\chapter{An inverse bound from the uncertainty principle}\label{s:inverse}

The entries of the matrix $\Mmat_N^\sharp(J)$ in \zcref{c:exact} involve the inverse of $\Id - K_J$. The goal of this section is to show that its norm grows at most exponentially in the length of $J$ measured on the microscopic scale. Given a bounded measurable set $J \subset \R$, we define its \emph{microscopic length}
\begin{equation}\label{e:Ldef}
L(J) = \sqrt{2N} \, |J| .
\end{equation}
In eigenvalue coordinates, we have $L(J) = N |A|$ for $J = \sqrt{N/2} \, A$, so $L(J)$ is comparable to the expected number of eigenvalues in $J$ when $J$ is in the bulk.

Since every eigenvalue of $K_J$ belongs to $[0, 1)$, we have $\| (\Id - K_J)^{-1} \| \le \det(\Id - K_J)^{-1}$. Since $\det(\Id - K_J)$ decays like $e^{-c L(J)^2}$ for an interval, this only gives a bound of the form $\| (\Id - K_J)^{-1} \| \le e^{C L(J)^2}$. However, this is not enough for our purposes. We will take $L(J)$ of order $\sqrt{\log N}$, where $e^{C L(J)^2}$ is a power of $N$ and would overwhelm the polynomially small errors in \zcref{s:bulk}. In this regime, the bound $e^{C L(J)}$ proved below is much smaller than any power of $N$. The main input to its proof is a quantitative uncertainty principle of Nazarov \cite{Nazarov1993}, in the form given by Jaming \cite{Jaming2007}.

\begin{proposition}\label{p:inverse}
There exist absolute constants $C, N_0 > 0$ such that the following holds. Let $N \ge N_0$, and let $J \subset \R$ be measurable with $|J| \le 1$. Then
\begin{equation}\label{e:inverse}
\| (\Id - K_J)^{-1} \|_{L^2(J) \to L^2(J)} \le C e^{C L(J)} .
\end{equation}
Further, for every measurable set $J \subset \R$ of finite measure, the integral operator $S_J$ on $L^2(J)$ with kernel $S(u - v)$, $u, v \in J$, satisfies $\| (\Id - S_J)^{-1} \| \le C e^{C |J|}$.
\end{proposition}

We will use the following one-dimensional form of Nazarov's uncertainty principle \cite{Nazarov1993}, as stated in \cite[Theorem]{Jaming2007}. Recall that $\mathcal F$ denotes the unitary Fourier transform from \eqref{e:fourierdef}.

\begin{lemma}\label{l:nazarov}
There exists an absolute constant $C_0 > 0$ such that for all measurable sets $A, \Sigma \subset \R$ of finite measure and all $f \in L^2(\R)$,
\begin{equation}\label{e:nazarov}
\| f \|_2^2 \le C_0 e^{C_0 |A| |\Sigma|} \Big( \int_{A^c} |f(x)|^2 \dd x + \int_{\Sigma^c} |(\mathcal F f)(\xi)|^2 \dd \xi \Big) .
\end{equation}
\end{lemma}

\begin{proof}
We define
\begin{equation}
\widehat f(\eta) = \int_\R f(x) e^{2 \pi i x \eta} \dd x .
\end{equation}
In \cite{Jaming2007}, the inequality is stated with a Fourier transform of this form, normalized so that it is unitary on $L^2(\R)$; the sign in the exponent is irrelevant, since replacing $\Sigma$ by $- \Sigma$ does not change $|\Sigma|$. In dimension one, the factor $e^{C \min(\cdot)}$ there is bounded by $e^{C |A| |\Sigma|}$, and the result is due to Nazarov \cite{Nazarov1993}. We set
\begin{equation}
\Sigma' = - \Sigma / (2 \pi) .
\end{equation}
Using
\begin{equation}
(\mathcal F f)(\xi) = (2 \pi)^{-1/2} \widehat f(- \xi / (2\pi))
\end{equation}
and changing variables, we have
\begin{equation}
\int_{\Sigma^c} |\mathcal F f|^2 = \int_{\Sigma'^c} |\widehat f|^2 .
\end{equation}
Together with $|\Sigma'| = |\Sigma| / (2\pi)$ and the Plancherel identity $\| \widehat f \|_2 = \| f \|_2$, this gives \eqref{e:nazarov}.
\end{proof}

\begin{proof}[Proof of \zcref{p:inverse}]
Set
\begin{equation}
\Sigma_N = [- 8 \sqrt N, 8 \sqrt N] .
\end{equation}
We first show that every $f \in V_N$ satisfies
\begin{equation}\label{e:VNtail}
\int_{\Sigma_N^c} |f(x)|^2 \dd x \le e^{-N} \| f \|_2^2 .
\end{equation}
Indeed, \eqref{e:KCS} and \zcref{l:hermitetail} imply that
\begin{equation}
|f(x)|^2 \le K_N(x, x) \| f \|_2^2 \le N e^{-x^2/4} \| f \|_2^2
\end{equation}
for $|x| \ge 8 \sqrt N$, and
\begin{equation}
\int_{|x| \ge 8 \sqrt N} N e^{-x^2/4} \dd x \le 2N \int_{8 \sqrt N}^\infty e^{- 2 \sqrt N x} \dd x \le e^{-N} ,
\end{equation}
where we used $x^2 / 4 \ge 2 \sqrt N x$ for $x \ge 8 \sqrt N$ in the first inequality, and $N \ge N_0$ in the second.

Now let $f \in V_N$. It follows from \eqref{e:hermitefourier} that the function $\mathcal F f$ belongs to the complex span of $h_0, \dots, h_{N - 1}$, and $\| \mathcal F f \|_2 = \| f \|_2$. The proof of \eqref{e:VNtail} applies verbatim to this complex span. Applying \zcref{l:nazarov} with $A = J$ and $\Sigma = \Sigma_N$, and then \eqref{e:VNtail} to $\mathcal F f$, we find
\begin{equation}
\| f \|_2^2 \le C_0 e^{16 C_0 \sqrt N |J|} \Big( \int_{J^c} |f|^2 + e^{-N} \| f \|_2^2 \Big) .
\end{equation}

Since $16 \sqrt N |J| \le 12 L(J)$ and $L(J) \le \sqrt{2N}$, we have
\begin{equation}
C_0 e^{16 C_0 \sqrt N |J| - N} \le 1/2
\end{equation}
for $N \ge N_0$. Therefore,
\begin{equation}\label{e:concentration}
\| \1_J f \|_2^2 \le \big( 1 - c \, e^{- C L(J)} \big) \| f \|_2^2 , \qquad f \in V_N ,
\end{equation}
with $c = (2 C_0)^{-1}$ and $C = 12 C_0$. This means that the operator $K_N \1_J K_N$ on $L^2(\R)$ has norm at most $1 - c e^{-C L(J)}$. The operator $K_J$ on $L^2(J)$ equals $(\1_J K_N)(\1_J K_N)^*$, while $K_N \1_J K_N = (\1_J K_N)^* (\1_J K_N)$, so the two operators have the same norm. Since $K_J$ is self-adjoint and nonnegative, we conclude that
\begin{equation}
\| (\Id - K_J)^{-1} \| \le c^{-1} e^{C L(J)} .
\end{equation}

Finally, we consider the sine kernel. Let $P$ denote the orthogonal projection of $L^2(\R)$ onto the functions $f$ with $\mathcal F f$ supported in $[-1, 1]$. Then $S_J = \1_J P \1_J$. Applying \zcref{l:nazarov} with $A = J$ and $\Sigma = [-1, 1]$ to such $f$, we find
\begin{equation}
\| \1_J f \|_2^2 \le (1 - C_0^{-1} e^{- 2 C_0 |J|}) \| f \|_2^2 ,
\end{equation}
and the rest of the argument is the same as above.
\end{proof}

We will use \zcref{p:inverse} through the following consequence for values at single points. For a bounded measurable set $J$ and a bounded function $f$ on $J$, the function $(\Id - K_J)^{-1} f$ is a priori only an element of $L^2(J)$. However, we have
\begin{equation}\label{e:resolventidentity}
(\Id - K_J)^{-1} f = f + K_J (\Id - K_J)^{-1} f ,
\end{equation}
and the second term on the right is the restriction to $J$ of a function in $V_N$. We therefore use the right side of \eqref{e:resolventidentity} to define $(\Id - K_J)^{-1} f$ at every point of the closure of $J$ at which $f$ is defined. In particular, if $J$ is a finite union of open intervals and $f$ is continuous on the closure of each of them, then $(\Id - K_J)^{-1} f$ has one-sided limits at all endpoints.

\begin{lemma}\label{l:pointvalues}
Let $\delta \in (0, 1/2]$. There exist $C, N_0 > 0$ depending only on $\delta$ such that the following holds for $N \ge N_0$. Let $J \subset \bulk_N(\delta)$ be measurable with $|J| \le 1$, let $f$ be a bounded function on $J$, and set $G = (\Id - K_J)^{-1} f$. Then
\begin{equation}\label{e:pointvalues}
\Big| \int_J K_N(t, z) G(z) \dd z \Big| \le C e^{C L(J)} \sup_J |f| , \qquad t \in \bulk_N(\delta) .
\end{equation}
Consequently, $|G(t)| \le C e^{C L(J)} \sup_J |f|$ for every $t \in J$, with $G(t)$ defined through \eqref{e:resolventidentity}. The same bound holds for boundary values obtained as limits from within $J$.
\end{lemma}

\begin{proof}
Using the Cauchy--Schwarz inequality, \zcref{l:pointwise}(b), and \zcref{p:inverse}, we find that the left side of \eqref{e:pointvalues} is bounded by
\begin{equation}
K_N(t, t)^{1/2} \, \| G \|_{L^2(J)} \le C N^{1/4} e^{C L(J)} |J|^{1/2} \sup_J |f| \le C e^{C L(J)} L(J)^{1/2} \sup_J |f| .
\end{equation}
Since $L(J)^{1/2} \le e^{L(J)}$, the last display implies \eqref{e:pointvalues}, and the bound for $G$ on $J$ follows from \eqref{e:resolventidentity}. Passing to limits gives the boundary assertion.
\end{proof}