J^c} \eps_{a_j} .
\end{equation}
Observe that $k_j \in V_N$ and $\ip{k_j}{f} = f(b_j) - f(a_j)$ for all $f \in V_N$. Further, since $\sgn(x - a_j) = \sgn(x - b_j)$ for $x \notin [a_j, b_j]$, we have $w_j = \1_{J^c} \eps_{b_j}$ almost everywhere. The function $w_j$ vanishes on $J$ and takes the value $-1/2$ on $J^c \cap (-\infty, a_j)$ and the value $1/2$ on $J^c \cap (b_j, \infty)$. We also define the function
\begin{equation}\label{e:Y0def}
Y_N(J) = \1_{J^c} \big( \omega_N - \eps(\1_J \varphi_N) \big) ,
\end{equation}
where we recall that $\omega_N$ was defined in \eqref{e:cNomegadef}, and the operator $T_N(J)$ on $V_N$ by
\begin{equation}\label{e:TNdef}
T_N(J) = \Id - K_N \1_J - \sum_{j = 1}^r k_j \otimes w_j + \psi_N \otimes Y_N(J) ,
\end{equation}
where $K_N \1_J$ denotes the operator $f \mapsto K_N(\1_J f)$, and the rank-one operators were defined in \eqref{e:tensordef}. Finally, we define
\begin{equation}\label{e:borderdef}
\begin{gathered}
X^\sharp_N(J) = - \frac12 \sum_{j = 1}^r k_j - \frac12 \Big( \int_J \varphi_N \Big) \psi_N , \qquad d_N(J) = 1 - c_N \int_J \psi_N , \\
Y^\sharp_N(J) = - 2 c_N \big( \1_J \eps \psi_N + \1_{J^c} \eps(\1_J \psi_N) \big) .
\end{gathered}
\end{equation}
Note that $X^\sharp_N(J) \in V_N$, and that $Y^\sharp_N(J)$ is bounded. For even $N$, we have $c_N = 0$, so $Y^\sharp_N(J) = 0$ and $d_N(J) = 1$.

\begin{proposition}\label{p:exact}
Let $\mathbb T_N(J)$ denote the operator on $V_N \oplus \R$ defined by
\begin{equation}\label{e:bigTdef}
\mathbb T_N(J) (f, a) = \big( T_N(J) f + a \, X^\sharp_N(J) , \ \ip{Y^\sharp_N(J)}{f} + a \, d_N(J) \big) .
\end{equation}
Then
\begin{equation}\label{e:exact}
\Pgoe_N(J)^2 = \det \mathbb T_N(J) .
\end{equation}
In particular, if $N$ is even, then $\Pgoe_N(J)^2 = \det T_N(J)$.
\end{proposition}

To state the consequence of \zcref{p:exact} that we will use, we write
\begin{equation}\label{e:QJdef}
Q_N(J) = \Id - K_N \1_J ,
\end{equation}
viewed as an operator on $V_N$. In the basis $(h_n)_{n < N}$, the matrix of this operator is the matrix in \eqref{e:Dfinite}, so $\det Q_N(J) = \Dgue_N(J) > 0$. In particular, $Q_N(J)$ is invertible. We define $X_0, \dots, X_r \in V_N$ and bounded functions $Y_0, \dots, Y_r$ by
\begin{equation}\label{e:XYdef}
X_0 = - \psi_N , \qquad X_j = k_j , \qquad Y_0 = Y_N(J) , \qquad Y_j = w_j , \qquad 1 \le j \le r ,
\end{equation}
so that $T_N(J) = Q_N(J) - \sum_{\alpha = 0}^r X_\alpha \otimes Y_\alpha$. We index the rows and columns of the following matrices by $\{0, 1, \dots, r\}$ and $\{0, 1, \dots, r + 1\}$, respectively.

\begin{definition}\label{d:Msharp}
We define the $(r + 1) \times (r + 1)$ matrix $\Mmat_N(J)$ and the $(r + 2) \times (r + 2)$ matrix $\Mmat_N^\sharp(J)$ by
\begin{equation}\label{e:Mdef}
\Mmat_N(J)_{\alpha \beta} = \1_{\alpha = \beta} - \ip{Y_\alpha}{Q_N(J)^{-1} X_\beta} , \qquad 0 \le \alpha, \beta \le r ,
\end{equation}
and
\begin{equation}\label{e:Msharpdef}
\Mmat_N^\sharp(J) = \begin{pmatrix} \Mmat_N(J) & v_N(J) \\ z_N(J)^{\mathsf T} & d_N(J) \end{pmatrix} ,
\end{equation}
where $v_N(J), z_N(J) \in \R^{r + 1}$ have the entries
\begin{equation}\label{e:vzdef}
v_N(J)_0 = \frac12 \int_J \varphi_N , \qquad v_N(J)_j = - \frac12 , \qquad z_N(J)_\beta = \ip{Y^\sharp_N(J)}{Q_N(J)^{-1} X_\beta} ,
\end{equation}
for $1 \le j \le r$ and $0 \le \beta \le r$.
\end{definition}

\begin{corollary}\label{c:exact}
For all $N \ge 2$ and $J$ as in \eqref{e:Jdef}, we have
\begin{equation}\label{e:exactmatrix}
\Pgoe_N(J)^2 = \Dgue_N(J) \det \Mmat_N^\sharp(J) .
\end{equation}
If $N$ is even, then $z_N(J) = 0$ and $d_N(J) = 1$, so $\det \Mmat_N^\sharp(J) = \det \Mmat_N(J)$.
\end{corollary}

\begin{proof}
We write $Q = Q_N(J)$ and $\widetilde X_\alpha = Q^{-1} X_\alpha$. By \eqref{e:borderdef} and \eqref{e:vzdef}, we have
\begin{equation}
X^\sharp_N(J) = \sum_{\alpha} v_N(J)_\alpha X_\alpha .
\end{equation}
Therefore, $\mathbb T_N(J) = (Q \oplus 1) \, \mathbb T'$, where
\begin{equation}
\mathbb T' (f, a) = \Big( f - \sum_{\alpha = 0}^r \widetilde X_\alpha \big( \ip{Y_\alpha}{f} - a \, v_N(J)_\alpha \big) , \ a + \ip{Y^\sharp_N(J)}{f} + a (d_N(J) - 1) \Big) .
\end{equation}

We now write $\mathbb T'$ as the identity minus a sum of $r + 2$ rank-one operators. Set $U_\beta = (\widetilde X_\beta, 0)$ for $\beta \le r$ and $U_{r + 1} = (0, 1)$. We define
\begin{equation}
\Phi_\alpha (f, a) = \ip{Y_\alpha}{f} - a \, v_N(J)_\alpha
\end{equation}
for $\alpha \le r$ and
\begin{equation}
\Phi_{r + 1}(f, a) = - \ip{Y^\sharp_N(J)}{f} - a (d_N(J) - 1) ,
\end{equation}
and we let $U \otimes \Phi$ denote the operator $u \mapsto \Phi(u) U$. Then
\begin{equation}
\mathbb T' = \Id - \sum_{\alpha = 0}^{r} (\widetilde X_\alpha, 0) \otimes \Phi_\alpha - (0, 1) \otimes \Phi_{r + 1} ,
\end{equation}
so Sylvester's determinant identity implies that $\det \mathbb T'$ is the determinant of the $(r + 2) \times (r + 2)$ matrix with entries $\1_{\alpha = \beta} - \Phi_\alpha(U_\beta)$. Comparing entries, we see that this matrix coincides with $\Mmat^\sharp_N(J)$. Since $\det(Q \oplus 1) = \Dgue_N(J)$, \zcref{p:exact} now gives \eqref{e:exactmatrix}. The last claim follows from $c_N = 0$ for even $N$.
\end{proof}

\section{De Bruijn's formula}

Let $A \subset \R$ be a measurable set. We define the skew-symmetric bilinear form
\begin{equation}\label{e:Omegadef}
\Omega_A(f, g) = \ip{\1_A f}{\eps(\1_A g)} , \qquad f, g \in V_{N + 1} ,
\end{equation}
and, for odd $N$, the skew-symmetric bilinear form on $V_N \oplus \R$ given by
\begin{equation}\label{e:Omegahatdef}
\widehat \Omega_A \big( (f, a), (g, b) \big) = \Omega_A(f, g) - \frac b2 \ip{\1_A}{f} + \frac a2 \ip{\1_A}{g} .
\end{equation}
With a slight abuse of notation, we also let $\Omega_A$ denote the matrix of $\Omega_A$ in the basis $(h_n)_{n < N}$ of $V_N$, and we let $\widehat \Omega_A$ denote the matrix of $\widehat \Omega_A$ in the basis of $V_N \oplus \R$ given by the vectors $(h_n, 0)$, $n < N$, and $(0, 1)$.

\begin{lemma}\label{l:deBruijn}
If $N$ is even, then $\det \Omega_\R \neq 0$ and $\Pgoe_N(J)^2 = \det \Omega_{J^c} / \det \Omega_\R$. If $N$ is odd, then $\det \widehat \Omega_\R \neq 0$ and $\Pgoe_N(J)^2 = \det \widehat \Omega_{J^c} / \det \widehat \Omega_\R$.
\end{lemma}

\begin{proof}
Since $h_n$ is a polynomial of degree $n$ times $e^{-x^2/2}$, there exists a constant $\varsigma_N \neq 0$ such that
\begin{equation}
\det( h_m(x_i) )_{m = 0, i = 1}^{N - 1, N} = \varsigma_N \prod_{i < j} (x_j - x_i) \prod_i e^{-x_i^2/2} .
\end{equation}
The product $\prod_{i < j} (x_j - x_i)$ is positive on $\{ x_1 < \dots < x_N \}$. Comparing with \eqref{e:goehermite}, we find that for every measurable $A$,
\begin{equation}
\Prob\Big( \sqrt{\frac N2} \, \lambda_i \in A \text{ for all } i \Big) = \frac{1}{\varsigma_N \widetilde Z_N} \int_{x_1 < \dots < x_N, \ x_i \in A} \det( h_m(x_i) ) \dd x_1 \cdots \dd x_N .
\end{equation}

For even $N$, de Bruijn's formula \cite{deBruijn1955} states that the integral on the right is the Pfaffian of the matrix with entries $\iint_{A \times A} \sgn(y - x) h_m(x) h_n(y) \dd x \dd y$. Observe that this entry equals $-2 \Omega_A(h_m, h_n)$. For odd $N$, the corresponding formula (see \cite[Theorem~6.1]{BaikRains2001} or \cite[(2.2)]{Sinclair2009}) states that the integral is the Pfaffian of the same matrix bordered by the column $( \int_A h_m )_{m < N}$, its negative transpose, and a zero. This bordered matrix coincides with $-2 \widehat \Omega_A$.

Taking $A = \R$ shows that the Pfaffian is nonzero in either case. Taking $A = J^c$, dividing by the case $A = \R$, and squaring, we obtain the claim, since the square of a Pfaffian is the determinant and the factors of $-2$ cancel.
\end{proof}

\section{The derivative trick}

To compute the determinants in \zcref{l:deBruijn}, we replace the first argument of $\Omega_A$ by derivatives. Here and below, we write $\partial f = f'$. The idea is that $\eps$ inverts the derivative, so that $\Omega_\R(\partial f, g) = - \ip{f}{g}$. However, $\partial f$ does not lie in $V_N$ in general, and its projection $K_N \partial f$ differs from $\partial f$ by a multiple of the top mode $h_N$. The following lemma carries out this computation for $A = \R$ and $A = J^c$. Recall from \eqref{e:Y0def} that $Y_N(J) = \1_{J^c}(\omega_N - \eps(\1_J \varphi_N))$.

\begin{lemma}\label{l:derivativetrick}
For all $f, g \in V_N$, we have
\begin{align}
\Omega_\R(K_N \partial f, g) &= - \ip{f}{g} - \ip{\psi_N}{f} \ip{\eps \varphi_N}{g} , \label{e:trickR} \\
\Omega_{J^c}(K_N \partial f, g) &= - \ip{f}{T_N(J) g} + c_N \ip{\psi_N}{f} \ip{\1_{J^c}}{g} . \label{e:trickJ}
\end{align}
\end{lemma}

\begin{proof}
By \eqref{e:recurrence}, we have $\partial f \in V_{N + 1}$ and
\begin{equation}
\ip{h_N}{\partial f} = - \ip{\partial h_N}{f} = - \sqrt{N/2} \, \ip{h_{N - 1}}{f} .
\end{equation}
Therefore,
\begin{equation}\label{e:KNpartial}
K_N \partial f = \partial f + \sqrt{\frac N2} \ip{h_{N - 1}}{f} h_N = \partial f + \ip{\psi_N}{f} \varphi_N .
\end{equation}
We first consider $A = \R$. Integrating by parts and using $\partial \eps g = g$, we find that
\begin{equation}
\Omega_\R(\partial f, g) = \ip{\partial f}{\eps g} = - \ip{f}{g} .
\end{equation}
Also, $\Omega_\R(\varphi_N, g) = - \ip{\eps \varphi_N}{g}$. Applying $\Omega_\R(\cdot, g)$ to \eqref{e:KNpartial} and inserting these identities, we obtain \eqref{e:trickR}.

We now consider $A = J^c$. We set
\begin{equation}
\Psi_g = \eps g - \1_{J^c} \eps(\1_{J^c} g) = \1_J \eps g + \1_{J^c} \eps(\1_J g) ,
\end{equation}
so that
\begin{equation}
\Omega_{J^c}(\partial f, g) = \ip{\partial f}{\eps g} - \ip{\partial f}{\Psi_g} = - \ip{f}{g} - \ip{\partial f}{\Psi_g} .
\end{equation}
The restriction of $\Psi_g$ to each interval $J_j$ extends continuously to the closure $\overline{J_j}$, and it has derivative $g$ on $J_j$. It is constant on each connected component of $J^c$, since $\eps(\1_J g)$ has derivative $\1_J g$, so its jumps occur only at the points $a_j$ and $b_j$. We have
\begin{equation}
\Psi_g(a_j^-) - \Psi_g(a_j^+) = (\eps(\1_J g))(a_j) - (\eps g)(a_j) = - (\eps(\1_{J^c} g))(a_j) = \ip{w_j}{g} ,
\end{equation}
where the last equality follows from $(\eps h)(t) = - \ip{\eps_t}{h}$ and $\1_{J^c} \eps_{a_j} = w_j$. In the same way, we find that
\begin{equation}
\Psi_g(b_j^-) - \Psi_g(b_j^+) = (\eps(\1_{J^c} g))(b_j) = - \ip{w_j}{g} ,
\end{equation}
where we used $\1_{J^c} \eps_{b_j} = w_j$ almost everywhere.

Integrating by parts on each interval where $\Psi_g$ is continuous, we find
\begin{equation}
\begin{split}
\ip{\partial f}{\Psi_g} & = \sum_{j = 1}^r \big( f(a_j) - f(b_j) \big) \ip{w_j}{g} - \ip{\1_J f}{g} \\
& = - \sum_{j = 1}^r \ip{f}{k_j} \ip{w_j}{g} - \ip{f}{K_N(\1_J g)} ,
\end{split}
\end{equation}
where in the last step we used $f \in V_N$. Further, the skew-symmetry of $\eps$ gives
\begin{equation}
\Omega_{J^c}(\varphi_N, g) = - \ip{\1_{J^c} \eps(\1_{J^c} \varphi_N)}{g} = - \ip{Y_N(J)}{g} + c_N \ip{\1_{J^c}}{g} ,
\end{equation}
where we used
\begin{equation}
\eps(\1_{J^c} \varphi_N) = \eps \varphi_N - \eps(\1_J \varphi_N) = \omega_N - c_N - \eps(\1_J \varphi_N) .
\end{equation}
Applying $\Omega_{J^c}(\cdot, g)$ to \eqref{e:KNpartial} and combining the identities above, we have
\begin{equation}
\begin{split}
\Omega_{J^c}(K_N \partial f, g) & = - \ip{f}{g} + \ip{f}{K_N(\1_J g)} + \sum_{j = 1}^r \ip{f}{k_j} \ip{w_j}{g} \\
& \quad - \ip{\psi_N}{f} \big( \ip{Y_N(J)}{g} - c_N \ip{\1_{J^c}}{g} \big) ,
\end{split}
\end{equation}
which is \eqref{e:trickJ} by the definition \eqref{e:TNdef}.
\end{proof}

\section{Proof of \texorpdfstring{\zcref{p:exact}}{Proposition 3.1}}

Let $B$ denote the $N \times N$ matrix with entries $B_{\ell m} = \ip{h_\ell}{\partial h_m}$. Then
\begin{equation}
K_N \partial h_m = \sum_\ell B_{\ell m} h_\ell ,
\end{equation}
so for every measurable $A$, the matrix with entries $\Omega_A(K_N \partial h_m, h_n)$ is given by $B^{\mathsf T} \Omega_A$.

Suppose first that $N$ is even. Then $c_N = 0$, and \zcref{l:derivativetrick} shows that the matrices $- B^{\mathsf T} \Omega_\R$ and $- B^{\mathsf T} \Omega_{J^c}$ are the matrices of the operators $\Id + \psi_N \otimes \eps \varphi_N$ and $T_N(J)$ in the basis $(h_n)_{n < N}$, respectively. The first operator has determinant $1 + \ip{\eps \varphi_N}{\psi_N} = 1$, by \zcref{l:parity}(c) with $c_N = 0$. In particular, $\det B \neq 0$, and \zcref{l:deBruijn} gives
\begin{equation}
\Pgoe_N(J)^2 = \frac{\det \Omega_{J^c}}{\det \Omega_\R} = \frac{\det(- B^{\mathsf T} \Omega_{J^c})}{\det(- B^{\mathsf T} \Omega_\R)} = \det T_N(J) .
\end{equation}

Suppose now that $N$ is odd, and set
\begin{equation}
\nu = 2 c_N (N/2)^{1/4} .
\end{equation}
We define the vectors
\begin{equation}
u_m = \big( K_N \partial h_m, \ - \nu \1_{m = N - 1} \big) , \quad 0 \le m < N , \qquad u_N = (\nu h_{N - 1}, 0) ,
\end{equation}
in $V_N \oplus \R$, and we set $e_n = (h_n, 0)$ for $n < N$ and $e_N = (0, 1)$. Let $\Gamma$ denote the matrix with
\begin{equation}
u_\alpha = \sum_\beta \Gamma_{\alpha \beta} e_\beta .
\end{equation}
Then the matrix with entries $- \widehat \Omega_A(u_\alpha, e_\beta)$ is given by $- \Gamma \widehat \Omega_A$. We now compute these entries for $A \in \{ \R, J^c \}$, using \zcref{l:derivativetrick} and the identity $\nu \1_{m = N - 1} / 2 = c_N \ip{\psi_N}{h_m}$.

We begin with the case $m, n < N$. Then \eqref{e:trickJ} gives
\begin{equation}
- \widehat \Omega_{J^c}(u_m, e_n) = - \Omega_{J^c}(K_N \partial h_m, h_n) + c_N \ip{\psi_N}{h_m} \ip{\1_{J^c}}{h_n} = \ip{h_m}{T_N(J) h_n} ,
\end{equation}
and in the same way, \eqref{e:trickR} gives $- \widehat \Omega_\R(u_m, e_n) = \ip{h_m}{(\Id + \psi_N \otimes \omega_N) h_n}$, where we used $\omega_N = \eps \varphi_N + c_N$.

Next, let $m < N$ and $n = N$. Then the definition \eqref{e:Omegahatdef} gives $- \widehat \Omega_A(u_m, e_N) = \ip{\1_A}{K_N \partial h_m} / 2$. When $A = \R$, this quantity equals
\begin{equation}
\ip{p_N}{\partial h_m} / 2 = - \ip{p_N'}{h_m} / 2 = 0 ,
\end{equation}
where we used \eqref{e:pNderivative}, $\int \varphi_N = 0$, and the orthogonality of $\varphi_N$ to $V_N$. When $A = J^c$, we combine the case $A = \R$ with \eqref{e:KNpartial} to obtain
\begin{equation}
\begin{split}
\frac12 \ip{\1_{J^c}}{K_N \partial h_m} & = - \frac12 \ip{\1_J}{\partial h_m + \ip{\psi_N}{h_m} \varphi_N} \\
& = - \frac12 \sum_{j = 1}^r \ip{k_j}{h_m} - \frac12 \ip{\psi_N}{h_m} \int_J \varphi_N = \ip{h_m}{X^\sharp_N(J)} .
\end{split}
\end{equation}

It remains to compute the entries with $\alpha = N$. For $n < N$, we have $- \widehat \Omega_A(u_N, e_n) = - \nu \, \Omega_A(h_{N - 1}, h_n)$. When $A = \R$, \zcref{l:antiderivatives}(c) implies that
\begin{equation}
\Omega_\R(h_{N - 1}, h_n) = - \ip{\eps h_{N - 1}}{h_n} = 0 ,
\end{equation}
so this entry vanishes. To treat $A = J^c$, we expand $\1_{J^c} = 1 - \1_J$ and use the skew-symmetry of $\eps$. This gives, for all $f, g \in V_N$,
\begin{equation}\label{e:OmegaJminusR}
\Omega_{J^c}(f, g) - \Omega_\R(f, g) = \ip{\1_J \eps f + \1_{J^c} \eps(\1_J f)}{g} .
\end{equation}
Taking $f = h_{N - 1}$ and using $\nu h_{N - 1} = 2 c_N \psi_N$, we find
\begin{equation}
- \widehat \Omega_{J^c}(u_N, e_n) = \ip{Y^\sharp_N(J)}{h_n} .
\end{equation}
Finally, we have
\begin{equation}
- \widehat \Omega_A(u_N, e_N) = (\nu/2) \ip{\1_A}{h_{N - 1}} = c_N \ip{\1_A}{\psi_N} ,
\end{equation}
which takes the value $1$ for $A = \R$ and the value $d_N(J)$ for $A = J^c$.

Combining these computations, we see that $- \Gamma \widehat \Omega_{J^c}$ is the matrix of $\mathbb T_N(J)$ in the basis $(e_\beta)$, and that $- \Gamma \widehat \Omega_\R$ is the matrix of $(\Id + \psi_N \otimes \omega_N) \oplus 1$, which has determinant $1 + \ip{\omega_N}{\psi_N} = 1$ by \zcref{l:parity}(c). In particular, $\det \Gamma \neq 0$, and \zcref{l:deBruijn} gives
\begin{equation}
\Pgoe_N(J)^2 = \frac{\det \widehat \Omega_{J^c}}{\det \widehat \Omega_\R} = \frac{\det(- \Gamma \widehat \Omega_{J^c})}{\det(- \Gamma \widehat \Omega_\R)} = \det \mathbb T_N(J) .
\end{equation}
This completes the proof. \qed