\chapter{Inputs and the interface theorem}\label{s:interface}

In Part~III, we prove the results of \zcref{s:intro} from the sigma model of \zcref{s:sigma}. We use three kinds of input.

The first input is the sigma model itself. We use the definitions of \zcref{s:sigma}, the weighted realization of the generator in \zcref{p:sigma-regular}, the multiplier bounds of \zcref{l:sigma-multipliers}, and the analyticity of the partition function in the sources in \zcref{l:sigma-entire}. We also use the supersymmetric localization \zcref{l:sigma-localization} and its consequences \zcref{c:sigma-normalization,c:sigma-ward}. Finally, we use the reduction of rank one in \zcref{s:sigma-radial} (\zcref{l:sigma-polar,l:sigma-radial-integral,l:res-singlet,l:sigma-scalar,l:sigma-singlet}), with the rank-one calculus of \zcref{s:int-rank-one} and the two source derivatives at rank one computed in \zcref{l:sigma-jets}.

The second input consists of facts about the band matrix $H$ that do not involve supersymmetry: the lower bounds on its variance profile, the factorial moment bounds for eigenvalue counts, the asymptotics of the expected density, and the scalar Gaussian component of $H$. We prove them in \zcref{s:iface-variance,s:iface-counts,s:iface-density,s:iface-gaussian}.

The third input is the interface theorem, \zcref{t:interface}, which collects the four statements about $H$ that Part~IV proves. We state it in terms of the matrix $H$ and of the objects of \zcref{s:sigma} only, so that Part~III can be read without Part~IV.

The sections of Part~III are organized as follows. \zcref[S]{s:process} identifies the limiting spectral point process and its source law. \zcref[S]{s:extract} shows how moments of eigenvector entries are extracted from local resolvent moments at positive damping. \zcref[S]{s:residue} computes the limits of the sigma-model kernels as the damping tends to zero, in terms of the Brownian functionals of \zcref{d:brownian}. \zcref[S]{s:intensity,s:coarse} prove the results on local intensities, the inverse participation ratio and the coarse eigenvector law. \zcref[S]{s:metallic,s:scales} analyze the function $I_X$ and the spectral fluctuations on all scales.

Throughout this section, $H$ is the Gaussian band matrix of \zcref{d:model}, with variance matrix $J = (\Id - W^2 \Delta)^{-1}$, and we use the scaling of \eqref{e:scaling}.

\section{The variance profile}\label{s:iface-variance}

We first verify the image formula \eqref{e:intro-images} for $J$, and we record two lower bounds on the entries of $J$ that follow from it. Recall from \zcref{s:intro-results} that $\theta_W = 2 \operatorname{arsinh}(1/(2W))$ and $J_{\Z}(m) = (1 + 4W^2)^{-1/2} e^{-\theta_W |m|}$ for $m \in \Z$.

\begin{lemma}
Let\label{l:iface-variance} $N \ge 2$ and $W > 0$. Then the image formula \eqref{e:intro-images} holds for all $i, j \in \{1, \dots, N\}$. In particular, $J_{ij} \ge J_{\Z}(i - j) > 0$ and $J_{ii} \ge (1 + 4W^2)^{-1/2}$. If $W \ge 1$, then
\begin{equation}\label{e:iface-band-lower}
 J_{ij} \ge \frac{c_J}{W} \quad \text{for } |i - j| \le W, \qquad c_J = \frac{e^{-1}}{\sqrt 5}.
\end{equation}
\end{lemma}

\begin{proof}
We first check that $J_{\Z}(i - j)$ is the kernel of $(\Id - W^2 \Delta_{\Z})^{-1}$. By the definition of $\theta_W$ and the identities $\cosh \theta - 1 = 2 \sinh^2(\theta/2)$ and $\sinh \theta = 2 \sinh(\theta/2) \cosh(\theta/2)$, we have
\[
 \cosh \theta_W = 1 + \frac{1}{2W^2}, \qquad \sinh \theta_W = \frac{\sqrt{1 + 4W^2}}{2W^2} .
\]
For $r \neq 0$, we therefore have
\[
 J_{\Z}(r) - W^2 \bigl( J_{\Z}(r + 1) + J_{\Z}(r - 1) - 2 J_{\Z}(r) \bigr)
 = J_{\Z}(r) \bigl( 1 - 2W^2 (\cosh \theta_W - 1) \bigr) = 0 ,
\]
and for $r = 0$ the left side is
\[
\begin{aligned}
 J_{\Z}(0) \bigl( 1 + 2W^2 (1 - e^{-\theta_W}) \bigr) &= J_{\Z}(0) \bigl( 1 + 2W^2 (1 - \cosh \theta_W + \sinh \theta_W) \bigr) \\
 &= J_{\Z}(0) \sqrt{1 + 4W^2} = 1 .
\end{aligned}
\]

Next, let $\widetilde J_{ij}$ denote the right side of \eqref{e:intro-images}, extended to all $i \in \Z$ by the same formula; the series converges absolutely. Since $J_{\Z}$ is even, $\widetilde J_{ij}$ is invariant under the reflections $i \mapsto 1 - i$ and $i \mapsto 2N + 1 - i$, that is,
\[
 \widetilde J_{1 - i, j} = \widetilde J_{ij} = \widetilde J_{2N + 1 - i, j}, \qquad i \in \Z .
\]
In particular, $\widetilde J_{0 j} = \widetilde J_{1 j}$ and $\widetilde J_{N + 1, j} = \widetilde J_{N j}$,
so for $1 \le i \le N$ the Neumann Laplacian $\Delta$ acts on $\widetilde J_{\cdot j}$ as the Laplacian $\Delta_{\Z}$ of $\Z$. By the first step, we have, for $i \in \Z$,
\[
\begin{aligned}
 (\Id - W^2 \Delta_{\Z}) \, J_{\Z}(\cdot - j + 2mN)(i) &= \1\{ i = j - 2mN \}, \\
 (\Id - W^2 \Delta_{\Z}) \, J_{\Z}(\cdot + j - 1 + 2mN)(i) &= \1\{ i = 1 - j - 2mN \} .
\end{aligned}
\]
Only one of these points lies in $\{1, \dots, N\}$, namely $i = j$, from the first family with $m = 0$. We conclude that
\[
 \bigl( (\Id - W^2 \Delta) \widetilde J \bigr)_{ij} = \delta_{ij}, \qquad i, j \in \{1, \dots, N\} ,
\]
which means that $\widetilde J = J$.

Keeping only the term $J_{\Z}(i - j)$ of \eqref{e:intro-images}, whose other terms are positive, we find
\[
 J_{ij} \ge J_{\Z}(i - j), \qquad J_{ii} \ge J_{\Z}(0) = (1 + 4W^2)^{-1/2} .
\]
Now let $W \ge 1$ and $|i - j| \le W$. Using $\operatorname{arsinh} y \le y$ for $y \ge 0$, we have $\theta_W \le 1/W$, and the first bound yields
\[
 J_{ij} \ge J_{\Z}(i - j) \ge \frac{e^{-\theta_W W}}{\sqrt{1 + 4W^2}} \ge \frac{e^{-1}}{\sqrt 5 \, W} = \frac{c_J}{W} ,
\]
where we used $1 + 4W^2 \le 5W^2$ in the last inequality.
\end{proof}

\section{Factorial moments of eigenvalue counts}\label{s:iface-counts}

For an interval $I \subset \R$, let $\mathcal N_H(I)$ denote the number of eigenvalues of $H$ in $I$, counted with multiplicity. For $a \in \R$ and an integer $m \ge 1$, let $(a)_m = a (a - 1) \cdots (a - m + 1)$ denote the falling factorial.

The following bound is due to Peled, Schenker, Shamis, and Sodin \cite{peled2019wegner}, who proved it for deformed block-Gaussian matrices. For $m = 1$ it is a Wegner estimate, and for $m \ge 2$ an estimate of Minami type. The bound depends on the blocks only through their total size, and this is what makes it uniform in $W$. We reduce our ensemble to theirs by splitting off a block-diagonal Gaussian matrix, as in their proof of \cite[Corollary~1.6]{peled2019wegner}. The only difference is that our variance profile is not translation invariant, and we use \zcref{l:iface-variance} in place of their assumption on the shape function.

\begin{lemma}\label{l:iface-counts}
There exists an absolute constant $C_{\mathrm{cnt}} > 0$ such that for all $N \ge W \ge 4$, all intervals $I \subset \R$ and all integers $m \ge 1$,
\begin{equation}\label{e:iface-counts}
 \E \bigl( \mathcal N_H(I) \bigr)_m \le \bigl( C_{\mathrm{cnt}} N |I| \bigr)^m .
\end{equation}
In particular, for all $E \in (-2, 2)$, all bounded intervals $A \subset \R$, and all $m \ge 1$, the unfolded process $\Xi_N$ of \zcref{s:intro-results} satisfies
\begin{equation}\label{e:iface-counts-unfolded}
 \E \bigl( \Xi_N(A) \bigr)_m \le \Bigl( \frac{C_{\mathrm{cnt}} |A|}{\rho(E)} \Bigr)^m .
\end{equation}
\end{lemma}

\begin{proof}
We first recall the result of \cite{peled2019wegner} in the form we use. In the normalization of \cite[Section~1 and Remark~1.5]{peled2019wegner}, a GUE matrix of size $n$ is a Hermitian matrix $V$ with density proportional to $\exp(-\frac n2 \Tr V^2)$. Its entries on and above the diagonal are independent, the diagonal entries are real centered Gaussians of variance $1/n$, and the off-diagonal entries have independent centered real and imaginary parts of variance $1/(2n)$ each. Let $V_1, \dots, V_\ell$ be independent GUE matrices of sizes $n_1, \dots, n_\ell$. Let $A$ be a deterministic Hermitian matrix of size $n_1 + \dots + n_\ell$. The unitary case of \cite[Theorem~3]{peled2019wegner} states that there exists an absolute constant $C_1$ such that for all intervals $I$ and all $m \ge 1$,
\begin{equation}\label{e:iface-psss}
 \E \bigl( \mathcal N_{A + V_1 \oplus \dots \oplus V_\ell}(I) \bigr)_m \le \Bigl( C_1 |I| \sum_{j = 1}^\ell n_j \Bigr)^m .
\end{equation}

We now write $H$ in this form. Let $\ell = \lceil N/\lfloor W \rfloor \rceil$, and partition $\{1, \dots, N\}$ into $\ell$ consecutive discrete intervals whose lengths $n_1, \dots, n_\ell$ satisfy $n_j \in \{ \lfloor N/\ell \rfloor, \lceil N/\ell \rceil \}$. We claim that
\begin{equation}\label{e:iface-blocks}
 \frac W4 \le n_j \le W, \qquad 1 \le j \le \ell .
\end{equation}
The upper bound follows from $N/\ell \le \lfloor W \rfloor \le W$. For the lower bound, the inequalities $\ell < N/\lfloor W \rfloor + 1$ and $N \ge \lfloor W \rfloor$ imply that
\[
 \frac N\ell > \frac{N \lfloor W \rfloor}{N + \lfloor W \rfloor} \ge \frac{\lfloor W \rfloor}{2}, \qquad \text{so that} \qquad n_j \ge \Bigl\lfloor \frac N\ell \Bigr\rfloor \ge \frac{\lfloor W \rfloor - 1}{2} \ge \frac{W - 2}{2} \ge \frac W4 ,
\]
where we used $W \ge 4$ in the last step. Two sites $i, i'$ in the $j$th interval satisfy
\[
 |i - i'| \le n_j - 1 < W ,
\]
and using \eqref{e:iface-band-lower} and \eqref{e:iface-blocks}, we obtain
\begin{equation}\label{e:iface-block-variance}
 J_{i i'} \ge \frac{c_J}{W} \ge \frac{\sigma^2}{n_j}, \qquad \sigma^2 = \frac{c_J}{4} .
\end{equation}

Let $V_1, \dots, V_\ell$ be independent GUE matrices of sizes $n_1, \dots, n_\ell$. We let $H_0$ be an independent Hermitian matrix with independent centered Gaussian entries on and above the diagonal, with real diagonal entries and with independent real and imaginary parts of equal variance off the diagonal, such that
\[
 \E |(H_0)_{i i'}|^2 = J_{i i'} - \frac{\sigma^2}{n_j} \1 \{ i, i' \text{ lie in the } j\text{th interval} \} .
\]
These variances are nonnegative by \eqref{e:iface-block-variance}. The matrix $H_0 + \sigma (V_1 \oplus \dots \oplus V_\ell)$ has independent centered Gaussian entries on and above the diagonal, with the same variances and the same real structure as $H$. It therefore has the law of $H$.

Conditionally on $H_0$, the matrix $\sigma^{-1} H_0 + V_1 \oplus \dots \oplus V_\ell$ has the form of \eqref{e:iface-psss}, with
\[
 A = \sigma^{-1} H_0, \qquad \sum_{j = 1}^\ell n_j = N .
\]
We apply \eqref{e:iface-psss} to it with the interval $\sigma^{-1} I$. Since $\mathcal N_H(I) = \mathcal N_{\sigma^{-1} H}(\sigma^{-1} I)$, we find
\[
 \E \bigl[ \bigl( \mathcal N_H(I) \bigr)_m \bigm| H_0 \bigr] \le \Bigl( \frac{C_1 N |I|}{\sigma} \Bigr)^m ,
\]
and taking the expectation over $H_0$ proves \eqref{e:iface-counts} with $C_{\mathrm{cnt}} = C_1/\sigma$. Finally, we have
\[
 \Xi_N(A) = \mathcal N_H \bigl( E + a_N^{-1} A \bigr), \qquad \bigl| E + a_N^{-1} A \bigr| = \frac{|A|}{N \rho(E)} ,
\]
and \eqref{e:iface-counts-unfolded} follows from \eqref{e:iface-counts}.
\end{proof}

\section{The expected density}\label{s:iface-density}

The case $m = 1$ of \eqref{e:iface-counts} shows that the expected eigenvalue distribution $N^{-1} \E \sum_\alpha \delta_{\lambda_\alpha}$ is absolutely continuous, with density bounded by $C_{\mathrm{cnt}}$. We let $\rho_{N, W}$ denote this density. We will use the following theorem of Disertori, Lohmann, and Sodin \cite{disertori2021density}, which identifies $\rho_{N, W}$ with the semicircle density up to an error $O(N^{-1} + W^{-2})$ for our ensemble. Their ensemble \cite[(1.4)]{disertori2021density} has the variance matrix $(-W^2 \Delta_N + 1_N)^{-1}$, where $\Delta_N$ has diagonal entries $-1$ at the two endpoints and $-2$ elsewhere, and off-diagonal entries $1$ between neighbors. We have $\Delta_N = \Delta$, so that this variance matrix is our $J$, and their ensemble is ours.\footnote{Disertori, Lohmann, and Sodin write the covariance in \cite[(1.2) and (1.4)]{disertori2021density} as $\E H_{x, y} H_{x', y'} = \delta_{x, x'} \delta_{y, y'} J_{xy}$. Read literally, this is not the covariance of a complex Hermitian matrix. We read it as $\E H_{x, y} \overline{H_{x', y'}} = \delta_{x, x'} \delta_{y, y'} J_{xy}$, which is \eqref{e:intro-covariance}. Their supersymmetric representation \cite[Proposition~2.1]{disertori2021density}, with one complex boson and one pair of Grassmann variables per site, confirms this reading, since it is the representation for complex Hermitian matrices.} Their finite-volume density of states is our $\rho_{N, W}$, and their $\rho_{\mathrm{s.c.}}$ is our $\rho$.

\begin{lemma}\label{l:iface-density}
For every compact interval $I \subset (-2, 2)$, there exists $C_I > 0$ such that for all $W \ge 2$ and $N \ge C_I W \log W$,
\begin{equation}\label{e:iface-dls}
 \sup_{\lambda \in I} |\rho_{N, W}(\lambda) - \rho(\lambda)| \le C_I \bigl( N^{-1} + W^{-2} \bigr) .
\end{equation}
Consequently, let $N, W \to \infty$ with $N/W^2$ bounded below, and let $E_0 \in (0, 2)$ and $R > 0$. Then, uniformly for $|E| \le E_0$, the density $\varrho_N(x) = \rho_{N, W}(E + x/a_N)/\rho(E)$ of $\E \Xi_N$ satisfies
\begin{equation}\label{e:iface-intensity}
 \sup_{|x| \le R} |\varrho_N(x) - 1| \to 0 ,
\end{equation}
and for every deterministic sequence $\bar r_N \downarrow 0$,
\begin{equation}\label{e:iface-window}
 \sup_{0 < r \le \bar r_N} \Bigl| \frac{A_N(E, r)}{N \rho(E)} - 1 \Bigr| \to 0, \qquad A_N(E, r) = \E \sum_\alpha \phi_{E, r}(\lambda_\alpha) ,
\end{equation}
where $A_N(E, r)$ is the denominator of the unrooted selection in \zcref{d:selection}.
\end{lemma}

\begin{proof}
The bound \eqref{e:iface-dls} combines the two estimates of \cite[Theorem~1]{disertori2021density},
\[
 |\rho_{N, W}(\lambda) - \rho_\infty(\lambda)| \le \frac{C(\lambda)}{N}, \qquad |\rho_\infty(\lambda) - \rho(\lambda)| \le \frac{C(\lambda)}{W^2} ,
\]
where $\rho_\infty$ is their infinite-volume density of states. These estimates hold for $|\lambda| < 2$ and $N \ge C(\lambda) W \log W$, and \cite[Remark~1.3]{disertori2021density} states that the constants $C(\lambda)$ are bounded on compact subintervals of $(-2, 2)$.

We now prove the consequences, and we set
\[
 I = \Bigl[ -\frac{2 + E_0}{2}, \frac{2 + E_0}{2} \Bigr] .
\]
The condition $N \ge C_I W \log W$ holds for all sufficiently large $W$, since $N/W^2$ is bounded below. Further, the bound $a_N \ge N \rho(E_0)$ implies that $E + x/a_N \in I$ for $|x| \le R$ and all sufficiently large $N$, and for such $x$ and $N$ we have
\[
\begin{aligned}
 |\varrho_N(x) - 1| &\le \frac{|\rho_{N, W}(E + x/a_N) - \rho(E + x/a_N)| + |\rho(E + x/a_N) - \rho(E)|}{\rho(E_0)} \\
 &\le C \Bigl( \frac1N + \frac1{W^2} + \frac RN \Bigr) ,
\end{aligned}
\]
where we used \eqref{e:iface-dls} and the fact that $\rho$ is Lipschitz on $I$, and \eqref{e:iface-intensity} follows. Next, the change of variables $\lambda = E + r v$ gives
\[
 \frac{A_N(E, r)}{N} = \int \phi_{E, r}(\lambda) \rho_{N, W}(\lambda) \dd \lambda = \int_{-1}^1 \phi(v) \rho_{N, W}(E + r v) \dd v .
\]
For $r \le \bar r_N$ and $|v| \le 1$, the argument above shows that
\[
 |\rho_{N, W}(E + r v) - \rho(E)| \le C \bigl( N^{-1} + W^{-2} + \bar r_N \bigr) .
\]
Using $\int \phi = 1$ and $\rho(E) \ge \rho(E_0) > 0$, we obtain
\[
 \Bigl| \frac{A_N(E, r)}{N \rho(E)} - 1 \Bigr| \le \frac{1}{\rho(E_0)} \int_{-1}^1 \phi(v) \, \bigl| \rho_{N, W}(E + r v) - \rho(E) \bigr| \dd v \le C \bigl( N^{-1} + W^{-2} + \bar r_N \bigr) ,
\]
which proves \eqref{e:iface-window}.
\end{proof}

\section{The scalar Gaussian component}\label{s:iface-gaussian}

The matrix $H$ contains an independent Gaussian multiple of the identity, whose unfolded standard deviation is of order $W^{1/2}$. Translating it moves all unfolded eigenvalues by the same amount and leaves the eigenvectors unchanged. We use the resulting approximate translation invariance for stationarity in \zcref{s:process}, and also in \zcref{s:extract}.

\begin{lemma}\label{l:iface-gaussian}
We set
\begin{equation}\label{e:iface-scalar}
 \sigma_N^2 = \Bigl( \sum_{i = 1}^N J_{ii}^{-1} \Bigr)^{-1}, \qquad
 \gamma_N = \sigma_N^2 \sum_{i = 1}^N \frac{H_{ii}}{J_{ii}}, \qquad
 \widetilde H = H - \gamma_N \Id .
\end{equation}
\begin{enumerate}[label=(\alph*)]
\item The variable $\gamma_N$ is a real centered Gaussian of variance $\sigma_N^2$, it is independent of $\widetilde H$, and $\sigma_N^{-2} \le N (1 + 4W^2)^{1/2}$.
\item For every bounded measurable function $F$ on $N \times N$ Hermitian matrices, all $E \in (-2, 2)$ and all $b \in \R$,
\begin{equation}\label{e:iface-shift}
 \bigl| \E F(H + b a_N^{-1} \Id) - \E F(H) \bigr| \le \sqrt{2/\pi} \, \|F\|_\infty \frac{|b|}{a_N \sigma_N} .
\end{equation}
\item For every nonnegative measurable function $\Phi$ on $N \times N$ Hermitian matrices and all $b \in \R$ and $K > 0$, with $\Phi_b(H) = \Phi(H + b a_N^{-1} \Id)$,
\begin{equation}\label{e:iface-shift-truncated}
 \bigl| \E \Phi_b(H) - \E \Phi(H) \bigr| \le \sqrt{2/\pi} \, \frac{K |b|}{a_N \sigma_N} + \frac{\E \Phi(H)^2 + \E \Phi_b(H)^2}{K} .
\end{equation}
\end{enumerate}
If $N, W \to \infty$ with $N/W^2$ bounded below and $|E| \le E_0 < 2$, then $(a_N \sigma_N)^{-1} \le C W^{-1/2}$, where $C$ depends only on $E_0$ and on the lower bound for $N/W^2$.
\end{lemma}

\begin{proof}
The diagonal entries $H_{ii}$ are independent real centered Gaussians of variance $J_{ii}$, and they are independent of the off-diagonal entries. As a linear combination of them, $\gamma_N$ is a real centered Gaussian. By \eqref{e:iface-scalar}, we have
\[
 \Var(\gamma_N) = \sigma_N^4 \sum_{i = 1}^N \frac{J_{ii}}{J_{ii}^2} = \sigma_N^2, \qquad
 \E [\gamma_N H_{ii}] = \sigma_N^2 \frac{J_{ii}}{J_{ii}} = \sigma_N^2 .
\]
In particular, the diagonal entries $\widetilde H_{ii} = H_{ii} - \gamma_N$ satisfy
\[
 \E[\gamma_N \widetilde H_{ii}] = \E[\gamma_N H_{ii}] - \Var(\gamma_N) = 0, \qquad 1 \le i \le N ,
\]
and $\gamma_N$ is uncorrelated with the off-diagonal entries of $\widetilde H$, which are those of $H$. Since the real coordinates of $(\gamma_N, \widetilde H)$ are jointly Gaussian, $\gamma_N$ and $\widetilde H$ are independent. Further, the bound $J_{ii} \ge (1 + 4W^2)^{-1/2}$ of \zcref{l:iface-variance} implies that
\[
 \sigma_N^{-2} = \sum_{i = 1}^N J_{ii}^{-1} \le N (1 + 4W^2)^{1/2} .
\]
This proves part (a).

We now prove part (b). Set $\beta = b/a_N$, and for $\sigma > 0$ let $\mathsf n_\sigma$ denote the centered Gaussian density on $\R$ of variance $\sigma^2$. Writing $H + \beta \Id = \widetilde H + (\gamma_N + \beta) \Id$ and using the independence of $\gamma_N$ and $\widetilde H$ from part (a), we have
\[
\begin{aligned}
 \bigl| \E F(H + \beta \Id) - \E F(H) \bigr|
 &= \Bigl| \E \int_\R F(\widetilde H + y \Id) \bigl( \mathsf n_{\sigma_N}(y - \beta) - \mathsf n_{\sigma_N}(y) \bigr) \dd y \Bigr| \\
 &\le \|F\|_\infty \, |\beta| \, \|\mathsf n_{\sigma_N}'\|_{L^1} ,
\end{aligned}
\]
and part (b) follows from
\[
 \|\mathsf n_{\sigma_N}'\|_{L^1} = 2 \mathsf n_{\sigma_N}(0) = \sqrt{2/\pi} \, \sigma_N^{-1} .
\]

For part (c), let $F = \min(\Phi, K)$, so that $\|F\|_\infty \le K$ and $0 \le \Phi - F = (\Phi - K)_+ \le \Phi^2/K$. The same holds for $\Phi_b$ and $F_b(H) = F(H + b a_N^{-1} \Id) = \min(\Phi_b(H), K)$. Using part (b) for $F$ and these bounds, we obtain
\[
\begin{aligned}
 \bigl| \E \Phi_b(H) - \E \Phi(H) \bigr| &\le \bigl| \E F_b(H) - \E F(H) \bigr| + \E \bigl( \Phi_b(H) - K \bigr)_+ + \E \bigl( \Phi(H) - K \bigr)_+ \\
 &\le \sqrt{2/\pi} \, \frac{K |b|}{a_N \sigma_N} + \frac{\E \Phi(H)^2 + \E \Phi_b(H)^2}{K} ,
\end{aligned}
\]
which is \eqref{e:iface-shift-truncated}.

Finally, the bound on $\sigma_N^{-2}$ in part (a) implies that
\[
 (a_N \sigma_N)^{-2} \le \frac{(1 + 4W^2)^{1/2}}{N \rho(E)^2} \le \frac{C}{W} ,
\]
where we used $a_N = N \rho(E)$, $\rho(E) \ge \rho(E_0)$, and $N \ge c W^2$ in the last step.
\end{proof}

\section{The interface theorem}\label{s:iface-theorem}

We now state the interface theorem. Its four parts concern the source ratios $T_N^{(k)}$ of \eqref{e:intro-ratio}, their derivatives with respect to local sources at finitely many sites, the diagonal resolvent moments at all sites, and the one-point function. We begin with the local sources.

Throughout this section, $k \ge 1$ and $\eps > 0$ are fixed, $\mathbf u = (u_1, \dots, u_k) \in \R^k$ satisfies $\sum_j u_j = 0$, and $\mathbf b = (b_1, \dots, b_k) \in (0, \infty)^k$. Recall from \zcref{d:rep-sources} the source supermatrix $\Gamma = \Gamma(\mathbf u, \mathbf b) = \diag(\eps \Lambda_{\mathrm B} \mid \Gamma_{\mathrm F})$, with
\[
 \Gamma_{\mathrm F} = \diag \bigl( b_1 - \iu u_1, \dots, b_k - \iu u_k, -b_1 - \iu u_1, \dots, -b_k - \iu u_k \bigr) ,
\]
and $\eta_0 = a_N^{-1}$, where the $2k$ flavors are ordered as $k$ retarded flavors followed by $k$ advanced flavors. Then
\begin{equation}\label{e:iface-Z}
\begin{gathered}
 E + \iu \eta_0 \Gamma = \diag(Z_{\mathrm B} \mid Z_{\mathrm F}) , \\
 Z_{\mathrm B} = \diag \bigl( z^{\mathrm B}_+ \Id_k, z^{\mathrm B}_- \Id_k \bigr), \quad
 Z_{\mathrm F} = \diag \bigl( z^{\mathrm F}_{1, +}, \dots, z^{\mathrm F}_{k, +}, z^{\mathrm F}_{1, -}, \dots, z^{\mathrm F}_{k, -} \bigr) ,
\end{gathered}
\end{equation}
where $z^{\mathrm B}_\pm = E \pm \iu \eps/a_N$ and $z^{\mathrm F}_{j, \pm} = E + (u_j \pm \iu b_j)/a_N$. Let $e_1, \dots, e_N$ denote the standard basis of $\C^N$.

\begin{definition}\label{d:iface-local}
Let $m \ge 0$, let $1 \le i_1 < \dots < i_m \le N$ be distinct sites, and for $1 \le a \le m$ let $Y_a = \diag(Y_a^{\mathrm B} \mid Y_a^{\mathrm F})$ be an even supermatrix with complex $2k \times 2k$ blocks $Y_a^{\mathrm B}$ and $Y_a^{\mathrm F}$. We write $\mathbf Y = (Y_1, \dots, Y_m)$, and we define the \emph{local source function}
\begin{equation}\label{e:iface-local}
 T_N^{(k)}(\mathbf u, \mathbf b; \mathbf Y)
 = \E \frac{\det \bigl( \Id_N \otimes Z_{\mathrm F} - H \otimes \Id_{2k} + \sum_{a = 1}^m e_{i_a} e_{i_a}^* \otimes Y_a^{\mathrm F} \bigr)}
 {\det \bigl( \Id_N \otimes Z_{\mathrm B} - H \otimes \Id_{2k} + \sum_{a = 1}^m e_{i_a} e_{i_a}^* \otimes Y_a^{\mathrm B} \bigr)} .
\end{equation}
\end{definition}

In \eqref{e:iface-local}, the local source $Y_a$ is added to $E + \iu \eta_0 \Gamma$ at the site $i_a$. In the notation of \zcref{s:rep}, we have
\[
 T_N^{(k)}(\mathbf u, \mathbf b; \mathbf Y) = \E \Sdet \Bigl( \Id_N \otimes (E + \iu \eta_0 \Gamma) - H \otimes \Id + \sum_{a = 1}^m e_{i_a} e_{i_a}^* \otimes Y_a \Bigr)^{-1} ,
\]
and the right side is $T_N^{(k)}(\Gamma; \mathbf Y)$ in \zcref{l:rep-dictionary}. Since the eigenvalues $z^{\mathrm B}_\pm - \lambda_\alpha$ of $\Id_N \otimes Z_{\mathrm B} - H \otimes \Id_{2k}$ have modulus at least $\eps/a_N$, the denominator in \eqref{e:iface-local} is bounded away from zero, uniformly in $H$, when $\max_a \|Y_a^{\mathrm B}\| \le \eps/(2 a_N)$. The numerator is a polynomial in the entries of $H$. The local source function is therefore finite and holomorphic in $\mathbf Y$ near zero, and we may differentiate it under the expectation. Only its derivatives at $\mathbf Y = 0$ are used.

The determinants at $\mathbf Y = 0$ factor over the eigenvalues of $H$. Using
\[
 (\lambda - z^{\mathrm F}_{j, +})(\lambda - z^{\mathrm F}_{j, -}) = \frac{(x - u_j)^2 + b_j^2}{a_N^2}, \qquad (\lambda - z^{\mathrm B}_+)(\lambda - z^{\mathrm B}_-) = \frac{x^2 + \eps^2}{a_N^2}
\]
for $x = a_N (\lambda - E)$, we find $T_N^{(k)}(\mathbf u, \mathbf b; 0) = T_N^{(k)}(\mathbf u, \mathbf b)$. The derivatives at $\mathbf Y = 0$ are expectations of the product in \eqref{e:intro-ratio} times polynomials in the resolvent entries $G_{i_a i_b}(z)$ with $z \in \{z^{\mathrm B}_\pm, z^{\mathrm F}_{j, \pm}\}$. For instance, let $k = 1$, $\mathbf u = 0$, and $\mathbf b = \eps$, so that $Z_{\mathrm F} = Z_{\mathrm B}$. Since
\[
 \det(z - H + t e_i e_i^*) = \det(z - H) \bigl( 1 - t G_{ii}(z) \bigr) ,
\]
a fermionic source $Y^{\mathrm F} = \diag(t, 0)$ at the site $i$, with $Y^{\mathrm B} = 0$, gives the first identity in
\begin{equation}\label{e:iface-examples}
 -\partial_t T_N^{(1)}(0, \eps; Y) \big|_{t = 0} = \E G_{ii}(z^{\mathrm B}_+), \qquad
 \partial_t \partial_{t'} T_N^{(1)}(0, \eps; Y_1, Y_2) \big|_{t = t' = 0} = \E G_{ii}(z^{\mathrm B}_+) G_{jj}(z^{\mathrm B}_-) ,
\end{equation}
and the fermionic sources $Y_1^{\mathrm F} = \diag(t, 0)$ at the site $i$ and $Y_2^{\mathrm F} = \diag(0, t')$ at a site $j > i$, with $Y_1^{\mathrm B} = Y_2^{\mathrm B} = 0$, give the second.

The limits of these derivatives are described by the sigma model of \zcref{s:sigma}. Recall the generator $\mathsf G_{X, \Gamma} = X \mathsf C_k + \pi \Str(Q \Gamma)$ of \eqref{e:sigma-generator}, realized as in \zcref{p:sigma-regular}, and the partition function
\[
 \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) = \mathcal B_k \bigl( e^{-s \mathsf G_{X, \Gamma}} 1, e^{-(1 - s) \mathsf G_{X, \Gamma}} 1 \bigr) = \mathcal B_k \bigl( 1, e^{-\mathsf G_{X, \Gamma}} 1 \bigr), \qquad 0 < s < 1 ,
\]
of \zcref{d:sigma-partition}, where $\Gamma = \Gamma(\mathbf u, \mathbf b)$; the second equality and the independence of $s$ are \zcref{p:sigma-regular}(d). In the sigma model, the local resolvent of the matrix at a site is replaced by the even supermatrix
\begin{equation}\label{e:iface-orbit-resolvent}
 \mathcal G(Q) = -\kappa_E \Id + \iu s_E Q, \qquad \kappa_E = E/2 .
\end{equation}
Its retarded and advanced diagonal entries at $Q = \Lambda$ are the limits $-\kappa_E \pm \iu s_E$ of $G_{ii}(E \pm \iu 0)$ for the semicircle law. Recall from \zcref{s:rep-heuristics} that the points of the balanced component of the saddle manifold are the supermatrices $S = \kappa_E - \iu s_E Q$, where $Q$ is the field on $\mathcal M_k$, and that $(E - S)^{-1} = \kappa_E - \iu s_E Q = -\mathcal G(Q)$ at these points, by \eqref{e:rep-saddle-inverse}. For an even supermatrix $Y$ as in \zcref{d:iface-local}, we define the \emph{orbit multiplier} $\mathfrak j_Y(Q) = \Sdet(\Id - \mathcal G(Q) Y)^{-1}$, as a formal power series in the entries of $Y$. By \eqref{e:rep-insertion}, a local source $Y$ multiplies the onsite weight by $\Sdet(\Id + (E - S)^{-1} Y)^{-1}$, and we have $\Sdet(\Id + (E - S)^{-1} Y)^{-1} = \mathfrak j_Y(Q)$ at the point $S = \kappa_E - \iu s_E Q$. For a multi-index $\alpha$ in the entries of $Y$, we let
\begin{equation}\label{e:iface-multiplier}
 \mathfrak j^{(\alpha)}(Q) = \partial_Y^\alpha \Sdet \bigl( \Id - \mathcal G(Q) Y \bigr)^{-1} \big|_{Y = 0}
\end{equation}
denote its Taylor coefficients. Each $\mathfrak j^{(\alpha)}$ is a polynomial in the entries of $Q$, hence a multiplier of finite degree on the weighted spaces $L^2_p$ of \zcref{s:sigma-weighted}, by \zcref{l:sigma-multipliers}.

\begin{theorem}\label{t:interface}
There exist energies $0 < E_* < E_{\mathrm{out}} \le 1/2$ such that the following holds for all $k \ge 1$ and $\eps > 0$. In parts (a) and (b), $N, W \to \infty$ with $|E| \le E_{\mathrm{out}}$ and $X_N(E) \to X \in (0, \infty)$, where the energy $E$ may depend on $N$.
\begin{enumerate}[label=(\alph*)]
\item \emph{Source limit.} There exists $r_{k, \eps} > 0$ such that
\begin{equation}\label{e:iface-source-limit}
 T_N^{(k)}(\mathbf u, \mathbf b) \to \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)
\end{equation}
uniformly on compact subsets of the set of real $(\mathbf u, \mathbf b)$ with $\sum_j u_j = 0$ and $\|\mathbf u\| + \|\mathbf b - \eps \1\| < r_{k, \eps}$.
\item \emph{Local jets.} Let $m \ge 1$ and $\delta \in (0, 1/2)$, let $1 \le i_1 < \dots < i_m \le N$ be sites with $\delta \le i_a/N \le 1 - \delta$ and $i_a/N \to x_a$ for all $a$, and let $\alpha = (\alpha_1, \dots, \alpha_m)$, where $\alpha_a$ is a multi-index in the entries of $Y_a$. Then, uniformly on compact subsets of the set in part (a),
\begin{equation}\label{e:iface-jets}
 \partial_{\mathbf Y}^\alpha T_N^{(k)}(\mathbf u, \mathbf b; \mathbf Y) \big|_{\mathbf Y = 0}
 \to \mathcal B_k \bigl( 1, \, e^{-x_1 \mathsf G_{X, \Gamma}} \mathfrak j^{(\alpha_1)} e^{-(x_2 - x_1) \mathsf G_{X, \Gamma}} \mathfrak j^{(\alpha_2)} \cdots \mathfrak j^{(\alpha_m)} e^{-(1 - x_m) \mathsf G_{X, \Gamma}} 1 \bigr) ,
\end{equation}
where $\Gamma = \Gamma(\mathbf u, \mathbf b)$ and $\mathfrak j^{(\alpha_a)}$ acts by multiplication. No lower bound on $i_{a + 1} - i_a$ is required. In particular, distinct sites whose scaled positions have the same limit $x_a = x_{a + 1}$ give two separate multipliers, with the factor $e^{-0 \cdot \mathsf G_{X, \Gamma}} = \Id$ between them, while all derivatives at one site enter through the single multiplier $\mathfrak j^{(\alpha_a)}$.
\item \emph{All-site moments.} For all integers $p \ge 1$, all $u_0 > 0$, and all $0 < X_- \le X_+ < \infty$, there exist $C$ and $N_0$ such that
\begin{equation}\label{e:iface-allsite}
 \sup_{1 \le i \le N} \, \sup_{|u| \le u_0} \E \bigl| G_{ii} \bigl( E + (u + \iu \eps)/a_N \bigr) \bigr|^{2p} \le C
\end{equation}
for all $N \ge N_0$ and all $(W, E)$ with $|E| \le E_*$ and $X_- \le X_N(E) \le X_+$. For $u = 0$, the same holds with $E_{\mathrm{out}}$ in place of $E_*$.
\item \emph{One-point normalization.} For all $\delta \in (0, 1/2)$ and $u_0 > 0$,
\begin{equation}\label{e:iface-onepoint}
 \E G_{ii} \bigl( E + (u + \iu \eps)/a_N \bigr) - \bigl( -\kappa_E + \iu s_E \bigr) \to 0, \qquad
 \E \ell^\eps_{i, N}(u) \to 1 ,
\end{equation}
uniformly in the critical regime of \zcref{d:intro-critical}, and uniformly for $\delta \le i/N \le 1 - \delta$ and $|u| \le u_0$.
\end{enumerate}
The energies $E_*$ and $E_{\mathrm{out}}$ do not depend on $k$, on the number $m$ of sites, or on the orders of the derivatives in part (b). These affect only the constants, the radius $r_{k, \eps}$, and the rate of convergence.
\end{theorem}

The proof of \zcref{t:interface} occupies Part~IV and is completed in \zcref{s:assembly}; its strategy is described in \zcref{s:overview}.

\begin{remark}\label{r:iface-energies}
We call $E_{\mathrm{out}}$ the \emph{outer energy bound} and $E_*$ the \emph{inner energy bound}. The outer bound is the minimum of $1/2$ and two energy thresholds up to which the estimates of Part~IV on the chain hold, and it is fixed in \eqref{e:asm-Eout}. Parts (a) and (b) hold for $|E| \le E_{\mathrm{out}}$, and so does part (c) at $u = 0$. In parts (c) and (d), the resolvent is evaluated at
\[
 E + \frac{u + \iu \eps}{a_N(E)} = E' + \frac{\iu \eps'}{a_N(E')}, \qquad E' = E + \frac{u}{a_N(E)}, \quad \eps' = \eps \, \frac{\rho(E')}{\rho(E)} ,
\]
and Part~IV deduces these parts from their case $u = 0$ at the energy $E'$, using \zcref{l:asm-recenter}. For $|E| \le E_*$ and $|u| \le u_0$, we have $|E'| \le E_{\mathrm{out}}$ for all sufficiently large $N$, and the margin $E_{\mathrm{out}} - E_*$ absorbs this change of energy. The inner bound is fixed in \zcref{s:asm-choices}, subject only to $0 < E_* < E_{\mathrm{out}}$. Their values are not explicit, since the thresholds of Part~IV depend on constants that are not computed.

Throughout Part~III, we use the inputs of \zcref{t:interface} for $|E| \le E_{\mathrm{out}}$, and we state conclusions for $|E| \le E_*$, as in \zcref{s:intro-results}. We also set $E_{\mathrm{mid}} = (E_* + E_{\mathrm{out}})/2$, so that
\begin{equation}\label{e:iface-energies}
 0 < E_* < E_{\mathrm{mid}} < E_{\mathrm{out}} \le \frac12 .
\end{equation}
Intermediate statements hold for $|E| \le E_{\mathrm{mid}}$, and the margin $E_{\mathrm{out}} - E_{\mathrm{mid}}$ is used to control the eigenvalues far from $E$.
\end{remark}

\begin{remark}\label{r:iface-uses}
Parts (a) and (b) are statements along sequences with $X_N(E) \to X$, while parts (c) and (d) are uniform over $|E| \le E_*$ (in (c) with $u = 0$, over $|E| \le E_{\mathrm{out}}$) and over $X_N(E)$ in compact subsets of $(0, \infty)$. Every sequence with $|E| \le E_{\mathrm{out}}$ and $N/W^2$ in a compact subset of $(0, \infty)$ has a subsequence along which $X_N(E)$ converges. By the argument of \zcref{r:extract-uniform}, the uniform form of (c) and (d) is therefore equivalent to the same statements along all such sequences, and Part~III uses them in either form, for instance in \zcref{l:extract-moments}.

Note that part~(a) identifies $T_N^{(k)}$ only near the base point $(\mathbf u, \mathbf b) = (0, \eps \1)$, at which both sides equal one. We extend the convergence to all real and complex fermionic source parameters, together with all derivatives, in \zcref{c:proc-complex-source}, using only part (a), \zcref{l:iface-counts}, and the analyticity of the partition function in \zcref{l:proc-entire}. Parts~(b)--(d) are used in \zcref{s:extract,s:intensity,s:coarse}. The first limit in \eqref{e:iface-onepoint} at $u = 0$ is a consequence of part (b) with $m = 1$, the first identity in \eqref{e:iface-examples}, and the localization \zcref{l:sigma-localization}; we list it separately because it is used in this form. The second limit in \eqref{e:iface-onepoint} follows from the first, since
\[
 \ell^\eps_{i, N}(u) = s_E^{-1} \Im G_{ii} \bigl( E + (u + \iu \eps)/a_N \bigr)
\]
by \zcref{s:intro-notation}.
\end{remark}

\begin{remark}\label{r:iface-onsite}
The derivatives of the local source function at one site are polynomials in the resolvent entries at that site, and the multiplier $\mathfrak j^{(\alpha)}$ is a Taylor coefficient of the complete local factor $\Sdet(\Id - \mathcal G(Q) Y)^{-1}$, including the off-diagonal entries of $\mathcal G(Q)$. For this reason, one site and two adjacent sites give different limits. For instance, let $k = 1$, $\mathbf u = 0$, and $\mathbf b = \eps$, and let $Y^{\mathrm B} = \diag(t, t')$ and $Y^{\mathrm F} = 0$ at the site $i$. Then
\[
\begin{gathered}
 \partial_t \partial_{t'} T_N^{(1)}(0, \eps; Y) \big|_{t = t' = 0} = \E G_{ii}(z^{\mathrm B}_+) G_{ii}(z^{\mathrm B}_-) , \\
 \partial_t \partial_{t'} \det \bigl( \Id - \mathcal G^{\mathrm{BB}} Y^{\mathrm B} \bigr)^{-1} \big|_{t = t' = 0} = \mathcal G^{\mathrm{BB}}_{\mathrm{RR}} \mathcal G^{\mathrm{BB}}_{\mathrm{AA}} + \mathcal G^{\mathrm{BB}}_{\mathrm{RA}} \mathcal G^{\mathrm{BB}}_{\mathrm{AR}} ,
\end{gathered}
\]
where $\mathcal G^{\mathrm{BB}}_{\mathrm{RR}}, \mathcal G^{\mathrm{BB}}_{\mathrm{RA}}, \mathcal G^{\mathrm{BB}}_{\mathrm{AR}}, \mathcal G^{\mathrm{BB}}_{\mathrm{AA}}$ are the entries of the boson-boson block $\mathcal G^{\mathrm{BB}}$ of $\mathcal G(Q)$, in the notation of \zcref{s:sigma-supermatrices}. When the two sources sit at two distinct sites instead, the limit contains the product $\mathcal G^{\mathrm{BB}}_{\mathrm{RR}} \mathcal G^{\mathrm{BB}}_{\mathrm{AA}}$ of two separate multipliers, with no term $\mathcal G^{\mathrm{BB}}_{\mathrm{RA}} \mathcal G^{\mathrm{BB}}_{\mathrm{AR}}$.
\end{remark}

\paragraph{The energies of Part III.} Recall the energies of \eqref{e:iface-energies} and their roles (\zcref{r:iface-energies}).