\chapter{Weighted moment measures and extraction of eigenvector moments}\label{s:extract}

In this section, we show how resolvent averages at a fixed positive damping, which are the quantities controlled by \zcref{t:interface}, determine expected products of eigenvector weights over arbitrarily small energy windows. The main result is \zcref{t:extract-main}. In \zcref{s:intensity}, we apply it to pairs of sites, which gives the local profile and the inverse participation ratio of \zcref{t:profile}. In \zcref{s:coarse}, it is applied to products over $k$ sites, which give the coarse eigenvector law.

We recall the mechanism from \zcref{s:ov-eigenvectors}. For two sites $i$ and $j$, the terms with $\alpha = \beta$ of the measure $N^2 \sum_{\alpha, \beta} w_\alpha(i) w_\beta(j) \delta_{(x_\alpha, x_\beta)}$ lie on the diagonal and form the eigenvector moment that we want. By \eqref{e:ov-poisson}, a diagonal part with constant density $c$ contributes $c/(2\pi\eps)$ to $\E \ell_{i, N}^\eps(0) \ell_{j, N}^\eps(0)$, while a measure that gives no mass to the diagonal contributes $o(\eps^{-1})$ as $\eps \downarrow 0$, so that the residue at $\eps = 0$ should identify $c$. In physics, eigenvector moments are computed from such residues \cite[Section~3.1]{mirlin2000statistics}.

Since the limits of \zcref{t:interface} are taken at fixed $\eps > 0$, and the residue only afterward, we pass to vague limits of the expected measures. \zcref[S]{s:extract-measures} defines these measures and derives their moment bounds from \zcref{t:interface}(c). In \zcref{s:extract-mixtures}, we show that the scalar Gaussian component of $H$ makes the expected diagonal measures Gaussian mixtures, for which vague convergence implies locally uniform convergence of the densities. The main step is \zcref{l:extract-limits}, which proves that every vague limit is stationary, that its part with distinct eigenvalue labels gives no mass to the diagonal, and that Poisson kernels may be integrated against it. For the second property, one might expect that products over $k$ sites require estimates on $k$ eigenvalues, but the bound on pairs of eigenvalues, which is the case $m = 2$ of \zcref{l:iface-counts}, is enough for every $k$. We deduce the extraction theorem from these properties in \zcref{s:extract-theorem}, and in \zcref{s:extract-windows} we pass from densities to energy windows of arbitrarily small width.

\paragraph{Conventions.} Recall the energies $E_* < E_{\mathrm{mid}} < E_{\mathrm{out}}$ of \eqref{e:iface-energies} and their roles (\zcref{r:iface-energies}). In this section, we consider sequences $N, W \to \infty$ with $N/W^2$ in a compact subset of $(0, \infty)$. The energy $E = E_N$, the sites and all other parameters may depend on $N$, and $a_N$ and $x_\alpha = a_N(\lambda_\alpha - E)$ refer to the energy $E$, as in \eqref{e:scaling}. Since our statements hold along every such sequence, they imply statements that are uniform over compact parameter sets; we spell this out in \zcref{r:extract-uniform}.

\section{Weighted moment measures}\label{s:extract-measures}

The local densities of \zcref{s:intro} are Poisson smoothings of the following random measures.

\begin{definition}\label{d:extract-measures}
For a site $i \in \{1, \dots, N\}$, the \emph{unfolded local spectral measure} at $i$ is the random measure
\begin{equation}\label{e:extract-local-measure}
 \ell_{i, N} = N \sum_{\alpha = 1}^N w_\alpha(i) \delta_{x_\alpha}
\end{equation}
on $\R$. Let $k \ge 1$. We call a random measure $\bm\Pi_N$ on $\R^k$ an \emph{eigenvector-weighted moment measure of order $k$} if it has the form
\[
 \bm\Pi_N = \sum_{\alpha_1, \dots, \alpha_k = 1}^N c_N(\alpha_1, \dots, \alpha_k) \delta_{(x_{\alpha_1}, \dots, x_{\alpha_k})},
\]
where each coefficient $c_N(\alpha_1, \dots, \alpha_k) \ge 0$ is a measurable function of the eigenprojections $u_{\alpha_1} u_{\alpha_1}^*, \dots, u_{\alpha_k} u_{\alpha_k}^*$. Let $\iota_k \colon \R \to \R^k$ denote the diagonal embedding $\iota_k(t) = (t, \dots, t)$, and let $r_k > 0$ be a normalization constant. We set
\[
 \bm\Pi_N^{\mathrm d} = \frac{1}{r_k} \sum_{\alpha = 1}^N c_N(\alpha, \dots, \alpha) \delta_{x_\alpha},
 \qquad
 \bm\Pi_N^{\ne} = \bm\Pi_N - r_k (\iota_k)_* \bm\Pi_N^{\mathrm d},
\]
and we call $r_k (\iota_k)_* \bm\Pi_N^{\mathrm d}$ the \emph{diagonal} (\emph{coincident-eigenvalue}) \emph{part} of $\bm\Pi_N$ and $\bm\Pi_N^{\ne}$ its \emph{off-diagonal part}. We write $\Pi_N = \E \bm\Pi_N$, $\Pi_N^{\mathrm d} = \E \bm\Pi_N^{\mathrm d}$, and $\Pi_N^{\ne} = \E \bm\Pi_N^{\ne}$. Finally, let $i_1, \dots, i_k$ be sites and $C_0 > 0$. We say that $\bm\Pi_N$ is \emph{dominated at the sites $i_1, \dots, i_k$}, with constant $C_0$, if for all nonnegative measurable functions $f_1, \dots, f_k$ on $\R$,
\begin{equation}\label{e:extract-domination}
 \bm\Pi_N(f_1 \otimes \dots \otimes f_k) \le C_0 \sum_{\pi \in S_k} \prod_{a = 1}^k \ell_{i_a, N}(f_{\pi(a)}).
\end{equation}
\end{definition}

The eigenprojections in \zcref{d:extract-measures} are almost surely well defined. Indeed, all entries of $J$ are positive by \zcref{l:iface-variance}, so that $H$ has a density with respect to Lebesgue measure on Hermitian matrices, and its eigenvalues are almost surely simple. By the definition of $\ell_{i, N}^\eps$ in \zcref{s:intro}, we have
\begin{equation}\label{e:extract-smoothing}
 \ell_{i, N}^\eps(u) = \int_\R P_\eps(x - u) \, \ell_{i, N}(\dd x) = (P_\eps * \ell_{i, N})(u),
\end{equation}
and $\ell_{i, N}(\R) = N$, since $u_\alpha$ is a unit vector. With $f_a = 1$, the domination \eqref{e:extract-domination} therefore implies that $\bm\Pi_N(\R^k) \le C_0 k! N^k$.

The basic example is the product $\bm\Pi_N = \ell_{i, N} \otimes \ell_{j, N}$, with coefficients $c_N(\alpha, \beta) = N^2 w_\alpha(i) w_\beta(j)$ and $r_2 = 1$. It is dominated at $(i, j)$ with $C_0 = 1$, and
\[
 \bm\Pi_N^{\mathrm d} = N^2 \sum_\alpha w_\alpha(i) w_\alpha(j) \delta_{x_\alpha} .
\]
In this case, $\Pi_N$ is a mixed second moment measure of the random measures $\ell_{i, N}$ and $\ell_{j, N}$. Its off-diagonal part $\Pi_N^{\ne}$, which is carried by pairs of distinct labels, is an eigenvector-weighted second factorial-moment measure; see \cite{daley2008introduction} for moment and factorial-moment measures of point processes. For $k \ge 3$, the off-diagonal part also contains label tuples with partial coincidences. In \zcref{s:coarse}, the coefficients are squared moduli of permanents of eigenvector entries, which depend on the eigenvectors only through the eigenprojections, and $r_k = k!$.

Under $H \mapsto H + b \Id$ with $b \in \R$, the eigenprojections and the order of the eigenvalues do not change, and every $\lambda_\alpha$ increases by $b$. Therefore the coefficients of an eigenvector-weighted moment measure are invariant under scalar translations of the matrix, and the measure itself is translated by $a_N b (1, \dots, 1)$.

All estimates in this section rest on bounds for moments of local densities, which we now derive from part (c) of \zcref{t:interface}.

\begin{lemma}\label{l:extract-moments}
Let $p \ge 1$ and let $\mathcal K \subset (0, \infty)$ be compact. There exists $C$ such that for all sufficiently large $N$, all sites $i$, all $\eps \in \mathcal K$, all $|E| \le E_{\mathrm{out}}$ and all $u \in \R$ with $|E + u/a_N| \le E_{\mathrm{out}}$,
\begin{equation}\label{e:extract-moments}
 \E \bigl[\ell_{i, N}^\eps(u)^p\bigr] \le C .
\end{equation}
For every $m \ge 1$, the same bound holds for $\E \prod_{a = 1}^m \ell_{i_a, N}^{\eps_a}(u_a)$, uniformly over sites, dampings, and centers as above.
\end{lemma}

\begin{proof}
We recenter at $E' = E + u/a_N$. Let $a_N(E') = N\rho(E')$, and let
\[
 x'_\alpha = a_N(E')(\lambda_\alpha - E'), \qquad\text{so that}\qquad x_\alpha - u = a_N(\lambda_\alpha - E') = \frac{\rho(E)}{\rho(E')} x'_\alpha .
\]
Since $P_\eps(cy) = c^{-1} P_{\eps/c}(y)$ for $c > 0$, we have
\[
 \ell_{i, N}^\eps(u) = \frac{\rho(E')}{\rho(E)} N \sum_\alpha w_\alpha(i) P_{\eps'}(x'_\alpha),
 \qquad \eps' = \frac{\eps \rho(E')}{\rho(E)} .
\]
Note that
\[
 \frac{\rho(E_{\mathrm{out}})}{\rho(0)} \le \frac{\rho(E')}{\rho(E)} \le \frac{\rho(0)}{\rho(E_{\mathrm{out}})} ,
\]
and in particular $\eps'$ ranges over a compact subset of $(0, \infty)$ when $\eps$ does.

By the elementary bound
\begin{equation}\label{e:extract-poisson-compare}
 P_{\eps'} \le \max\Bigl(\frac{\eps'}{\eps''}, \frac{\eps''}{\eps'}\Bigr) P_{\eps''}, \qquad \eps', \eps'' > 0,
\end{equation}
with $\eps'' = 1$, we have
\[
 \ell_{i, N}^\eps(u) \le C \, \frac{1}{s_{E'}} \Im G_{ii}\bigl(E' + \iu/a_N(E')\bigr) \le C \bigl|G_{ii}\bigl(E' + \iu/a_N(E')\bigr)\bigr| .
\]
Further, $X_N(E') = N/((4 - E'^2) W^2)$ ranges over a compact subset of $(0, \infty)$. We apply \zcref{t:interface}(c) with $u = 0$, damping one and moment order $2p$, at the energy $E'$. Passing to subsequences along which $X_N(E')$ converges, we find
\[
 \sup_{1 \le i \le N} \, \sup_{|E'| \le E_{\mathrm{out}}} \E \bigl|G_{ii}\bigl(E' + \iu/a_N(E')\bigr)\bigr|^{2p} \le C .
\]
Combining this with the previous display and Jensen's inequality, we obtain \eqref{e:extract-moments}. The bound for products follows from H\"older's inequality.
\end{proof}

\section{Gaussian mixtures}\label{s:extract-mixtures}

The matrix $H$ contains a scalar Gaussian component. Recall from \zcref{l:iface-gaussian} that $H = \widetilde H + \gamma_N \Id$, where $\gamma_N$ is a centered Gaussian variable of variance $\sigma_N^2$ that is independent of $\widetilde H$. We set $\tau_N = a_N \sigma_N$, the standard deviation of $a_N \gamma_N$. By \zcref{l:iface-gaussian}, there exists $C$, depending only on $E_{\mathrm{out}}$ and the compact range of $N/W^2$, such that
\begin{equation}\label{e:extract-tau}
 \tau_N^{-1} \le C W^{-1/2}, \qquad |E| \le E_{\mathrm{out}} .
\end{equation}
We let $\mathsf n_\tau$ denote the centered Gaussian density on $\R$ with variance $\tau^2$. A change of $\gamma_N$ shifts every unfolded eigenvalue by one common amount and does not change the eigenvectors. As a consequence, expected eigenvector-weighted spectral measures are Gaussian mixtures.

\begin{lemma}\label{l:extract-gaussian}
Let $\bm\mu_N = \sum_\alpha c_N(\alpha) \delta_{x_\alpha}$, where $c_N(\alpha) \ge 0$ is a measurable function of $u_\alpha u_\alpha^*$, and suppose that $\E \bm\mu_N(\R) < \infty$. Then there exists a finite positive measure $m_N$ on $\R$ with $m_N(\R) = \E \bm\mu_N(\R)$ such that $\E \bm\mu_N = m_N * \mathsf n_{\tau_N}$. In particular, $\E \bm\mu_N$ has a smooth density.
\end{lemma}

\begin{proof}
Let $\tilde\lambda_\alpha$ denote the eigenvalues of $\widetilde H$, in increasing order, and set $\tilde x_\alpha = a_N(\tilde\lambda_\alpha - E)$. The eigenprojections of $H$ and $\widetilde H$ coincide, and $x_\alpha = \tilde x_\alpha + a_N \gamma_N$. Therefore $\bm\mu_N$ is the translate by $a_N \gamma_N$ of the measure
\[
 \widetilde{\bm\mu}_N = \sum_\alpha c_N(\alpha) \delta_{\tilde x_\alpha},
\]
which is a function of $\widetilde H$. Since $a_N \gamma_N$ has density $\mathsf n_{\tau_N}$ and is independent of $\widetilde H$, Fubini's theorem gives $\E \bm\mu_N = m_N * \mathsf n_{\tau_N}$ with $m_N = \E \widetilde{\bm\mu}_N$.
\end{proof}

By \zcref{l:extract-gaussian}, the expected diagonal parts $\Pi_N^{\mathrm d}$ are Gaussian mixtures. We will need the following regularity property of such mixtures.

\begin{lemma}\label{l:extract-mixture}
Let $m$ be a finite positive measure on $\R$, let $\tau > 0$, and set $f = m * \mathsf n_\tau$. Then $u \mapsto \log f(u) + u^2/(2\tau^2)$ is convex, and for all $u_0 \in \R$,
\begin{equation}\label{e:extract-mixture-bound}
 f(u_0) \le \frac12 e^{1/(2\tau^2)} \int_{u_0 - 1}^{u_0 + 1} f(u) \dd u .
\end{equation}
Further, let $f_N = m_N * \mathsf n_{\tau_N}$ for finite positive measures $m_N$ and numbers $\tau_N$ with $\inf_N \tau_N > 0$. If $f_N(u) \dd u \to c \dd u$ vaguely for some $c \ge 0$, then $f_N \to c$ locally uniformly.
\end{lemma}

\begin{proof}
We may assume that $m \ne 0$. We have
\[
 f(u) e^{u^2/(2\tau^2)} = \frac{1}{\sqrt{2\pi}\, \tau} \int_\R e^{ut/\tau^2} e^{-t^2/(2\tau^2)} \, m(\dd t) .
\]
The right side is the Laplace transform of the positive measure $e^{-t^2/(2\tau^2)} m(\dd t)$, evaluated at $u/\tau^2$. Laplace transforms of positive measures are log-convex by H\"older's inequality, and the first claim follows. Next, fix $u_0$ and set
\[
 \psi(u) = \log f(u) + \frac{(u - u_0)^2}{2\tau^2} ,
\]
which differs from $\log f(u) + u^2/(2\tau^2)$ by an affine function and is therefore convex. For $0 \le h \le 1$, the inequality $2\psi(u_0) \le \psi(u_0 + h) + \psi(u_0 - h)$ gives
\[
 f(u_0) \le e^{h^2/(2\tau^2)} \sqrt{f(u_0 + h) f(u_0 - h)} \le \frac12 e^{1/(2\tau^2)} \bigl(f(u_0 + h) + f(u_0 - h)\bigr) .
\]
Integrating over $h \in [0, 1]$, we obtain \eqref{e:extract-mixture-bound}.

We now prove the second claim. Fix $R > 0$ and set $\tau_0 = \inf_N \tau_N$. The indices with $m_N = 0$ satisfy $f_N = 0$, and we may discard them. Using \eqref{e:extract-mixture-bound} and the vague convergence, we see that $f_N$ is bounded on $[-R - 1, R + 1]$, uniformly in $N$. Hence there exists $M$ such that the convex functions
\[
 \psi_N(u) = \log f_N(u) + \frac{u^2}{2\tau_N^2}
\]
satisfy $\psi_N \le M$ on $[-R - 1, R + 1]$ for all $N$.

Let $|u| \le R$. By convexity, the one-sided derivatives $\psi_N'(u \pm)$ satisfy
\[
 \psi_N(u) - M \le \psi_N(u) - \psi_N(u - 1) \le \psi_N'(u \pm) \le \psi_N(u + 1) - \psi_N(u) \le M - \psi_N(u) .
\]
Using $x(M - \log x) \le e^{M - 1}$ for $0 < x \le e^M$, we find that $e^{\psi_N}$ has Lipschitz constant at most $e^{M - 1}$ on $[-R, R]$. As $\tau_N \ge \tau_0$, the functions $f_N = e^{\psi_N} e^{-u^2/(2\tau_N^2)}$ are then uniformly bounded and equi-Lipschitz on $[-R, R]$. By the Arzel\`a--Ascoli theorem, every subsequence has a further subsequence that converges uniformly on $[-R, R]$, and the vague convergence identifies the limit as the constant $c$, also when $c = 0$.
\end{proof}

\section{Stationary limits, collisions, and Poisson tails}\label{s:extract-limits}

We now prove the three properties of vague limits that were described at the beginning of the section. Recall that a positive Radon measure $\nu$ on $\R^d$ is translation bounded if $\sup_{s \in \R^d} \nu(s + [0, 1]^d) < \infty$.

\begin{lemma}\label{l:extract-limits}
Let $k \ge 1$, and let $\bm\Pi_N$ be eigenvector-weighted moment measures of order $k$ that are dominated at sites $i_1, \dots, i_k$, with a constant $C_0$ independent of $N$. Suppose that $|E| \le E_{\mathrm{mid}}$.
\begin{enumerate}[label=(\alph*)]
\item For every compact set $V \subset \R$, we have $\sup_N \E[\bm\Pi_N(V^k)^2] < \infty$. In particular, the sequences $\Pi_N$, $\Pi_N^{\mathrm d}$, and $\Pi_N^{\ne}$ are vaguely relatively compact.
\item Suppose that $\Pi_N \to \Pi$, $\Pi_N^{\mathrm d} \to \Pi^{\mathrm d}$, and $\Pi_N^{\ne} \to \Pi^{\ne}$ vaguely along a subsequence. Then $\Pi = r_k (\iota_k)_* \Pi^{\mathrm d} + \Pi^{\ne}$, the measure $\Pi^{\ne}$ gives no mass to the diagonal $\iota_k(\R)$, and $\Pi^{\mathrm d} = c \dd t$ for a constant $c \ge 0$. Moreover, in the coordinates $(t_1, \mathbf y)$ with $\mathbf y = (t_2 - t_1, \dots, t_k - t_1) \in \R^{k - 1}$,
\begin{equation}\label{e:extract-limits}
 \Pi(\dd \mathbf t) = \dd t_1 \, \nu(\dd \mathbf y) ,
 \qquad
 \nu(\{0\}) = r_k c ,
\end{equation}
where $\nu$ is a translation-bounded positive Radon measure on $\R^{k - 1}$. For $k = 1$, the space $\R^{k - 1}$ is a point, and \eqref{e:extract-limits} reads $\Pi = r_1 c \dd t$.
\item Let $\eps > 0$ and $u \in \R$, and let $g_b$ and $g_{b, N}$, $1 \le b \le k$, be continuous functions on $\R$ such that $|g_b|, |g_{b, N}| \le C P_\eps(\cdot - u)$ for all $N$ and $g_{b, N} \to g_b$ locally uniformly. Then, along the subsequence of (b),
\begin{equation}\label{e:extract-tail}
 \int_{\R^k} \prod_{b = 1}^k g_{b, N}(t_b) \, \Pi_N(\dd \mathbf t) \longrightarrow \int_{\R^k} \prod_{b = 1}^k g_b(t_b) \, \Pi(\dd \mathbf t) ,
\end{equation}
and the same holds with $(\Pi_N, \Pi)$ replaced by $(\Pi_N^{\ne}, \Pi^{\ne})$, or by $((\iota_k)_* \Pi_N^{\mathrm d}, (\iota_k)_* \Pi^{\mathrm d})$.
\end{enumerate}
\end{lemma}

By \eqref{e:extract-poisson-compare}, the functions in (c) include products of Poisson kernels $P_{\eps_b}(\cdot - u_b)$ whose dampings and centers converge to a positive damping and a real center, respectively. They also include squared Cauchy kernels $(\cdot - u_b \pm \iu\eps_b)^{-2}$, since $|x - u_b \pm \iu\eps_b|^{-2} = \pi P_{\eps_b}(x - u_b)/\eps_b$, and derivatives of Poisson kernels in their parameters.

\begin{proof}
We begin with (a). Let $V \subset \R$ be compact, and let $m_V > 0$ be a lower bound for $P_1$ on $V$. By \eqref{e:extract-smoothing},
\[
 \ell_{i, N}(V) \le \frac{1}{m_V} \int_\R P_1(x) \, \ell_{i, N}(\dd x) = \frac{\ell_{i, N}^1(0)}{m_V} .
\]
Inserting this bound into the domination \eqref{e:extract-domination} with $f_a = \1_V$, we obtain
\[
 \bm\Pi_N(V^k) \le C_0 \, k! \, m_V^{-k} \prod_{a = 1}^k \ell_{i_a, N}^1(0) .
\]
Using \zcref{l:extract-moments} with $m = 2k$, we find that the right side has a second moment that is bounded uniformly in $N$. Further, $r_k (\iota_k)_* \bm\Pi_N^{\mathrm d} \le \bm\Pi_N$ and $\bm\Pi_N^{\ne} \le \bm\Pi_N$ as measures, which implies $\Pi_N^{\mathrm d}(V) \le r_k^{-1} \Pi_N(V^k)$ and $\Pi_N^{\ne}(V^k) \le \Pi_N(V^k)$. A sequence of positive Radon measures whose masses on compact sets are bounded is vaguely relatively compact, and (a) follows.

We now turn to (b). The relation $\Pi = r_k(\iota_k)_* \Pi^{\mathrm d} + \Pi^{\ne}$ follows from the corresponding relation for $\Pi_N$, since $\varphi \circ \iota_k$ is continuous with compact support for all $\varphi \in C_c(\R^k)$.

To show that $\Pi^{\ne}$ gives no mass to the diagonal, we use the pair count bound, which is the case $m = 2$ of \zcref{l:iface-counts} and an estimate of Minami type \cite{minami1996local}. Fix $R \ge 1$ and $\gamma \in (0, 1)$. Let $\mathcal E_\gamma$ be the event that there exist labels $\alpha \ne \beta$ such that $x_\alpha, x_\beta \in [-R, R]$ and $|x_\alpha - x_\beta| < \gamma$. By the covering \eqref{e:proc-cover} in the proof of \zcref{p:proc-subsequences}, applied with $A = R$ and $\delta = \gamma$, two points $x \le y$ of $[-R, R]$ with $y - x < \gamma$ lie in the interval $I_l$ with $l = \lfloor x/\gamma \rfloor$, where
\[
 I_l = \bigl((l - 1)\gamma, (l + 2)\gamma\bigr), \qquad l \in \Z ,
\]
as there, and this $l$ satisfies $|l| \le R/\gamma + 1$. Therefore, on $\mathcal E_\gamma$, one of the $2R/\gamma + 3 \le 5R/\gamma$ intervals $I_l$ with $|l| \le R/\gamma + 1$ contains two eigenvalues, counted with multiplicity. Using $\1\{n \ge 2\} \le (n)_2/2$ for integers $n \ge 0$, \eqref{e:iface-counts-unfolded} with $m = 2$, and $\rho(E) \ge \rho(E_{\mathrm{out}})$, we find
\[
 \Prob(\mathcal E_\gamma) \le \frac12 \sum_{|l| \le R/\gamma + 1} \E(\Xi_N(I_l))_2 \le \frac{5R}{2\gamma} \Bigl(\frac{3 C_{\mathrm{cnt}} \gamma}{\rho(E_{\mathrm{out}})}\Bigr)^2 \le C_R \gamma .
\]

Next, we consider the band around the diagonal,
\[
 B_\gamma = \Bigl\{\mathbf t \in [-R, R]^k : \max_a t_a - \min_a t_a < \gamma\Bigr\} .
\]
An atom of $\bm\Pi_N^{\ne}$ in $B_\gamma$ sits at a point $(x_{\alpha_1}, \dots, x_{\alpha_k})$ with $\alpha_a \ne \alpha_b$ for some $a, b$, and such a point can occur only on $\mathcal E_\gamma$. By the Cauchy--Schwarz inequality, (a), and the bound on $\Prob(\mathcal E_\gamma)$, we have
\begin{equation}\label{e:extract-collision}
 \Pi_N^{\ne}(B_\gamma) \le \E\bigl[\1_{\mathcal E_\gamma} \bm\Pi_N([-R, R]^k)\bigr] \le \Prob(\mathcal E_\gamma)^{1/2} \E\bigl[\bm\Pi_N([-R, R]^k)^2\bigr]^{1/2} \le C_R \sqrt\gamma .
\end{equation}
Let $\varphi \in C_c(\R^k)$ satisfy $0 \le \varphi \le \1_{B_\gamma}$ and $\varphi = 1$ on $\iota_k([-R + 1, R - 1])$. Then \eqref{e:extract-collision} implies
\[
 \Pi^{\ne}\bigl(\iota_k([-R + 1, R - 1])\bigr) \le \int \varphi \dd \Pi^{\ne} = \lim_N \int \varphi \dd \Pi_N^{\ne} \le C_R \sqrt\gamma .
\]
Letting $\gamma \downarrow 0$, and then $R \to \infty$, we find that $\Pi^{\ne}(\iota_k(\R)) = 0$.

Next, we show that the three limits are invariant under the translations $\mathbf t \mapsto \mathbf t + b\mathbf 1$, where $\mathbf 1 = (1, \dots, 1)$, and $t \mapsto t + b$ for $\Pi^{\mathrm d}$. Let $\bm M_N$ be one of $\bm\Pi_N$, $\bm\Pi_N^{\mathrm d}$, and $\bm\Pi_N^{\ne}$, let $\varphi \ge 0$ be continuous with compact support, and let $\Phi = \bm M_N[\varphi]$, viewed as a function of $H$. By the remarks after \zcref{d:extract-measures}, we have $\Phi_b = \bm M_N[\varphi(\cdot + b\mathbf 1)]$ in the notation of \zcref{l:iface-gaussian}(c). We also have $\E \Phi_0^2 + \E \Phi_b^2 \le C_{\varphi, b}$ uniformly in $N$, by (a). We apply \eqref{e:iface-shift-truncated} with $K = \tau_N^{1/2}$, recalling that $\tau_N = a_N \sigma_N$, and we find
\[
 \bigl| \E \bm M_N[\varphi(\cdot + b\mathbf 1)] - \E \bm M_N[\varphi] \bigr| \le \sqrt{2/\pi} \, \frac{|b|}{\tau_N^{1/2}} + \frac{C_{\varphi, b}}{\tau_N^{1/2}} \longrightarrow 0 , \qquad b \in \R ,
\]
where we used $\tau_N \to \infty$, by \eqref{e:extract-tau}.
By linearity, the same holds for all real $\varphi \in C_c$, and the translation invariance of the limits follows.

We now derive \eqref{e:extract-limits}. The linear map $\mathbf t \mapsto (t_1, \mathbf y)$ has determinant one, and it carries the diagonal translations to translations in the first coordinate. A positive Radon measure on $\R \times \R^{k - 1}$ that is invariant under translations in the first coordinate has the form $\dd t_1 \, \nu(\dd\mathbf y)$ for a unique positive Radon measure $\nu$. (For $\psi \in C_c(\R^{k - 1})$ with $\psi \ge 0$, the Radon measure on $\R$ that integrates $\1_A(t_1)\psi(\mathbf y)$ is translation invariant and hence a multiple $\nu(\psi)$ of Lebesgue measure, and $\psi \mapsto \nu(\psi)$ is a positive linear functional.) The same argument with $k = 1$ shows that $\Pi^{\mathrm d} = c \dd t$. Finally, we have $\nu(\{0\}) = r_k c$, since $(\iota_k)_*(c \dd t) = c \dd t_1 \, \delta_0(\dd \mathbf y)$ and $\Pi^{\ne}$ gives no mass to $\{\mathbf y = 0\}$.

It remains to show that $\nu$ is translation bounded. Let $\mathbf s \in \R^{k - 1}$, and let $A_1 = [0, 1]$ and $A_b = [s_b, s_b + 2]$ for $2 \le b \le k$, so that $\nu(\mathbf s + [0, 1]^{k - 1}) \le \Pi(A_1 \times \dots \times A_k)$. The rectangle is compact, and $P_1 \ge P_1(1)$ on $[-1, 1]$. Using \eqref{e:extract-domination} and H\"older's inequality, we have
\[
 \Pi(A_1 \times \dots \times A_k) \le \limsup_N \Pi_N(A_1 \times \dots \times A_k) \le C_0 \, k! \, P_1(1)^{-k} \limsup_N \sup_{i, u} \bigl\|\ell_{i, N}^1(u)\bigr\|_{L^k}^k ,
\]
where the supremum is over all sites $i$ and all centers $u$ with $|E + u/a_N| \le E_{\mathrm{out}}$. For fixed $\mathbf s$, the midpoints of the intervals $A_b$ are such centers for all sufficiently large $N$ (recall that $|E| \le E_{\mathrm{mid}}$). The right side is bounded independently of $\mathbf s$, by \zcref{l:extract-moments}.

Finally, we prove (c), and we begin with the tails of $\ell_{i, N}^\eps(u)$. We set
\[
 c_1 = \frac{(E_{\mathrm{out}} - E_{\mathrm{mid}})\rho(E_{\mathrm{out}})}{2} ,
\]
so that $|E + t/a_N| \le E_{\mathrm{out}}$ for all $|t| \le c_1N + |u| + 2$ and all sufficiently large $N$. For $R \ge 3$, we define
\[
 T_R = \int_{|x - u| > R} P_\eps(x - u) \, \ell_{i, N}(\dd x) .
\]
The points with $|x - u| > c_1N$ contribute at most
\[
 N \sum_\alpha w_\alpha(i) \frac{\eps}{\pi c_1^2 N^2} = \frac{\eps}{\pi c_1^2 N}
\]
to $T_R$, since $u_\alpha$ is a unit vector.

On the remaining range, we bound $T_R$ as in the tail bound \eqref{e:proc-tail}, with $\ell_{i, N}$ in place of $\Xi^{(n)}$ and the $L^{2k}$ norm in place of the expectation. We cover this range by the unit intervals $u + [m, m + 1)$ with integers $R - 1 \le |m| \le c_1 N + 1$, on which $P_\eps(x - u) \le 4\eps/(\pi m^2)$. In place of the bound on $\E \Xi^{(n)}(I_j)$ used there, we use
\[
 \ell_{i, N}\bigl(u + [m, m + 1)\bigr) \le P_1(1)^{-1} \ell_{i, N}^1(u + m + 1/2)
\]
together with \zcref{l:extract-moments}, which applies at the centers $u + m + 1/2$ by the choice of $c_1$. By Minkowski's inequality, we obtain
\begin{equation}\label{e:extract-tail-bound}
 \|T_R\|_{L^{2k}} \le \sum_{|m| \ge R - 1} \frac{4\eps}{\pi m^2 P_1(1)} \sup_{u'} \bigl\|\ell_{i, N}^1(u')\bigr\|_{L^{2k}} + \frac{\eps}{\pi c_1^2 N} \le C_\eps\Bigl(\frac1R + \frac1N\Bigr) ,
\end{equation}
where the supremum is over centers $u'$ with $|E + u'/a_N| \le E_{\mathrm{out}}$.

Let $\chi_R \in C_c(\R)$ satisfy $0 \le \chi_R \le 1$, $\chi_R = 1$ on $[u - R, u + R]$ and $\supp \chi_R \subset [u - R - 1, u + R + 1]$. We expand each factor as $g_{b, N} = g_{b, N}\chi_R + g_{b, N}(1 - \chi_R)$. The product of the truncated factors is continuous, supported in a compact set that does not depend on $N$, and converges uniformly to $\prod_b g_b(t_b)\chi_R(t_b)$. Its integral against $\Pi_N$ therefore converges, by the vague convergence and (a).

Every other term has at least one factor bounded by $C P_\eps(\cdot - u) \1_{| \cdot - u| > R}$, while the remaining factors are bounded by $C P_\eps(\cdot - u)$. By \eqref{e:extract-domination} and H\"older's inequality, the integral of its modulus against $\Pi_N$ is at most
\[
 C^k C_0 \, k! \, \max_{a} \|T_R(i_a)\|_{L^k} \, \max_a \bigl\|\ell^\eps_{i_a, N}(u)\bigr\|_{L^k}^{k - 1} \le C' \Bigl(\frac1R + \frac1N\Bigr) ,
\]
where $T_R(i)$ denotes $T_R$ at the site $i$, and we used \eqref{e:extract-tail-bound} and \zcref{l:extract-moments}. The same bound $CR^{-1}$ holds for the limit $\Pi$. Indeed, for every nonnegative lower semicontinuous function $f$ on $\R^k$, vague convergence implies
\[
 \int f \dd\Pi \le \liminf_N \int f \dd\Pi_N .
\] Letting $N \to \infty$, and then $R \to \infty$, we obtain \eqref{e:extract-tail}. The measures $\Pi_N^{\ne}$ and $r_k(\iota_k)_*\Pi_N^{\mathrm d}$ are bounded by $\Pi_N$, so the same argument applies to them.
\end{proof}

\section{The extraction theorem}\label{s:extract-theorem}

We now identify the diagonal density from the residue of the source limits. For $\mathbf y = (y_2, \dots, y_k) \in \R^{k - 1}$, we set $y_1 = 0$ and define
\begin{equation}\label{e:extract-overlap}
 \mathsf P_k(\mathbf y) = \int_\R \prod_{b = 1}^k P_1(t + y_b) \dd t ,
 \qquad
 \varkappa_{k, r} = r (2\pi)^{k - 1} \mathsf P_k(0) , \quad r > 0 .
\end{equation}
For $k = 1$, we set $\mathsf P_1 = 1$, so that $\varkappa_{1, r} = r$. The beta integral $\int_\R (1 + t^2)^{-k} \dd t = \sqrt\pi\, \Gamma(k - 1/2)/\Gamma(k)$ gives
\begin{equation}\label{e:extract-constant}
 \mathsf P_k(0) = \frac{(2k - 3)!!}{2^{k - 1} \pi^{k - 1} (k - 1)!} ,
 \qquad
 \varkappa_{k, r} = \frac{r (2k - 3)!!}{(k - 1)!} ,
\end{equation}
with the convention $(-1)!! = 1$. For example, $\varkappa_{2, 1} = 1$ and $\varkappa_{k, k!} = k(2k - 3)!!$.

\begin{theorem}\label{t:extract-main}
Let $k \ge 1$ and $r_k > 0$. For each $N$, let $|E| \le E_{\mathrm{mid}}$, let $i_1, \dots, i_k$ be sites, and let $\bm\Pi_N$ be an eigenvector-weighted moment measure of order $k$ that is dominated at $i_1, \dots, i_k$ with a constant $C_0$ independent of $N$. Suppose that there exist a function $\mathfrak f \colon (0, \infty) \to \R$ and a constant $c_* \ge 0$ such that for all $\eps > 0$,
\begin{equation}\label{e:extract-source}
 \int_{\R^k} \prod_{b = 1}^k P_\eps(t_b) \, \Pi_N(\dd\mathbf t) \longrightarrow \mathfrak f(2\pi\eps) ,
 \qquad
 \lim_{z \downarrow 0} z^{k - 1} \mathfrak f(z) = \varkappa_{k, r_k} c_* .
\end{equation}
Then $\Pi_N^{\mathrm d} \to c_* \dd u$ vaguely, and the densities of $\Pi_N^{\mathrm d}$ converge to $c_*$ locally uniformly.
\end{theorem}

The matrix limit in \eqref{e:extract-source} is used only at fixed positive damping, which is the form in which \zcref{t:interface} provides it. When $\bm\Pi_N$ is a tensor product of the measures $\ell_{i, N}$, the left side of \eqref{e:extract-source} is an expected product of local densities, by \eqref{e:extract-smoothing}.

\begin{proof}
By \zcref{l:extract-limits}(a), it suffices to show that $\Pi^{\mathrm d} = c_* \dd u$ for every vague limit point $\Pi^{\mathrm d}$ of $\Pi_N^{\mathrm d}$. Passing to a further subsequence, we may assume that $\Pi_N$, $\Pi_N^{\mathrm d}$, and $\Pi_N^{\ne}$ converge vaguely along it, and we let $\Pi^{\mathrm d} = c \dd t$ and $\nu$ be as in \zcref{l:extract-limits}(b). Applying \zcref{l:extract-limits}(c) with $g_{b, N} = g_b = P_\eps$ and using \eqref{e:extract-limits}, we find that for all $\eps > 0$,
\[
 \mathfrak f(2\pi\eps) = \int_{\R^{k - 1}} \int_\R \prod_{b = 1}^k P_\eps(t_1 + y_b) \dd t_1 \, \nu(\dd\mathbf y) = \eps^{1 - k} \int_{\R^{k - 1}} \mathsf P_k(\mathbf y/\eps) \, \nu(\dd\mathbf y) ,
\]
where in the second step we used $P_\eps(x) = \eps^{-1} P_1(x/\eps)$ and substituted $t_1 = \eps t$. For $k = 1$, this reads $\mathfrak f(2\pi\eps) = r_1 c$, and \eqref{e:extract-source} implies that $c = c_*$. We assume $k \ge 2$ from now on.

To let $\eps \downarrow 0$, we need three properties of $\mathsf P_k$. First, we insert the representation
\[
 P_1(t) = \frac1\pi \int_0^\infty e^{-s(1 + t^2)} \dd s
\]
into \eqref{e:extract-overlap} and compute the Gaussian integral in $t$, which gives
\[
 \mathsf P_k(\mathbf y) = \pi^{1/2 - k} \int_{(0, \infty)^k} \frac{e^{-|\mathbf s|}}{\sqrt{|\mathbf s|}} \exp\Bigl(-\frac{1}{|\mathbf s|} \sum_{1 \le a < b \le k} s_a s_b (y_a - y_b)^2\Bigr) \dd\mathbf s ,
 \qquad |\mathbf s| = \sum_{b = 1}^k s_b .
\]
It follows that $\mathsf P_k(v\mathbf y) \le \mathsf P_k(\mathbf y)$ for $v \ge 1$, and, by dominated convergence, that $\mathsf P_k(v\mathbf y) \to 0$ as $v \to \infty$ for $\mathbf y \ne 0$. Second, integrating first over $y_2, \dots, y_k$ and using that $P_1$ is a probability density, we find
\[
 \int_{\R^{k - 1}} \mathsf P_k(\mathbf y) \dd\mathbf y = 1 .
\]
Third, we have $|P_1'| \le P_1$, and hence $|\partial_{y_b} \mathsf P_k| \le \mathsf P_k$ for $2 \le b \le k$. The third property implies that on every unit cube, the supremum of $\mathsf P_k$ is bounded by $e^{k - 1}$ times its infimum. Together with the second property, this shows that the suprema of $\mathsf P_k$ over the cubes $\mathbf s + [0, 1)^{k - 1}$, $\mathbf s \in \Z^{k - 1}$, have sum at most $e^{k - 1}$.

Since $\nu$ is translation bounded, the last statement implies that $\mathsf P_k$ is $\nu$-integrable, with
\[
 \int_{\R^{k - 1}} \mathsf P_k \dd\nu \le e^{k - 1} \sup_{\mathbf s \in \R^{k - 1}} \nu\bigl(\mathbf s + [0, 1]^{k - 1}\bigr) .
\] The first property shows that $\mathsf P_k$ dominates $\mathsf P_k(\cdot /\eps)$ for $\eps \le 1$. Using dominated convergence, we obtain
\[
 \lim_{z \downarrow 0} z^{k - 1} \mathfrak f(z) = (2\pi)^{k - 1} \lim_{\eps \downarrow 0} \int_{\R^{k - 1}} \mathsf P_k(\mathbf y/\eps) \, \nu(\dd\mathbf y) = (2\pi)^{k - 1} \mathsf P_k(0) \nu(\{0\}) = \varkappa_{k, r_k} c ,
\]
where we used \eqref{e:extract-limits} in the last step. Comparing with \eqref{e:extract-source}, we find $c = c_*$. For $k = 2$, we have $\mathsf P_2 = P_2$, and this limit is \zcref{l:tools-poisson}(c) with $a = 2\eps$.

It remains to prove the convergence of densities. We have $\E \bm\Pi_N^{\mathrm d}(\R) \le r_k^{-1} C_0 k! N^k$, so \zcref{l:extract-gaussian} applies to $\bm\Pi_N^{\mathrm d}$, and the measure $\Pi_N^{\mathrm d}$ has the density $m_N * \mathsf n_{\tau_N}$ for a finite positive measure $m_N$. Since $\tau_N \to \infty$ by \eqref{e:extract-tau}, and since $\Pi_N^{\mathrm d} \to c_* \dd u$ vaguely, we conclude from \zcref{l:extract-mixture} that the densities converge to $c_*$ locally uniformly.
\end{proof}

\begin{remark}\label{r:extract-uniform}
Since all parameters in \zcref{t:extract-main} may depend on $N$, the theorem implies uniform statements. Suppose that for each $N$ the parameters (the energy, $N/W^2$, the sites, and any further parameters) range over a set $\Theta_N$, and that $c_*(\theta)$ is a bounded function of the parameters. Suppose further that every sequence $\theta_N \in \Theta_N$ has a subsequence along which the hypotheses of \zcref{t:extract-main} hold, for some $\mathfrak f$ and some $c_*$ such that $c_*(\theta_N) \to c_*$. Let $f_\theta$ denote the density of $\Pi_N^{\mathrm d}$ at the parameter $\theta$. Then for all $R > 0$,
\[
 \sup_{\theta \in \Theta_N} \sup_{|u| \le R} |f_\theta(u) - c_*(\theta)| \longrightarrow 0 .
\]
Indeed, otherwise there exist $\delta_0 > 0$ and parameters $\theta_N \in \Theta_N$ with $\sup_{|u| \le R} |f_{\theta_N}(u) - c_*(\theta_N)| \ge \delta_0$ for infinitely many $N$, and the theorem, applied along a suitable subsequence, yields a contradiction.
\end{remark}

\section{Densities and arbitrarily small windows}\label{s:extract-windows}

The case $k = 1$ of \zcref{t:extract-main} normalizes the local spectral measures.

\begin{corollary}\label{c:extract-onepoint}
For each $N$, let $|E| \le E_{\mathrm{mid}}$ and let $i$ be a site. If $\E \ell_{i, N}^\eps(0) \to 1$ for all $\eps > 0$, then the densities of $\E \ell_{i, N}$ converge to one locally uniformly.
\end{corollary}

\begin{proof}
We apply \zcref{t:extract-main} with $k = 1$, $\bm\Pi_N = \ell_{i, N}$, $r_1 = 1$, $\mathfrak f = 1$, and $c_* = 1$. Then $\bm\Pi_N^{\mathrm d} = \ell_{i, N}$ and $\bm\Pi_N^{\ne} = 0$, the domination holds with $C_0 = 1$, and the source limit is the hypothesis, by \eqref{e:extract-smoothing}.
\end{proof}

We now pass from densities to energy windows. Let $\chi \ge 0$ be an integrable function with compact support and $\int \chi > 0$; the window $\phi$ of \zcref{d:selection} is an example. Let $k \ge 1$, and let $F_\alpha = F_{\alpha, N} \ge 0$ be bounded measurable functions of $u_\alpha u_\alpha^*$. For a physical energy $e$, we set $a_N(e) = N\rho(e)$, and we let $f_{N, e}$ denote the density of the measure
\begin{equation}\label{e:extract-unfolded-at-e}
 N^k \E \sum_\alpha F_\alpha \, \delta_{a_N(e)(\lambda_\alpha - e)} ,
\end{equation}
which exists by \zcref{l:extract-gaussian}, applied with the energy $e$ in place of $E$.

\begin{corollary}
Let\label{c:extract-windows} $k$, $F_\alpha$, and $f_{N, e}$ be as above, and let $\bar c_N$ be functions on $[-E_{\mathrm{mid}}, E_{\mathrm{mid}}]$ that are bounded and equicontinuous, uniformly in $N$. Suppose that
\[
 \sup_{|e| \le E_{\mathrm{mid}}} |f_{N, e}(0) - \bar c_N(e)| \longrightarrow 0 .
\]
Then for every sequence $\bar r_N \downarrow 0$, uniformly for $|E| \le E_*$ and $0 < r \le \bar r_N$,
\begin{equation}\label{e:extract-window}
 N^k \frac{\E \sum_\alpha F_\alpha \, \chi((\lambda_\alpha - E)/r)}{\E \sum_\alpha \chi((\lambda_\alpha - E)/r)} - \bar c_N(E) \longrightarrow 0 .
\end{equation}
Let $i = i(N)$ be sites, and suppose in addition that the densities $g_{N, e}$ of the measures $N \E \sum_\alpha w_\alpha(i) \delta_{a_N(e)(\lambda_\alpha - e)}$ satisfy $\sup_{|e| \le E_{\mathrm{mid}}} |g_{N, e}(0) - 1| \to 0$. Then, uniformly in the same sense,
\begin{equation}\label{e:extract-rooted-window}
 N^{k - 1} \frac{\E \sum_\alpha F_\alpha \, \chi((\lambda_\alpha - E)/r)}{\E \sum_\alpha w_\alpha(i) \chi((\lambda_\alpha - E)/r)} - \bar c_N(E) \longrightarrow 0 .
\end{equation}
\end{corollary}

There is no lower bound on $r$, so the ratios in \eqref{e:extract-window} and \eqref{e:extract-rooted-window} are controlled also for windows that typically contain no eigenvalue.

\begin{proof}
The measure $N^k \E \sum_\alpha F_\alpha \delta_{\lambda_\alpha}$ has a density $m_N$ in the physical variable, by \zcref{l:extract-gaussian}. The change of variables $u = a_N(e)(\lambda - e)$ gives
\[
 m_N(e) = a_N(e) f_{N, e}(0) = N\rho(e) f_{N, e}(0)
\]
for all $e$. Similarly, the measure $\E \sum_\alpha w_\alpha(i) \delta_{\lambda_\alpha}$ has the density $\rho(e) g_{N, e}(0)$. Recall from \zcref{l:iface-density} that $\E \sum_\alpha \delta_{\lambda_\alpha}$ has the density $N\rho_{N, W}(e)$, and that $\rho_{N, W}(e) = \rho(e)(1 + o(1))$, uniformly for $|e| \le E_{\mathrm{mid}}$.

Let $\supp\chi \subset [-R_\chi, R_\chi]$. For $|E| \le E_*$ and $r \le \bar r_N$, all energies $e$ with $\chi((e - E)/r) \ne 0$ satisfy $|e - E| \le R_\chi \bar r_N$, and in particular $|e| \le E_{\mathrm{mid}}$ for $N$ large. For such $e$, the hypothesis and the equicontinuity of $\bar c_N$ imply that $f_{N, e}(0) = \bar c_N(E) + o(1)$, uniformly. We obtain
\[
 N^k \frac{\E \sum_\alpha F_\alpha \chi((\lambda_\alpha - E)/r)}{\E \sum_\alpha \chi((\lambda_\alpha - E)/r)}
 = \frac{\int \rho(e) f_{N, e}(0) \chi((e - E)/r) \dd e}{\int \rho_{N, W}(e) \chi((e - E)/r) \dd e}
 = \frac{\bar c_N(E) + o(1)}{1 + o(1)} ,
\]
where we used that
\[
 \int \rho(e)\chi((e - E)/r) \dd e \ge \rho(E_{\mathrm{mid}}) \, r \int\chi > 0 .
\]
Since $\bar c_N$ is bounded, \eqref{e:extract-window} follows. The proof of \eqref{e:extract-rooted-window} is the same, with
\[
\begin{gathered}
 N^{k - 1} \E \sum_\alpha F_\alpha \chi\Bigl(\frac{\lambda_\alpha - E}{r}\Bigr) = \int \rho(e) f_{N, e}(0) \chi\Bigl(\frac{e - E}{r}\Bigr) \dd e , \\
 \E \sum_\alpha w_\alpha(i) \chi\Bigl(\frac{\lambda_\alpha - E}{r}\Bigr) = \int \rho(e) g_{N, e}(0) \chi\Bigl(\frac{e - E}{r}\Bigr) \dd e
\end{gathered}
\]
in place of the identities used above.
\end{proof}

In the applications, we verify the hypothesis of \zcref{c:extract-windows} with \zcref{t:extract-main} and \zcref{r:extract-uniform}. For each energy $e$, the measure \eqref{e:extract-unfolded-at-e} is the expected diagonal part, unfolded at $e$, of an eigenvector-weighted moment measure. Suppose that the hypotheses of \zcref{t:extract-main} hold along all sequences of energies $e = e_N$ with $|e_N| \le E_{\mathrm{mid}}$, with targets $c_*$ such that $\bar c_N(e_N) - c_* \to 0$. Then \zcref{r:extract-uniform} implies that $\sup_{|e| \le E_{\mathrm{mid}}} |f_{N, e}(0) - \bar c_N(e)| \to 0$.