\chapter{Local intensities and the inverse participation ratio}\label{s:intensity}

In this section, we prove \zcref{t:intensity,t:profile}. In \zcref{l:int-contact-residue}, we also prove that $z \widetilde K_X(x; z) \to 2\mathcal C_X(x, x)$ as $z \downarrow 0$, which, together with \zcref{t:res-scalar}, identifies the eigenvector correlations as the residues of the kernels of \zcref{d:kernels}.

The proof has three steps. First, \zcref{t:interface}(b) expresses the limits of the expected products of two resolvent entries at a fixed damping as pairings in the rank-one sigma model, with explicit multipliers at the sites. We evaluate these pairings with the supersymmetric localization of \zcref{l:sigma-localization} and with the rank-one calculus of \zcref{s:int-rank-one}, which reduces retarded-covariant matrices of superfunctions to a value at the corner of $D$ and a function that evolves by the radial diffusion of \zcref{s:sigma-scalar}. As explained in \zcref{r:iface-onsite}, one site and two distinct sites have different multipliers, and this is the origin of the contact kernel $\widetilde K_X$. For an infinite wire, the corresponding distinction between coincident and nearby points, with the weight $B/(B - F)$ of \eqref{e:intro-Kc}, was found by Skvortsov and Ostrovsky \cite[(7)--(8)]{skvortsov2007local}. Second, the extraction theorem \zcref{t:extract-main} converts the limits at fixed damping into the expected densities of $N^2 \sum_\alpha w_\alpha(i) w_\alpha(j) \delta_{x_\alpha}$, which give the profile. The full two-energy intensity requires the separation measure of the pair also away from its atom. We identify it from its values at coinciding energies, using that it is even, which we prove in \zcref{l:int-even}. Third, the inverse participation ratio is a sum over all sites, and the sites near the ends of the chain are controlled by \zcref{t:interface}(c). Its limit is evaluated exactly in \zcref{p:int-exact}.

In the physics literature, eigenvector statistics of band matrices in this regime were derived from the one-dimensional supersymmetric sigma model by Fyodorov and Mirlin; see \cite[Section~3.2]{mirlin2000statistics} and the references there. In particular, the limit $2 + 2X/3$ of $N\langle \IPR(u_\alpha) \rangle_{E, r}$ agrees with their prediction \cite{fyodorov1992analytical,fyodorov1994statistical}, and its expansion to second order in $X$ appears in \cite[(12), (17)]{fyodorov1995mesoscopic}.

\paragraph{Conventions.} We keep the conventions of \zcref{s:extract}, and in particular the energies $E_* < E_{\mathrm{mid}} < E_{\mathrm{out}}$ of \eqref{e:iface-energies}. We write $s = s_E$, so that $\kappa_E^2 + s^2 = 1$ with $\kappa_E = E/2$, and we set $z = 2\pi\eps$ for a damping $\eps > 0$. We fix $\delta \in (0, 1/2)$, and we call a site $i$ \emph{interior} if $\delta \le i/N \le 1 - \delta$. Statements are along sequences as in \zcref{s:extract}; uniform versions follow by \zcref{r:extract-uniform} or by the same argument.

\section{Local sources}\label{s:int-sources}

We use \zcref{t:interface} at rank $k = 1$, with $\mathbf u = 0$ and $\mathbf b = \eps$, and with local sources at one or two interior sites. We abbreviate $T(\mathbf Y) = T_N^{(1)}(0, \eps; \mathbf Y)$ for the local source function of \zcref{d:iface-local}, and we write
\[
 z_\pm = E \pm \iu\eta, \qquad \eta = \frac{\eps}{a_N}, \qquad G^\pm = G(z_\pm) .
\]
Recall the supermatrix $\mathcal G(Q) = -\kappa_E \Id + \iu s Q$ of \eqref{e:iface-orbit-resolvent} and the multipliers $\mathfrak j^{(\alpha)}$ of \eqref{e:iface-multiplier}. For a $(2|2)$ supermatrix $M$, whose indices are a flavor ($\mathrm R$ or $\mathrm A$) and a type ($\mathrm B$ or $\mathrm F$), we write $M^{\mathrm{BB}}_{\mathrm{RA}}$ for the entry with boson-boson type, retarded row, and advanced column, and similarly for the other entries. We let $E_{\mathrm{RR}}$ and $E_{\mathrm{AA}}$ denote the matrix units of the two flavors.

\begin{lemma}\label{l:int-sources}
Let $i < j$ be sites, and let $t, t'$ be complex parameters. The following derivatives at $\mathbf Y = 0$ hold, where the sources not listed vanish, and each line gives the multipliers at the sites, in the notation of \eqref{e:iface-jets}.
\begin{enumerate}[label=(\alph*)]
\item For $Y^{\mathrm F} = t E_{\mathrm{RR}}$ at $i$, we have $\partial_t T = -\E G^+_{ii}$, with the multiplier $-\mathcal G^{\mathrm{FF}}_{\mathrm{RR}}$.
\item For $Y^{\mathrm F} = t E_{\mathrm{RR}}$ at $i$ and $Y^{\mathrm F} = t' E_{\mathrm{AA}}$ at $j$, we have $\partial_t \partial_{t'} T = \E G^+_{ii} G^-_{jj}$, with the multipliers $-\mathcal G^{\mathrm{FF}}_{\mathrm{RR}}$ at $i$ and $-\mathcal G^{\mathrm{FF}}_{\mathrm{AA}}$ at $j$, whose signs cancel in the product. Exchanging $E_{\mathrm{RR}}$ and $E_{\mathrm{AA}}$ gives $\E G^-_{ii} G^+_{jj}$, with the multipliers exchanged.
\item For $Y^{\mathrm F} = t E_{\mathrm{RR}}$ at $i$ and $Y^{\mathrm F} = t' E_{\mathrm{RR}}$ at $j$, we have $\partial_t \partial_{t'} T = \E (G^+_{ii} G^+_{jj} - G^+_{ij} G^+_{ji})$, with the multiplier $-\mathcal G^{\mathrm{FF}}_{\mathrm{RR}}$ at both sites. For $Y^{\mathrm B} = t E_{\mathrm{RR}}$ at $i$ and $Y^{\mathrm B} = t' E_{\mathrm{RR}}$ at $j$, we have $\partial_t \partial_{t'} T = \E (G^+_{ii} G^+_{jj} + G^+_{ij} G^+_{ji})$, with the multiplier $\mathcal G^{\mathrm{BB}}_{\mathrm{RR}}$ at both sites.
\item For $Y^{\mathrm B} = \diag(t, t')$ at $i$, we have $\partial_t \partial_{t'} T = \E G^+_{ii} G^-_{ii}$, with the multiplier
\begin{equation}\label{e:int-contact-multiplier}
 \mathcal G^{\mathrm{BB}}_{\mathrm{RR}} \mathcal G^{\mathrm{BB}}_{\mathrm{AA}} + \mathcal G^{\mathrm{BB}}_{\mathrm{RA}} \mathcal G^{\mathrm{BB}}_{\mathrm{AR}} .
\end{equation}
For $Y^{\mathrm B} = t E_{\mathrm{RR}}$ at $i$, we have $\frac12 \partial_t^2 T = \E (G^+_{ii})^2$, with the multiplier $(\mathcal G^{\mathrm{BB}}_{\mathrm{RR}})^2$.
\end{enumerate}
\end{lemma}

\begin{proof}
At $\mathbf u = 0$ and $\mathbf b = \eps$, the energies and the matrices of \eqref{e:iface-Z} are
\[
 z^{\mathrm B}_\pm = z^{\mathrm F}_{1, \pm} = z_\pm , \qquad Z_{\mathrm F} = Z_{\mathrm B} = \diag(z_+, z_-) .
\]
With the flavors ordered as $\mathrm R, \mathrm A$, we have $t E_{\mathrm{RR}} = \diag(t, 0)$ and $t' E_{\mathrm{AA}} = \diag(0, t')$. Therefore (a) and the first identity in (b) are the two identities of \eqref{e:iface-examples}, and the first identity in (d) is the first identity of \zcref{r:iface-onsite}. The second identity in (b) is the computation of \eqref{e:iface-examples} with the two flavors exchanged.

For (c) and the last identity in (d), we compute in the same way. All sources listed are diagonal in the flavors, so that the numerator and the denominator in \eqref{e:iface-local} factor over the two flavors, and the factors without sources cancel. The sources are added to $z$, as in \zcref{d:iface-local}. For a matrix $A = t e_i e_i^* + t' e_j e_j^*$ and $z \in \{z_+, z_-\}$, we have $\det(z - H + A) = \det(z - H) \det(\Id - G(z) A)$, since $G(z) = -(z - H)^{-1}$, and
\begin{equation}\label{e:int-rank-two}
 \det\bigl(\Id - G(z) A\bigr) = 1 - t G_{ii} - t' G_{jj} + t t' (G_{ii} G_{jj} - G_{ij} G_{ji}) ,
\end{equation}
with all entries at $z$. Fermionic sources appear in the numerator, and the first part of (c) follows from \eqref{e:int-rank-two} directly. Bosonic sources appear in the denominator, and expanding the reciprocal of \eqref{e:int-rank-two} to second order, we find
\[
 \partial_t \partial_{t'} \det\bigl(\Id - G(z) A\bigr)^{-1} \big|_{t = t' = 0} = -(G_{ii} G_{jj} - G_{ij} G_{ji}) + 2 G_{ii} G_{jj} = G_{ii} G_{jj} + G_{ij} G_{ji} ,
\]
which is the second part of (c). When $Y^{\mathrm B} = t E_{\mathrm{RR}}$, the denominator contains the factor $1 - t G^+_{ii}$, and the last identity in (d) follows from $\frac12 \partial_t^2 (1 - t G^+_{ii})^{-1} = (G^+_{ii})^2$ at $t = 0$.

The multipliers come from the block formula for the superdeterminant of \zcref{l:grass-sdet}. The supermatrix $\Id - \mathcal G Y$ of \eqref{e:iface-multiplier} is block upper triangular when $Y = \diag(0 \mid Y^{\mathrm F})$ and block lower triangular when $Y = \diag(Y^{\mathrm B} \mid 0)$, which gives
\[
\begin{gathered}
 \Sdet\bigl(\Id - \mathcal G \diag(0 \mid Y^{\mathrm F})\bigr)^{-1} = \det\bigl(\Id - \mathcal G^{\mathrm{FF}} Y^{\mathrm F}\bigr) , \\
 \Sdet\bigl(\Id - \mathcal G \diag(Y^{\mathrm B} \mid 0)\bigr)^{-1} = \det\bigl(\Id - \mathcal G^{\mathrm{BB}} Y^{\mathrm B}\bigr)^{-1} ,
\end{gathered}
\]
where $\mathcal G^{\mathrm{FF}}$ and $\mathcal G^{\mathrm{BB}}$ are the diagonal blocks of $\mathcal G$. Distinct sites carry separate multipliers, and the multipliers follow by differentiating these expressions. The multiplier \eqref{e:int-contact-multiplier} for $Y^{\mathrm B} = \diag(t, t')$ is the second identity of \zcref{r:iface-onsite}. The fermionic factors have the first derivatives $-\mathcal G^{\mathrm{FF}}_{\mathrm{RR}}$ and $-\mathcal G^{\mathrm{FF}}_{\mathrm{AA}}$, and these signs cancel in pairs in (b) and (c). The same holds on the matrix side, where each fermionic first derivative carries the factor $-1$.
\end{proof}

\section{Pairings at rank one}\label{s:int-pairings}

We now evaluate the pairings that \zcref{t:interface}(b) produces from the multipliers of \zcref{l:int-sources}, with the rank-one calculus of \zcref{s:int-rank-one} at $z = 2\pi\eps$. Recall from there the generator and the semigroup
\[
 \mathsf G_{X, z} = X \mathsf C_1 + z \, d_1(Q) , \qquad d_1(Q) = \tfrac12 \Str(\Lambda Q) , \qquad U_t = e^{-t \mathsf G_{X, z}} .
\]
At $\mathbf u = 0$ and $\mathbf b = \eps$, we have $\Gamma(0, \eps \1) = \eps \Lambda$, so that $\pi \Str(Q \Gamma) = z \, d_1(Q)$, and $\mathsf G_{X, z}$ is the generator $\mathsf G_{X, \Gamma}$ of \eqref{e:iface-jets}. Using the symmetry of $\mathcal B_1$ under $U_t$ of \zcref{p:sigma-regular}(c), we see that the limits in \eqref{e:iface-jets} with one site $i$ and with two sites $i < j$, where $i/N \to x$ and $j/N \to y$, are
\begin{equation}\label{e:int-jet-limits}
 \mathcal B_1\bigl(U_x 1, \mathfrak j \, U_{1 - x} 1\bigr) ,
 \qquad
 \mathcal B_1\bigl(U_x 1, \mathfrak j \, U_{y - x} \, \mathfrak j' \, U_{1 - y} 1\bigr) ,
\end{equation}
where $\mathfrak j$ and $\mathfrak j'$ are the multipliers at the sites, which act by multiplication.

Recall that $U_t 1$ is an invariant superfunction with the radial representative $\Psi_t$ and the value one at the corner, by fact \ref{i:int-radial} of \zcref{s:int-rank-one}, and we denote it by $\Psi_t$, as in the proof of \zcref{l:sigma-jets}. We write $f_0 = f_\emptyset(\Lambda)$ for the value at the corner of a superfunction $f$. For the multipliers of \zcref{l:int-sources}(a)--(c), which are diagonal entries of $\mathcal G$, the pairings are those of \zcref{l:int-pairings}. For the contact multiplier of \zcref{l:int-sources}(d), we use the radial integration formula \zcref{l:sigma-radial-integral}, with the top coefficients $[A]_{\mathrm{top}}$ and their phase averages $\langle A \rangle_{\mathrm{top}}$ of \zcref{l:sigma-polar}\ref{i:sigma-polar-top}, and the localization fact \ref{i:int-localization}.

The contact multiplier contains the off-diagonal entries of the boson-boson block. By \zcref{l:sigma-polar}\ref{i:sigma-polar-top}, the average over the phases of their top coefficient is
\begin{equation}\label{e:int-offdiag-top}
 \bigl\langle Q^{\mathrm{BB}}_{\mathrm{RA}} Q^{\mathrm{BB}}_{\mathrm{AR}} \bigr\rangle_{\mathrm{top}} = -(B^2 - F^2) = -d(B + F) .
\end{equation}
Only this average enters the radial integration formula \zcref{l:sigma-radial-integral}.

\begin{lemma}\label{l:int-contact}
Let $\delta \le x \le 1 - \delta$. Then
\begin{equation}\label{e:int-contact}
\begin{gathered}
 \mathcal B_1\Bigl(\Psi_x, \bigl(\mathcal G^{\mathrm{BB}}_{\mathrm{RR}} \mathcal G^{\mathrm{BB}}_{\mathrm{AA}} + \mathcal G^{\mathrm{BB}}_{\mathrm{RA}} \mathcal G^{\mathrm{BB}}_{\mathrm{AR}}\bigr) \Psi_{1 - x}\Bigr) = 1 + 2s^2 \widetilde K_X(x; z) , \\
 \mathcal B_1\bigl(\Psi_x, (\mathcal G^{\mathrm{BB}}_{\mathrm{RR}})^2 \Psi_{1 - x}\bigr) = (-\kappa_E + \iu s)^2 .
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
Let $\mathfrak c$ denote the multiplier \eqref{e:int-contact-multiplier}. Inserting $\mathcal G = -\kappa_E\Id + \iu s Q$, we have
\[
 \mathfrak c = \kappa_E^2 - \iu s\kappa_E \bigl(Q^{\mathrm{BB}}_{\mathrm{RR}} + Q^{\mathrm{BB}}_{\mathrm{AA}}\bigr) - s^2 \bigl(Q^{\mathrm{BB}}_{\mathrm{RR}} Q^{\mathrm{BB}}_{\mathrm{AA}} + Q^{\mathrm{BB}}_{\mathrm{RA}} Q^{\mathrm{BB}}_{\mathrm{AR}}\bigr) .
\]
At $\Lambda$, we have $Q^{\mathrm{BB}}_{\mathrm{RR}} = 1$, $Q^{\mathrm{BB}}_{\mathrm{AA}} = -1$, and $Q^{\mathrm{BB}}_{\mathrm{RA}} = 0$, so that
\[
 \mathfrak c_0 = \kappa_E^2 - \iu s \kappa_E (1 - 1) + s^2 = 1 .
\] By \zcref{l:sigma-polar}\ref{i:sigma-polar-entries}, the second term of $\mathfrak c$ is a sum of terms with $n_{\mathrm R}$ or $n_{\mathrm A}$ alone, and $[Q^{\mathrm{BB}}_{\mathrm{RR}} Q^{\mathrm{BB}}_{\mathrm{AA}}]_{\mathrm{top}} = -d^2$. Combining this with \eqref{e:int-offdiag-top}, we find
\[
 \langle \mathfrak c \rangle_{\mathrm{top}} = s^2 \bigl(d^2 + d(B + F)\bigr) = 2s^2 B d .
\]

The functions $\Psi_x$ and $\Psi_{1 - x}$ are invariant, so they depend only on $(B, F)$ in the polar chart, and
\[
 \frac{\langle \Psi_x \mathfrak c \Psi_{1 - x} \rangle_{\mathrm{top}}}{d^2} = 2s^2 \, \frac{B}{d} \, \Psi_x \Psi_{1 - x} .
\]
The right side is integrable on $D$. Indeed, $B/d$ is integrable near the corner, and $\Psi_x \Psi_{1 - x} \le C(1 + B)^{-3}$ by \zcref{l:sigma-scalar}\ref{i:sigma-scalar-tails}, for every $z > 0$. Applying \zcref{l:sigma-radial-integral}, with the corner term $\mathfrak c_0 = 1$, and using the definition \eqref{e:intro-Kc} of $\widetilde K_X$, we obtain the first identity in \eqref{e:int-contact}. The second integrand is advanced-invariant, with the value $(-\kappa_E + \iu s)^2$ at $\Lambda$, and fact \ref{i:int-localization} of \zcref{s:int-rank-one} applies.
\end{proof}

\begin{remark}\label{r:int-gue}
The constant $2s^2$ and the weight $B/d$ can be checked against the GUE. If we set $X = 0$ in \zcref{d:kernels}, then $\Psi_t = e^{-tzd}$, and a direct computation gives
\[
 K_0(x, y; z) = \frac{1 - e^{-2z}}{z^2} = 2 I_{\GUE}(z) , \qquad \widetilde K_0(x; z) = \int_D \frac{B}{d} e^{-zd} \dd B \dd F = \frac1z + I_{\GUE}(z) .
\]
A single site carries the additional term $1/z$, whose residue doubles the coincident-eigenvalue contribution, in agreement with the exact identity $\E |u(i)|^4 = \frac{2N}{N + 1} (\E |u(i)|^2)^2$ for Haar-distributed eigenvectors, whose factor tends to two as $N \to \infty$.
\end{remark}

The contact kernel and the diagonal value of $K_X$ have the same residue at $z = 0$, up to the factor two of \zcref{t:res-scalar}.

\begin{lemma}\label{l:int-contact-residue}
Let $0 < X_- \le X_+ < \infty$. Uniformly for $X \in [X_-, X_+]$ and $\delta \le x \le 1 - \delta$,
\begin{equation}\label{e:int-contact-residue}
 \lim_{z \downarrow 0} z \widetilde K_X(x; z) = 2\mathcal C_X(x, x) .
\end{equation}
\end{lemma}

\begin{proof}
The definitions \eqref{e:intro-K} at $y = x$ and \eqref{e:intro-Kc} give
\[
 \widetilde K_X(x; z) - K_X(x, x; z) = \int_D \frac{F}{d} \Psi_x \Psi_{1 - x} \dd B \dd F .
\]
Let $0 < z \le X_-$. Recall that $\Psi_x \Psi_{1 - x}$ takes values in $[0, 1]$, and note that $1/d$ is integrable on $[1, 2] \times [-1, 1]$. For $B \ge 2$, we have
\[
 \frac{|F|}{d} \le \frac{1}{B - 1} \le \frac2B , \qquad \Psi_x \Psi_{1 - x} \le \min\bigl(1, \bar\Upsilon(zB/X)\bigr) ,
\]
where the second bound is \eqref{e:res-envelope-original}. Substituting $q = zB/X$, we obtain
\[
 \bigl| \widetilde K_X(x; z) - K_X(x, x; z) \bigr| \le C + 4 \int_{2z/X}^\infty \min\bigl(1, \bar\Upsilon(q)\bigr) \frac{\dd q}{q} \le C\Bigl(1 + \log\frac{X_+}{z}\Bigr) ,
\]
where we used
\[
 \int_1^\infty \bar\Upsilon(q) \frac{\dd q}{q} \le \int_0^\infty \bar\Upsilon(q) \dd q < \infty ,
\]
which holds by \zcref{l:res-envelope}. Multiplying by $z$ and letting $z \downarrow 0$, we see that $z(\widetilde K_X - K_X(x, x; \cdot)) \to 0$ uniformly. Now \zcref{t:res-scalar} at $y = x$ implies \eqref{e:int-contact-residue}.
\end{proof}

\section{Local limits at positive damping}\label{s:int-balanced}

We now combine \zcref{t:interface}(b) with the pairings above. The result is the limit of the local densities at a fixed damping, at one or two interior sites, with no lower bound on the distance between the sites.

\begin{proposition}\label{p:int-balanced}
Let $|E| \le E_{\mathrm{out}}$, and let $i, j$ be interior sites such that $i/N \to x$, $j/N \to y$, $X_N(E) \to X \in (0, \infty)$, and $\1_{i = j}$ is constant. Let $\eps > 0$ and $z = 2\pi\eps$. Then $\E G^+_{ii} - (-\kappa_E + \iu s) \to 0$ and $\E \ell^\eps_{i, N}(0) \to 1$. If $i \ne j$, then
\begin{align}
 \E G^+_{ii} G^-_{jj} - 1 - s^2 K_X(x, y; z) &\longrightarrow 0, \qquad
 \E G^+_{ii} G^+_{jj} - (-\kappa_E + \iu s)^2 \longrightarrow 0, \label{e:int-balanced-distinct} \\
 \E \ell^\eps_{i, N}(0) \ell^\eps_{j, N}(0) &\longrightarrow 1 + \tfrac12 K_X(x, y; z) . \label{e:int-balanced-ldos}
\end{align}
If $i = j$, then
\begin{equation}\label{e:int-balanced-contact}
\begin{gathered}
 \E G^+_{ii} G^-_{ii} - 1 - 2s^2 \widetilde K_X(x; z) \to 0 , \\
 \E (G^+_{ii})^2 - (-\kappa_E + \iu s)^2 \to 0 , \\
 \E \ell^\eps_{i, N}(0)^2 \to 1 + \widetilde K_X(x; z) .
\end{gathered}
\end{equation}
\end{proposition}

\begin{proof}
Passing to subsequences, we may assume that $E$ converges, so that $\kappa_E$ and $s$ converge. We first consider two distinct sites, and we assume $i < j$ and $x \le y$; the case $i > j$ is the same, with the second form of \zcref{l:int-sources}(b) and of \eqref{e:int-pairing-cross}, and with $K_X$ extended symmetrically. By \zcref{t:interface}(b), \zcref{l:int-sources}(b), and \eqref{e:int-jet-limits}, $\E G^+_{ii} G^-_{jj}$ has the limit
\begin{multline*}
 \mathcal B_1\bigl(\Psi_x, (-\kappa_E + \iu s Q^{\mathrm{FF}}_{\mathrm{RR}}) U_{y - x} (-\kappa_E + \iu s Q^{\mathrm{FF}}_{\mathrm{AA}}) \Psi_{1 - y}\bigr) \\
 = \kappa_E^2 - \iu s\kappa_E (1 - 1) + (\iu s)^2 \bigl(-1 - K_X(x, y; z)\bigr) = 1 + s^2 K_X(x, y; z) ,
\end{multline*}
where we expanded the product, used \zcref{l:int-pairings}, and used $\kappa_E^2 + s^2 = 1$ in the last step.

Next, let $c \in \{\mathrm{BB}, \mathrm{FF}\}$. Expanding in the same way and using \eqref{e:int-pairing-one} and \eqref{e:int-pairing-same}, we find
\[
 \mathcal B_1\bigl(\Psi_x, (-\kappa_E + \iu s Q^c_{\mathrm{RR}}) U_{y - x} (-\kappa_E + \iu s Q^c_{\mathrm{RR}}) \Psi_{1 - y}\bigr) = \kappa_E^2 - 2\iu s\kappa_E - s^2 = (-\kappa_E + \iu s)^2 .
\]
By \zcref{l:int-sources}(c), with $c = \mathrm{FF}$ and $c = \mathrm{BB}$, the expectations $\E(G^+_{ii} G^+_{jj} - G^+_{ij} G^+_{ji})$ and $\E(G^+_{ii} G^+_{jj} + G^+_{ij} G^+_{ji})$ converge to $(-\kappa_E + \iu s)^2$. Taking their average, we find the second limit in \eqref{e:int-balanced-distinct}. For one site, the first two limits in \eqref{e:int-balanced-contact} follow from \zcref{l:int-sources}(d) and \zcref{l:int-contact}. Similarly, \zcref{l:int-sources}(a) and \eqref{e:int-pairing-one} imply that $\E G^+_{ii} \to -\kappa_E + \iu s$.

It remains to pass to the local densities. Recall that $\ell^\eps_{i, N}(0) = s^{-1} \Im G^+_{ii}$, and that $\overline{G_{jj}(z_+)} = G_{jj}(z_-)$ since $H$ is Hermitian. The identity $\Im a \Im b = \frac12 \Re(a\bar b - ab)$ gives
\[
 \E \ell^\eps_{i, N}(0) \ell^\eps_{j, N}(0) = \frac{1}{2s^2} \Re\bigl( \E G^+_{ii} G^-_{jj} - \E G^+_{ii} G^+_{jj} \bigr) .
\]
The kernels are real for real $z$, and
\[
 \Re(-\kappa_E + \iu s)^2 = \kappa_E^2 - s^2 = 1 - 2s^2 .
\]
Inserting the limits above, we find for $i \ne j$ and $i = j$, respectively,
\[
 \frac{1 + s^2 K_X - (1 - 2s^2)}{2s^2} = 1 + \frac12 K_X(x, y; z) , \qquad
 \frac{1 + 2s^2 \widetilde K_X - (1 - 2s^2)}{2s^2} = 1 + \widetilde K_X(x; z) ,
\]
which are \eqref{e:int-balanced-ldos} and the last limit in \eqref{e:int-balanced-contact}. Finally, $\E \ell^\eps_{i, N}(0) = s^{-1} \Im \E G^+_{ii}$ tends to one.
\end{proof}

We also need these limits at shifted centers, where the damping of \zcref{t:interface} is not fixed. The next lemma provides the necessary equicontinuity.

\begin{lemma}\label{l:int-equicontinuity}
For sites $i, j$, an energy $|E| \le E_{\mathrm{mid}}$, $u \in \R$, and $\eps > 0$, let
\[
 F_N(u, \eps) = \E G_{ii}\Bigl(E + \frac{u + \iu\eps}{a_N}\Bigr) G_{jj}\Bigl(E + \frac{u - \iu\eps}{a_N}\Bigr) .
\]
Then $F_N$ is Lipschitz on compact subsets of $\R \times (0, \infty)$, uniformly in $N$, the sites, and $E$. If $i, j$, $x, y$, and $X$ are as in \zcref{p:int-balanced} and $E = E_N \to E_0$, then, locally uniformly in $(u, \eps)$,
\[
 F_N(u, \eps) \longrightarrow
 \begin{cases}
  1 + s_{E_0}^2 K_X(x, y; 2\pi\eps) , & i \ne j , \\
  1 + 2s_{E_0}^2 \widetilde K_X(x; 2\pi\eps) , & i = j .
 \end{cases}
\]
\end{lemma}

\begin{proof}
For $w = E + (u + \iu\eps)/a_N$, we have
\[
 \partial_u G_{ii}(w) = \frac{(G(w)^2)_{ii}}{a_N} , \qquad \partial_\eps G_{ii}(w) = \frac{\iu (G(w)^2)_{ii}}{a_N} .
\] By the Cauchy--Schwarz inequality and the spectral decomposition,
\begin{equation}\label{e:int-ward-identity}
 \bigl|(G(w)^2)_{ii}\bigr| \le \sum_k |G_{ik}(w)|^2 = \frac{\Im G_{ii}(w)}{\Im w} = \frac{a_N}{\eps} \Im G_{ii}(w) .
\end{equation}
We first show that
\[
 \phi(\eps) = \Bigl(\E \bigl|G_{jj}(E' + \iu\eps/a_N(E'))\bigr|^2\Bigr)^{1/2}
\]
is bounded for $\eps$ in compact subsets of $(0, \infty)$, uniformly in $N$, $j$, and $|E'| \le E_{\mathrm{out}}$. Using \eqref{e:int-ward-identity} and the Cauchy--Schwarz inequality, we find
\[
 \bigl|\partial_\eps \phi(\eps)^2\bigr| \le \frac{2}{\eps} \, \phi(\eps) \, \Bigl(\E \bigl(\Im G_{jj}(E' + \iu\eps/a_N(E'))\bigr)^2\Bigr)^{1/2} .
\] The last factor is bounded by \zcref{l:extract-moments}, since $s_{E'}^{-1} \Im G_{jj}(E' + \iu\eps/a_N(E')) = \ell^\eps_{j, N}(0)$ at the energy $E'$. Therefore $\phi$ is Lipschitz on compact sets. At $\eps = 1$, $\phi$ is bounded by \zcref{t:interface}(c). The recentering of the proof of \zcref{l:extract-moments} extends this bound to the arguments in $F_N$.

Let $w_\pm = E + (u \pm \iu\eps)/a_N$. Using \eqref{e:int-ward-identity} once more, at $w_+$ and at $w_-$, where $\Im w_- = -\eps/a_N$, we find
\[
 |\partial_u F_N| + |\partial_\eps F_N| \le \frac{2}{\eps} \Bigl( \E\bigl[\Im G_{ii}(w_+) \, |G_{jj}(w_-)|\bigr] + \E\bigl[|G_{ii}(w_+)| \, \Im G_{jj}(w_+)\bigr] \Bigr) ,
\]
where we used that $|\Im G_{jj}(w_-)| = \Im G_{jj}(w_+)$, since $w_- = \overline{w_+}$ and $G_{jj}(\bar w) = \overline{G_{jj}(w)}$. The right side is bounded on compact sets, by the Cauchy--Schwarz inequality and the bounds just proved, and the Lipschitz bound follows.

For the convergence, let $E' = E + u/a_N$ and $\eps' = \eps\rho(E')/\rho(E)$, so that
\[
 E + \frac{u \pm \iu\eps}{a_N} = E' \pm \frac{\iu\eps'}{a_N(E')} .
\] Then $|E'| \le E_{\mathrm{out}}$ for $N$ large, and $\eps' - \eps$, $X_N(E') - X_N(E)$, and $s_{E'} - s_E$ tend to zero. The Lipschitz bound allows us to replace $\eps'$ by $\eps$ with an error $o(1)$. Since $E' \to E_0$, we also have $s_{E'} \to s_{E_0}$. Now \zcref{p:int-balanced} at the energy $E'$ implies the stated convergence for each $(u, \eps)$, and the Lipschitz bound makes it locally uniform.
\end{proof}

\section{The profile}\label{s:int-profile}

We now apply the extraction theorem to the pairs of sites.

\begin{proposition}\label{p:int-diagonal}
For sites $i, j$ and an energy $e$, let $f^{ij}_{N, e}$ denote the density of $N^2 \E \sum_\alpha w_\alpha(i) w_\alpha(j) \delta_{a_N(e)(\lambda_\alpha - e)}$. Then for every $R > 0$,
\begin{equation}\label{e:int-diagonal}
 \sup \Bigl| f^{ij}_{N, e}(u) - (1 + \1_{i = j}) \, \mathcal C_{X_N(e)}(i/N, j/N) \Bigr| \longrightarrow 0 ,
\end{equation}
where the supremum is over all interior sites $i, j$, all $|e| \le E_{\mathrm{mid}}$ and all $|u| \le R$. The convergence is uniform for $N/W^2$ in compact subsets of $(0, \infty)$.
\end{proposition}

\begin{proof}
By \zcref{r:extract-uniform}, it suffices to consider sequences of sites $i, j$ and energies $e$ along which $i/N \to x$, $j/N \to y$, $X_N(e) \to X$, and $\1_{i = j}$ is constant, and to prove that $f^{ij}_{N, e}$ converges locally uniformly to $c_* = (1 + \1_{i = j})\mathcal C_X(x, y)$. Indeed, $\mathcal C_X(x, y)$ is continuous in $(X, x, y)$, by dominated convergence in \zcref{d:brownian}, so that $c_* - (1 + \1_{i = j})\mathcal C_{X_N(e)}(i/N, j/N) \to 0$.

We apply \zcref{t:extract-main} at the energy $e$ with $k = 2$, $r_2 = 1$, and $\bm\Pi_N = \ell_{i, N} \otimes \ell_{j, N}$, which is dominated with $C_0 = 1$ and has the diagonal part
\[
 \bm\Pi_N^{\mathrm d} = N^2 \sum_\alpha w_\alpha(i) w_\alpha(j) \delta_{x_\alpha} .
\]
Using \eqref{e:extract-smoothing}, we write the left side of \eqref{e:extract-source} as
\[
 \int_{\R^2} P_\eps(t) P_\eps(v) \, \Pi_N(\dd t, \dd v) = \E \ell^\eps_{i, N}(0) \ell^\eps_{j, N}(0) ,
\]
and \zcref{p:int-balanced} shows that it converges to $\mathfrak f(2\pi\eps)$, where
\[
 \mathfrak f(z) = 1 + \tfrac12 K_X(x, y; z) \quad (i \ne j), \qquad \mathfrak f(z) = 1 + \widetilde K_X(x; z) \quad (i = j) .
\]
By \zcref{t:res-scalar} and \zcref{l:int-contact-residue}, we have $\lim_{z \downarrow 0} z \mathfrak f(z) = c_*$ in both cases. Since $\varkappa_{2, 1} = 1$, \zcref{t:extract-main} implies that the densities $f^{ij}_{N, e}$ of $\Pi_N^{\mathrm d}$ converge to $c_*$ locally uniformly.
\end{proof}

\begin{proof}[Proof of \zcref{t:profile}, first part]
Let $\bar r_N \downarrow 0$. We apply \zcref{c:extract-windows} with $k = 2$, $\chi = \phi$, $F_\alpha = w_\alpha(i) w_\alpha(j)$ and
\[
 \bar c_N(e) = (1 + \1_{i = j}) \, \mathcal C_{X_N(e)}(i/N, j/N) .
\]
These functions are bounded and equicontinuous in $|e| \le E_{\mathrm{mid}}$, uniformly in $N$ and in the sites, since $X_N(e) = N/((4 - e^2)W^2)$ is Lipschitz in $e$ uniformly for $N/W^2$ in a compact set, and $\mathcal C_X(x, y)$ is uniformly continuous on compact sets. The hypothesis of \zcref{c:extract-windows} is \eqref{e:int-diagonal} at $u = 0$.

For the rooted ratio, the densities $g_{N, e}$ of $N \E \sum_\alpha w_\alpha(i) \delta_{a_N(e)(\lambda_\alpha - e)}$ converge to one, uniformly for $|e| \le E_{\mathrm{mid}}$ and interior sites $i$. This follows from \zcref{c:extract-onepoint} and \zcref{r:extract-uniform}, since $\E \ell^\eps_{i, N}(0) \to 1$ by \zcref{p:int-balanced}. With these choices, \eqref{e:extract-window} becomes the statement for $N^2 \langle w_\alpha(i) w_\alpha(j) \rangle_{E, r}$, and \eqref{e:extract-rooted-window} yields \eqref{e:intro-profile} for $P_N(E, r; i, j)$. These limits hold uniformly over interior pairs, since the hypotheses do.
\end{proof}

\section{The two-energy intensity}\label{s:int-twoenergy}

We now turn to the full two-energy intensity, which requires the part of the separation measure of the pair away from zero. Let $i, j$ be interior sites and $|E| \le E_{\mathrm{mid}}$, and set $\Pi_N = \E[\ell_{i, N} \otimes \ell_{j, N}]$. By \zcref{l:extract-limits}, every vague limit of $\Pi_N$ along a subsequence with $i/N \to x$, $j/N \to y$, $X_N(E) \to X$, and constant $\1_{i = j}$ has the form
\begin{equation}\label{e:int-stationary}
 \Pi(\dd t, \dd v) = \dd t \, \nu_{x, y}(\dd(v - t)) ,
\end{equation}
where $\nu_{x, y}$ is a positive translation-bounded measure on $\R$. We call $\nu_{x, y}$ the \emph{separation measure} of the limit. Exchanging the two sites reflects the separation measure, so that the limit for $(j, i)$ has the separation measure $\nu_{x, y}(-\dd r)$. Therefore evenness of $\nu_{x, y}$ is equivalent to its symmetry in the two sites. This symmetry does not follow from exchanging the eigenvalue labels, and we derive it from the limits at coinciding energies.

\begin{lemma}\label{l:int-even}
Every separation measure $\nu = \nu_{x, y}$ in \eqref{e:int-stationary} is even.
\end{lemma}

\begin{proof}
Fix a subsequence along which $\Pi_N \to \Pi$ vaguely, with separation measure $\nu$ as in \eqref{e:int-stationary}. Passing to a further subsequence, we may assume that $E = E_N \to E_0$; this preserves $\Pi$ and $\nu$. Let $F_N$ be as in \zcref{l:int-equicontinuity}, which now shows that $F_N \to A$ locally uniformly on $\R \times (0, \infty)$, where
\[
 A(\eps) = 1 + s_{E_0}^2 K_X(x, y; 2\pi\eps) \quad (i \ne j) , \qquad A(\eps) = 1 + 2s_{E_0}^2 \widetilde K_X(x; 2\pi\eps) \quad (i = j) .
\]
The function $A$ is real and independent of $u$. With $w_\pm = u \pm \iu\eps$, we have $F_N(u, \eps) = \Phi_N(w_+, w_-)$, where
\[
 \Phi_N(w, w') = \E\, G_{ii}(E + w/a_N) \, G_{jj}(E + w'/a_N)
\]
is holomorphic in $(w, w')$ for $\Im w > 0 > \Im w'$, and $\partial_u^2 + \partial_\eps^2 = 4\partial_{w_+}\partial_{w_-}$. Using the spectral decomposition
\[
 G_{ii}(E + w/a_N) = \sum_\alpha \frac{a_N w_\alpha(i)}{x_\alpha - w} ,
\]
we find
\begin{equation}\label{e:int-laplacian}
 (\partial_u^2 + \partial_\eps^2) F_N(u, \eps) = 4\rho(E)^2 \int_{\R^2} \frac{\Pi_N(\dd t, \dd v)}{(t - u - \iu\eps)^2 (v - u + \iu\eps)^2} .
\end{equation}

As $N \to \infty$, the left side converges to $A''(\eps)$ in the sense of distributions on $\R \times (0, \infty)$, and $A''$ is real. By \zcref{l:extract-limits}(c), applied to the squared Cauchy kernels, and $\rho(E) \to \rho(E_0) > 0$, the right side converges locally uniformly along the subsequence to $4\rho(E_0)^2$ times the corresponding integral against $\Pi$. This continuous limit equals $A''$ as a distribution. Since $4\rho(E_0)^2$ is positive and real, the integral against $\Pi$ is real.

We compute this integral with \eqref{e:int-stationary}. For $a = 2\eps$, the residue theorem, applied in the upper half-plane at the double pole $t = \iu\eps$, gives
\[
 \int_\R \frac{\dd t}{(t - \iu\eps)^2 (t + r + \iu\eps)^2} = -\frac{4\pi\iu}{(r + \iu a)^3} , \qquad r \in \R .
\]
Substituting $t \mapsto t + u$, we conclude that
\[
 \Re \int_\R \frac{\nu(\dd r)}{(r + \iu a)^3} = 0 , \qquad a > 0 .
\]

Let $\nu_\pm$ denote the even and the odd part of $\nu$, so that $\nu_- = \frac12(\nu - \check\nu)$ with $\check\nu(\dd r) = \nu(-\dd r)$. Since
\[
 \overline{(r + \iu a)^{-3}} = -(-r + \iu a)^{-3} , \qquad r \in \R ,
\]
the integral of $(r + \iu a)^{-3}$ against the even measure $\nu_+$ is purely imaginary, and against the odd measure $\nu_-$ it is real. Therefore the function
\[
 T(w) = \int_\R \frac{\nu_-(\dd r)}{(r + w)^3}
\]
vanishes on the positive imaginary axis. Since $\nu$ is translation bounded, $T$ is holomorphic in the upper half-plane, and the identity theorem implies that $T = 0$ there. As $\nu_-$ is real, we have $T(\bar w) = \overline{T(w)}$, and $T = 0$ also in the lower half-plane.

Finally, write the Poisson convolution as
\[
 (P_a * \nu_-)(u) = \frac1\pi \int_\R \Im \frac{1}{r - u - \iu a} \, \nu_-(\dd r) .
\]
For fixed $a>0$, this real kernel and its first two derivatives in $u$ are bounded by $C(1+r^2)^{-1}$, locally uniformly in $u$. The translation boundedness of $|\nu_-|$ therefore makes these integrals absolutely convergent and justifies differentiating twice under the integral. Thus
\[
 \partial_u^2 (P_a * \nu_-)(u) = \frac2\pi \Im T(-u - \iu a) = 0 .
\]
The function $P_a * \nu_-$ is bounded, since $\nu$ is translation bounded. Being affine and bounded, it is constant, and being odd, it vanishes. Therefore $P_a * \nu = P_a * \check\nu$ for all $a > 0$, and \zcref{l:tools-poisson}(b) implies that $\nu = \check\nu$.
\end{proof}

We now identify the separation measure from its values at coinciding energies. For an even positive translation-bounded measure $\nu$ on $\R$, we define
\begin{equation}\label{e:int-even-transform}
 \mathcal F_\nu(w) = \int_\R \frac{w}{\pi(w^2 + r^2)} \, \nu(\dd r) , \qquad \Re w > 0 .
\end{equation}
The integrand is holomorphic in $w$ and bounded by $C(1 + r^2)^{-1}$ locally uniformly in $\Re w > 0$, so $\mathcal F_\nu$ is holomorphic. Using the evenness of $\nu$ and
\[
 \frac{w}{w^2 + r^2} = \frac12 \Bigl(\frac{1}{w - \iu r} + \frac{1}{w + \iu r}\Bigr) ,
\]
we find for $a > 0$ and $d \in \R$
\begin{equation}\label{e:int-even-values}
 \mathcal F_\nu(a) = (P_a * \nu)(0) , \qquad \Re \mathcal F_\nu(a - \iu d) = (P_a * \nu)(d) .
\end{equation}
In particular, the values $(P_a * \nu)(0)$ for $a > 0$ determine $\mathcal F_\nu$ by the identity theorem, hence all functions $P_a * \nu$ by \eqref{e:int-even-values}, hence $\nu$ by \zcref{l:tools-poisson}(b).

The uniform statement of \zcref{t:intensity} also needs the continuity of the kernels in all their arguments.

\begin{lemma}\label{l:int-continuity}
The map $(X, x, y, z) \mapsto K_X(x, y; z)$ is continuous on $(0, \infty) \times \{0 < x \le y < 1\} \times \{\Re z > 0\}$, and the map $(X, x, z) \mapsto \widetilde K_X(x; z)$ is continuous on $(0, \infty) \times (0, 1) \times \{\Re z > 0\}$.
\end{lemma}

\begin{proof}
We first remove $X$ by a change of time. The diffusion with generator $X\Delta_D$ is the diffusion with generator $\Delta_D$ at time $Xt$. With $w = z/X$, the definitions in \zcref{d:kernels} therefore imply that
\begin{equation}\label{e:int-time-change}
 e^{t(X\Delta_D - zd)} = e^{Xt(\Delta_D - wd)} , \qquad \Psi^{X, z}_t = \Psi^{1, w}_{Xt} .
\end{equation}
Inserting \eqref{e:int-time-change} into the definitions \eqref{e:intro-K} and \eqref{e:intro-Kc}, with $t_1 = Xx$, $t_2 = X(y - x)$, and $t_3 = X(1 - y)$, we have
\[
 K_X(x, y; z) = \int_D \Psi^{1, w}_{t_1} \, e^{t_2(\Delta_D - wd)} \Psi^{1, w}_{t_3} \dd B \dd F , \qquad
 \widetilde K_X(x; z) = \int_D \frac Bd \, \Psi^{1, w}_{t_1} \Psi^{1, w}_{X(1 - x)} \dd B \dd F .
\]

Let
\[
 f(t, p, w) = \bigl(e^{t(\Delta_D - wd)} 1\bigr)(p) .
\]
For fixed $w$ with $\Re w > 0$, $f$ is continuous in $(t, p)$ on $[0, \infty) \times D$, as shown in the proof of \zcref{l:sigma-scalar}\ref{i:sigma-scalar-singlet}. For fixed $(t, p)$, $f$ is holomorphic in $w$ by dominated convergence in the Feynman--Kac formula, and $|f| \le 1$. Cauchy's estimate therefore shows that $|\partial_w f| \le 1/\alpha$ on $\{\Re w \ge 2\alpha\}$, uniformly in $(t, p)$. Together with the continuity in $(t, p)$, this implies that $f$ is jointly continuous in $(t, p, w)$. By \eqref{e:intro-singlet},
\[
 \Psi^{1, w}_t(p) = 1 - w \, d(p) \int_0^t f(v, p, w) \dd v
\]
is then jointly continuous on $[0, \infty) \times D \times \{\Re w > 0\}$.

Next, let $(\tilde B_t, \tilde F_t)$ be the radial diffusion of \zcref{s:sigma-scalar} with $X = 1$, with generator $\Delta_D$. By the Markov property,
\[
 \bigl(e^{t_2(\Delta_D - wd)} \Psi^{1, w}_{t_3}\bigr)(p) = \E_p\Bigl[ \exp\Bigl(-w \int_0^{t_2} d(\tilde B_u, \tilde F_u) \dd u\Bigr) \Psi^{1, w}_{t_3}(\tilde B_{t_2}, \tilde F_{t_2}) \Bigr] .
\] When $(t_2, t_3, w)$ converges, the integrand converges almost surely, by the previous paragraph and the continuity of the paths, and it is bounded by one. The bounded convergence theorem then shows that this function is continuous in $(t_2, t_3, w)$ for every $p$.

Finally, fix a point in the domain of $K$ and a compact neighborhood of it on which $t_1, t_3 \ge \sigma > 0$, $t_1, t_2, t_3 \le T$ and $\Re w \ge a > 0$. Using \zcref{l:sigma-scalar}\ref{i:sigma-scalar-singlet},\ref{i:sigma-scalar-tails}, we bound the integrand of $K$ there by
\[
 \bigl|\Psi^{1, w}_{t_1}\bigr| \le \Psi^{1, a}_{t_1} \le \Psi^{1, a}_\sigma \le C(1 + B)^{-3} ,
\]
which is integrable on $D$. The continuity of $K$ follows by dominated convergence. For $\widetilde K$, we use the dominating function $(B/d) C (1 + B)^{-3}$, which is integrable since $B/d$ is integrable near the corner.
\end{proof}

\begin{proof}[Proof of \zcref{t:intensity}]
By the argument by contradiction of \zcref{r:extract-uniform}, it suffices to consider sequences of interior sites $i, j$ and of parameters $\eps_1, \eps_2 > 0$ and $u, v \in \R$, with $|E| \le E_*$, along which $i/N \to x$, $j/N \to y$, $X_N(E) \to X$, $\1_{i = j}$ is constant, and the parameters converge to positive dampings $\eps_1, \eps_2$ and real centers $u, v$. Passing to a further subsequence, we may assume that $\Pi_N = \E[\ell_{i, N} \otimes \ell_{j, N}]$ and its diagonal and off-diagonal parts converge vaguely, with the separation measure $\nu$ of \eqref{e:int-stationary}. It suffices to identify the limits along this subsequence.

By \eqref{e:extract-smoothing} and \zcref{l:extract-limits}(c), applied to the Poisson kernels with the converging parameters, $\E \ell^{\eps_1}_{i, N}(u) \ell^{\eps_2}_{j, N}(v)$ converges to
\[
 \int_\R \int_\R P_{\eps_1}(t - u) P_{\eps_2}(t + r - v) \dd t \, \nu(\dd r) = \int_\R P_{\eps_1 + \eps_2}(v - u - r) \, \nu(\dd r) = (P_{\eps_1 + \eps_2} * \nu)(u - v) ,
\]
where we used the semigroup property of the Poisson kernels and the evenness of $\nu$ from \zcref{l:int-even}. For $\eps_1 = \eps_2 = a/2$ and $u = v = 0$, \zcref{p:int-balanced} identifies the same limit. Comparing the two limits and using \eqref{e:int-even-values}, we find
\[
 \mathcal F_\nu(a) = 1 + \tfrac12 K_X(x, y; \pi a) \quad (i \ne j), \qquad \mathcal F_\nu(a) = 1 + \widetilde K_X(x; \pi a) \quad (i = j), \qquad a > 0 .
\]
The kernels are holomorphic in $\Re z > 0$ by \zcref{l:sigma-scalar}\ref{i:sigma-scalar-kernels}, and the identity theorem implies that $\mathcal F_\nu(w) = 1 + \frac12 K_X(x, y; \pi w)$, or $1 + \widetilde K_X(x; \pi w)$, for $\Re w > 0$. Applying \eqref{e:int-even-values} with $a = \eps_1 + \eps_2$ and $d = u - v$, so that $\pi(a - \iu d) = z$ in the notation of \zcref{t:intensity}, we obtain \eqref{e:intro-int-distinct} and \eqref{e:intro-int-contact}.

For the same-eigenvalue part, the diagonal part $\Pi_N^{\mathrm d}$ converges vaguely to $(1 + \1_{i = j})\mathcal C_X(x, y) \dd t$ by \zcref{p:int-diagonal}. We apply \zcref{l:extract-limits}(c) to $(\iota_2)_*\Pi_N^{\mathrm d}$, and we use
\[
 \int_\R P_{\eps_1}(t - u) P_{\eps_2}(t - v) \dd t = P_{\eps_1 + \eps_2}(u - v) .
\]
This shows that the sum in \eqref{e:intro-int-same} converges to $(1 + \1_{i = j})\mathcal C_X(x, y) P_{\eps_1 + \eps_2}(u - v)$. Since the limits do not depend on the subsequence and, by \zcref{l:int-continuity} and the continuity of $\mathcal C_X(x, y)$ in $(X, x, y)$, are continuous in all the parameters, the theorem follows.
\end{proof}

\section{Boundary control and the inverse participation ratio}\label{s:int-ipr}

The inverse participation ratio also involves the sites within distance $\delta N$ of the two ends, at which \zcref{t:interface}(b) is not available. We only need a uniform bound on the diagonal densities there.

\begin{lemma}\label{l:int-density}
For a site $i$ and an energy $e$, let $f^{ii}_{N, e}$ be as in \zcref{p:int-diagonal}. Then
\begin{equation}\label{e:int-density}
 \sup_N \sup_{1 \le i \le N} \sup_{|e| \le E_{\mathrm{mid}}} f^{ii}_{N, e}(0) < \infty .
\end{equation}
\end{lemma}

\begin{proof}
Let $\Pi^{\mathrm d}_N$ denote the measure $N^2 \E \sum_\alpha w_\alpha(i)^2 \delta_{x_\alpha}$, unfolded at $e$. Using
\[
 \sum_\alpha w_\alpha(i)^2 \1_{|x_\alpha| \le 1} \le \Bigl(\sum_\alpha w_\alpha(i) \1_{|x_\alpha| \le 1}\Bigr)^2
\]
and the lower bound $P_1 \ge P_1(1)$ on $[-1, 1]$, we find
\[
 \Pi^{\mathrm d}_N([-1, 1]) \le \E \bigl[\ell_{i, N}([-1, 1])^2\bigr] \le P_1(1)^{-2} \, \E\bigl[\ell^1_{i, N}(0)^2\bigr] \le C ,
\]
uniformly in all sites and in $|e| \le E_{\mathrm{mid}}$, by \zcref{l:extract-moments}. The density of $\Pi^{\mathrm d}_N$ is a Gaussian mixture with variance $\tau_N^2$, by \zcref{l:extract-gaussian}. Using \eqref{e:extract-mixture-bound} with $u_0 = 0$ and \eqref{e:extract-tau}, we conclude that
\[
 f^{ii}_{N, e}(0) \le \frac12 e^{1/(2\tau_N^2)} \, \Pi^{\mathrm d}_N([-1, 1]) \le C . \qedhere
\]
\end{proof}

\begin{proof}[Proof of \zcref{t:profile}, second part]
For a physical energy $e$, the measure
\[
 \frac1N \sum_{i = 1}^N N^2 \E \sum_\alpha w_\alpha(i)^2 \delta_{a_N(e)(\lambda_\alpha - e)} = N^2 \E \sum_\alpha F_\alpha \, \delta_{a_N(e)(\lambda_\alpha - e)} , \qquad F_\alpha = \frac1N \IPR(u_\alpha) ,
\]
has the density
\[
 f_{N, e} = \frac1N \sum_{i = 1}^N f^{ii}_{N, e} .
\] By \zcref{p:int-diagonal}, $f^{ii}_{N, e}(0) - 2\mathcal C_{X_N(e)}(i/N, i/N) \to 0$ uniformly over interior sites and $|e| \le E_{\mathrm{mid}}$, and by \zcref{l:int-density}, the remaining $2\delta N + O(1)$ sites contribute at most $C(\delta + N^{-1})$ to $f_{N, e}(0)$. The function $(X, x) \mapsto \mathcal C_X(x, x)$ is continuous and bounded on $[X_-, X_+] \times [0, 1]$ for $0 < X_- \le X_+$. Taking the Riemann sum over the interior sites, and then letting $\delta \downarrow 0$, we obtain
\[
 \sup_{|e| \le E_{\mathrm{mid}}} \Bigl| f_{N, e}(0) - 2 \int_0^1 \mathcal C_{X_N(e)}(x, x) \dd x \Bigr| \longrightarrow 0 .
\]

We set $\bar c_N(e) = 2 + 2X_N(e)/3$. By \zcref{p:int-exact} below, we have
\[
 \bar c_N(e) = 2 \int_0^1 \mathcal C_{X_N(e)}(x, x) \dd x ,
\]
and these functions are bounded and equicontinuous in $|e| \le E_{\mathrm{mid}}$. We apply \eqref{e:extract-window} with $k = 2$, $\chi = \phi$, and these $F_\alpha$. Since $N^2 F_\alpha = N \IPR(u_\alpha)$, we conclude that $N \langle \IPR(u_\alpha) \rangle_{E, r} - 2 - 2X_N(E)/3 \to 0$, uniformly for $|E| \le E_*$ and $0 < r \le \bar r_N$, which is \eqref{e:intro-ipr}.
\end{proof}

\section{The exact participation formula}\label{s:int-exact}

We now evaluate the integrated diagonal of $\mathcal C_X$. We use the notation of \zcref{s:residue}, with
\[
 g_1(u) = \sqrt2 \beta_u - u , \qquad A_t = \int_0^t e^{g_1(u)} \dd u , \qquad \Upsilon_t(q) = \E e^{-q A_t} , \qquad t \ge 0 , \ q > 0 .
\]
We also set
\[
 A^+_t = \int_0^t e^{\sqrt2 \beta_u + u} \dd u , \qquad \Upsilon^+_t(q) = \E e^{-q A^+_t} ,
\]
which correspond to the opposite drift, and we write $\langle f, g \rangle = \int_0^\infty f(q) g(q) \dd q$ for nonnegative functions on $(0, \infty)$.

For $\lambda > 2$, we set
\[
 \upsilon_\lambda(q) = \int_0^\infty e^{-\lambda t} \Upsilon_t(q) \dd t , \qquad \upsilon^+_\lambda(q) = \int_0^\infty e^{-\lambda t} \Upsilon^+_t(q) \dd t ,
\]
which are the resolvents at $\lambda$, applied to the constant one, of the geometric Brownian motions with drifts $-1$ and $+1$, killed at rate $q$. We will need the following properties of $\upsilon_\lambda$.

\begin{lemma}\label{l:int-resolvent}
Let $\lambda > 2$. The function $\upsilon_\lambda$ is smooth on $(0, \infty)$ and satisfies
\begin{equation}\label{e:int-resolvent-ode}
 q^2 \upsilon_\lambda'' - (q + \lambda) \upsilon_\lambda = -1 .
\end{equation}
Moreover, $\upsilon_\lambda(0+) = \lambda^{-1}$, $\upsilon_\lambda'(0+) = -\lambda^{-2}$, and for all $q > 0$,
\begin{equation}\label{e:int-resolvent-bounds}
 0 \le \upsilon_\lambda(q) \le \min\bigl(\lambda^{-1}, q^{-1}\bigr) , \qquad 0 \le -\upsilon_\lambda'(q) \le \frac{4}{eq^2} , \qquad \upsilon^+_\lambda(q) = \frac{1 - \lambda \upsilon_\lambda(q)}{q} .
\end{equation}
\end{lemma}

\begin{proof}
We have
\[
 \partial_q^m \Upsilon_t(q) = \E\bigl[(-A_t)^m e^{-qA_t}\bigr] , \qquad \bigl| q^m \partial_q^m \Upsilon_t(q) \bigr| \le \sup_{a \ge 0} a^m e^{-a} = m^m e^{-m} ,
\]
for all $m \ge 0$, uniformly in $t$. The process $q^0_u = q e^{g_1(u)}$ satisfies $\dd q^0_u = \sqrt2 q^0_u \dd\beta_u$, and $A_t$ splits at $r \in (0, t)$ as $A_t = A_r + e^{g_1(r)} \tilde A_{t - r}$, where $\tilde A_{t - r}$ is independent of $(\beta_u)_{u \le r}$ and has the law of $A_{t - r}$. Conditioning on $(\beta_u)_{u \le r}$, we find
\[
 \Upsilon_t(q) = \E\bigl[e^{-qA_r} \Upsilon_{t - r}(q^0_r)\bigr] .
\] Applying It\^o's formula to $e^{-qA_u} \Upsilon_{t - r}(q^0_u)$ on $[0, r]$, whose martingale part is a true martingale by the bound on $q\partial_q \Upsilon_{t - r}$, and letting $r \downarrow 0$, we obtain the backward equation
\[
 \partial_t \Upsilon_t = q^2 \partial_q^2 \Upsilon_t - q \Upsilon_t , \qquad \Upsilon_0 = 1 .
\]
Multiplying by $e^{-\lambda t}$ and integrating over $t \in (0, \infty)$, with differentiation under the integral justified by the bounds above, we obtain
\[
 \lambda \upsilon_\lambda - 1 = \int_0^\infty e^{-\lambda t} \partial_t \Upsilon_t \dd t = q^2 \upsilon_\lambda'' - q \upsilon_\lambda ,
\]
which is \eqref{e:int-resolvent-ode}. Since $\E A_t = t$, dominated differentiation gives $\upsilon_\lambda(0+) = \lambda^{-1}$ and
\[
 \upsilon_\lambda'(0+) = -\int_0^\infty e^{-\lambda t} t \dd t = -\lambda^{-2} ,
\]
and $0 \le \Upsilon_t \le 1$ implies that $\upsilon_\lambda \le \lambda^{-1}$.

To prove $\upsilon_\lambda \le 1/q$, note that the generator $q^2\partial_q^2 - q$ of the killed process satisfies
\[
 \big( \lambda - q^2\partial_q^2 + q \big) \frac1q = 1 + \frac{\lambda - 2}{q} \ge 1 .
\]
We apply Dynkin's formula to $1/q$ and the killed process up to the exit time of $[n^{-1}, n]$, and we discard the nonnegative terminal term. Letting $n \to \infty$, which is allowed since geometric Brownian motion reaches neither $0$ nor $\infty$ in finite time, we find that $\upsilon_\lambda(q) \le 1/q$. For the derivative, we use $a e^{-qa/2} \le 2/(eq)$ for $a \ge 0$, which gives
\[
 -\partial_q \Upsilon_t(q) = \E\bigl[A_t e^{-qA_t}\bigr] \le \frac{2}{eq} \Upsilon_t(q/2) .
\] Integrating against $e^{-\lambda t} \dd t$ and using $\upsilon_\lambda(q/2) \le 2/q$, we obtain the bound on $\upsilon_\lambda'$.

Finally, dominated differentiation gives
\[
 \partial_t \Upsilon_t(q) = -q \, \E\bigl[e^{g_1(t)} e^{-qA_t}\bigr] .
\] The density $e^{g_1(t)}$ has mean one, and by the Cameron--Martin theorem, under the measure it defines, $(\beta_u - \sqrt2 u)_{u \le t}$ is a Brownian motion, so that $g_1(u)$ becomes $\sqrt2\beta_u + u$ for $u \le t$. We conclude that $\partial_t \Upsilon_t = -q \Upsilon^+_t$. Integrating by parts in $t$, and using $0 \le \Upsilon_t \le 1$, we obtain
\[
 q \, \upsilon^+_\lambda(q) = -\int_0^\infty e^{-\lambda t} \partial_t \Upsilon_t(q) \dd t = 1 - \lambda \upsilon_\lambda(q) . \qedhere
\]
\end{proof}

\begin{proposition}\label{p:int-exact}
For all $X > 0$,
\begin{equation}\label{e:int-exact}
 \int_0^1 \mathcal C_X(x, x) \dd x = 1 + \frac X3 .
\end{equation}
\end{proposition}

The proof rests on two identities, which come from one Markov decomposition of the exponential functional $A_X$, and on two integrated identities for $\upsilon_\lambda$, which we call the virial identities.

\begin{proof}
We set
\[
 \mathcal A(X) = \int_0^1 \mathcal C_X(x, x) \dd x .
\]
We first express $\mathcal A(X)$ and the number one as integrals of products of $\Upsilon$ and $\Upsilon^+$. By \eqref{e:res-pair} with $y = x$, we have
\[
 \mathcal C_X(x, x) = X \langle \Upsilon_{Xx}, \Upsilon_{X(1 - x)} \rangle ,
\]
and the substitution $t = Xx$ gives
\begin{equation}\label{e:int-trace-split}
 \mathcal A(X) = \int_0^X \langle \Upsilon_t, \Upsilon_{X - t} \rangle \dd t .
\end{equation}

For the second identity, we use Brownian scaling: $(g_X(x))_{x \le 1}$ has the law of $(g_1(Xx))_{x \le 1}$, under which $e^{g_X(x)}/Z_X$ becomes $X e^{g_1(Xx)}/A_X$. Taking the expectation of the total mass of the probability measure $\mu_X$ and substituting $t = Xx$, we find
\[
 \int_0^X \E\Bigl[\frac{e^{g_1(t)}}{A_X}\Bigr] \dd t = 1 .
\] Next, for $0 \le t \le X$, we split the integral defining $A_X$ at $t$ and divide by $e^{g_1(t)}$, which gives
\begin{equation}\label{e:int-split}
 \frac{A_X}{e^{g_1(t)}} = \int_0^t e^{\sqrt2(\beta_{t - r} - \beta_t) + r} \dd r + \int_0^{X - t} e^{\sqrt2(\beta_{t + r} - \beta_t) - r} \dd r .
\end{equation}
The increments of $\beta$ before and after $t$ are independent, and $r \mapsto \beta_{t - r} - \beta_t$ is a Brownian motion on $[0, t]$. The two integrals in \eqref{e:int-split} are therefore independent, with the laws of $A^+_t$ and $A_{X - t}$. Inserting the identity
\[
 \frac1a = \int_0^\infty e^{-qa} \dd q , \qquad a = \frac{A_X}{e^{g_1(t)}} ,
\]
and using the independence of the two integrals in \eqref{e:int-split}, we find
\begin{equation}\label{e:int-mass-split}
 1 = \int_0^X \langle \Upsilon^+_t, \Upsilon_{X - t} \rangle \dd t .
\end{equation}

Next, we take Laplace transforms in $X$. All integrands in \eqref{e:int-trace-split} and \eqref{e:int-mass-split} are nonnegative. Using Tonelli's theorem, we find, for $\lambda > 2$,
\begin{equation}\label{e:int-laplace}
 \int_0^\infty e^{-\lambda X} \mathcal A(X) \dd X = \mathfrak v_1(\lambda) , \qquad \frac1\lambda = \mathfrak v_2(\lambda) ,
\end{equation}
where
\[
 \mathfrak v_1 = \langle \upsilon_\lambda, \upsilon_\lambda \rangle , \qquad \mathfrak v_2 = \langle \upsilon_\lambda, \upsilon^+_\lambda \rangle .
\] We also set
\[
 \mathfrak v_3 = \int_0^\infty q \, \upsilon_\lambda'(q)^2 \dd q .
\] The bounds \eqref{e:int-resolvent-bounds} imply that the integrals $\mathfrak v_1$, $\mathfrak v_2$, and $\mathfrak v_3$ are finite, and that the functions $q^2 (\upsilon_\lambda')^2$, $q \upsilon_\lambda^2$, and $q \upsilon_\lambda \upsilon_\lambda'$ tend to zero as $q \downarrow 0$ and as $q \to \infty$.

We now derive the virial identities, writing $\upsilon = \upsilon_\lambda$ for brevity. First, we multiply \eqref{e:int-resolvent-ode} by $\upsilon'$ and integrate over $(0, \infty)$. Integrating by parts, and using $\upsilon(0+) = \lambda^{-1}$ from \zcref{l:int-resolvent}, $\upsilon(\infty) = 0$, and the vanishing boundary terms, we have
\[
\begin{gathered}
 \int_0^\infty q^2 \upsilon' \upsilon'' \dd q = -\mathfrak v_3 , \qquad
 -\int_0^\infty q \upsilon \upsilon' \dd q = \frac{\mathfrak v_1}{2} , \\
 -\lambda \int_0^\infty \upsilon \upsilon' \dd q = \frac{1}{2\lambda} , \qquad
 -\int_0^\infty \upsilon' \dd q = \frac1\lambda ,
\end{gathered}
\]
and therefore
\[
 -\mathfrak v_3 + \frac{\mathfrak v_1}{2} + \frac{1}{2\lambda} = \frac1\lambda .
\]

Second, by \eqref{e:int-resolvent-bounds}, we may write \eqref{e:int-resolvent-ode} in the form
\[
 q \upsilon_\lambda'' - \upsilon_\lambda + \upsilon^+_\lambda = 0 .
\]
We multiply by $\upsilon$ and integrate. Here we keep $\upsilon \upsilon^+_\lambda = \upsilon(1 - \lambda \upsilon)/q$ together, since its two terms are not separately integrable at zero. Integrating by parts, we have
\[
 \int_0^\infty q \upsilon \upsilon'' \dd q = -\int_0^\infty \bigl(\upsilon \upsilon' + q (\upsilon')^2\bigr) \dd q = \frac{1}{2\lambda^2} - \mathfrak v_3 ,
\]
and we obtain $1/(2\lambda^2) - \mathfrak v_3 - \mathfrak v_1 + \mathfrak v_2 = 0$. Together, the two identities are
\begin{equation}\label{e:int-virial}
 \mathfrak v_1 - 2\mathfrak v_3 = \frac1\lambda , \qquad \mathfrak v_2 = \mathfrak v_1 + \mathfrak v_3 - \frac{1}{2\lambda^2} .
\end{equation}
Recall from \eqref{e:int-laplace} that $\mathfrak v_2 = \lambda^{-1}$. Eliminating $\mathfrak v_2$ and $\mathfrak v_3$ from \eqref{e:int-virial}, we obtain
\[
 \int_0^\infty e^{-\lambda X} \mathcal A(X) \dd X = \mathfrak v_1(\lambda) = \frac1\lambda + \frac{1}{3\lambda^2} = \int_0^\infty e^{-\lambda X} \Bigl(1 + \frac X3\Bigr) \dd X , \qquad \lambda > 2 .
\]
The function $\mathcal A$ is continuous on $(0, \infty)$, by dominated convergence in \zcref{d:brownian}. By the uniqueness of the Laplace transform, $\mathcal A(X) = 1 + X/3$ for all $X > 0$.
\end{proof}

\begin{remark}\label{r:int-metallic}
Let $G_{\mathrm N}$ be the Neumann Green kernel of \zcref{p:metallic}. Since $\Tr G_{\mathrm N} = \int_0^1 (x^2 - x + 1/3) \dd x = 1/6$, the identity \eqref{e:int-exact} reads $\int_0^1 \mathcal C_X(x, x) \dd x = 1 + 2X \Tr G_{\mathrm N}$ for all $X > 0$. Compared with the expansion $\mathcal C_X = 1 + 2XG_{\mathrm N} + O(X^2)$ of \zcref{p:metallic}, the expansion of the diagonal integrated over the chain terminates at first order. The first-order term is the physics prediction \cite[(3.55)]{mirlin2000statistics}, and the vanishing of the second-order term for the inverse participation ratio is visible in \cite[(12), (17)]{fyodorov1995mesoscopic}. The limit $2$ of $2 + 2X/3$ as $X \downarrow 0$ is the value for the GUE, and the slope $2/3$ as $X \to \infty$ agrees with $X^{-1}\mathcal C_X(x, x) \to 1/3$, which is proved in \zcref{s:coarse}.
\end{remark}