\chapter{The product contour}\label{s:contour}

In this appendix, we prove the contour statements used in \zcref{s:rep}. These are the Gaussian integral \eqref{e:rep-hs-flat} on the flat cycle, the deformation of \zcref{p:rep-contour} onto the translated product of Sch\"afer--Wegner cycles, the facts about the heat kernel in \zcref{l:rep-heat}, and the deformation used in the proof of \zcref{p:rep-anchored}. In \zcref{s:contour-kick}, we also bound the source factor of the onsite weight on the saddle manifold, which Part IV uses. As explained after \zcref{p:rep-contour}, the earlier results on the Sch\"afer--Wegner domain concern a single matrix variable, and the decay in the noncompact directions comes there from the coupling to the vector variables \cite{zirnbauer1996riemannian} or from the condition $A s > 0$ on the source \cite{fyodorov2008hyperbolic,mullerhill2011equivalence}. Here we need $N$ matrix variables coupled by a nearest-neighbor Laplacian, with a source of rank one at each site, and the decay must come from the damping, which translates the cycle. The new ingredients are the identity \eqref{e:contour-edge-identity} for the link terms, which shows that the map $X \mapsto f_0(X)$ is a contraction, and the bound of \zcref{l:contour-deform}, which is uniform along a deformation that ends on the translated cycle. All statements concern integrals of fixed finite dimension, and the constants may depend on all the parameters. The only delicate point is the integrability along the deformations, since on the untranslated cycle the Gaussian weight does not decay in the noncompact directions; the rest of the argument is elementary.

We use the notation of \zcref{s:rep}, in particular $f_0(X)$ and $\mathsf g_X$ of \eqref{e:rep-f0} and their properties in \zcref{l:rep-cycle}. We let $\mathcal S^{\mathrm B}$ denote the set of matrices $P - \iu \Lambda_{\mathrm B}(f_0(X) + X)$ with $P \in \mathrm{Herm}_{\rm d}(2k)$ and $X \in \mathrm{Herm}_{\rm o}(2k)$, which is the set of BB blocks of the cycle $\mathcal S$.

\section{A Stokes lemma for real cycles}\label{s:contour-stokes}

The deformations of this appendix move the BB and FF blocks of the integration variables in complex space and leave the odd variables unchanged. Each coefficient of the integrand, in the odd variables, is then an ordinary integral of a holomorphic function over a real cycle of half dimension. We will use the following form of Cauchy's theorem.

\begin{lemma}\label{l:contour-stokes}
Let $d \ge 1$, let $F$ be an entire function on $\C^d$, and let $(t, x) \mapsto \mathbf z_t(x)$ be a continuously differentiable map from $[0, 1] \times \R^d$ to $\C^d$. Let $D \mathbf z_t(x)$ denote the complex $d \times d$ matrix of partial derivatives of $\mathbf z_t$ in $x$, and set $j_t(x) = \det D \mathbf z_t(x)$. Suppose that there exists $F^\sharp \in L^1(\R^d)$ such that, for all $t \in [0, 1]$ and $x \in \R^d$,
\begin{equation}\label{e:contour-stokes-majorant}
|F(\mathbf z_t(x))| \big( |j_t(x)| + |\partial_t \mathbf z_t(x)| \, \|D \mathbf z_t(x)\|_{\rm op}^{d - 1} \big) \le F^\sharp(x) .
\end{equation}
Then $\int_{\R^d} F(\mathbf z_t(x)) j_t(x) \dd x$ does not depend on $t \in [0, 1]$.
\end{lemma}

\begin{proof}
The holomorphic $d$-form $F(z) \dd z_1 \wedge \dots \wedge \dd z_d$ on $\C^d$ is closed, and so is its pullback by $(t, x) \mapsto \mathbf z_t(x)$ to $[0, 1] \times \R^d$. We apply Stokes' theorem to this pullback on $[0, 1] \times \{|x| \le r\}$, where $r > 0$, and we obtain
\[
\int_{|x| \le r} F(\mathbf z_1) j_1 \dd x - \int_{|x| \le r} F(\mathbf z_0) j_0 \dd x = \pm \int_{[0, 1] \times \{|x| = r\}} \mathbf z^* \big( F \dd z_1 \wedge \dots \wedge \dd z_d \big) .
\]
Restricted to $[0, 1] \times \{|x| = r\}$, the pulled-back form is the product of $F(\mathbf z_t)$ and a sum of $d \times d$ minors of the derivative of $(t, x) \mapsto \mathbf z_t(x)$ that contain the column $\partial_t \mathbf z_t$. The density of this form with respect to $\dd t$ and the surface measure is therefore bounded by
\[
C_d |F(\mathbf z_t)| \, |\partial_t \mathbf z_t| \, \|D \mathbf z_t\|_{\rm op}^{d - 1} \le C_d F^\sharp .
\]
Let $\sigma_r$ denote the surface measure on the sphere $\{|x| = r\}$. Since
\[
\int_0^\infty \int_{\{|x| = r\}} F^\sharp \dd \sigma_r \dd r = \int_{\R^d} F^\sharp \dd x < \infty ,
\]
there exist $r_j \to \infty$ such that the integral of $F^\sharp$ over the sphere $\{|x| = r_j\}$ tends to zero. Letting $r = r_j \to \infty$ and using dominated convergence for the two integrals on the left, we find the claim.
\end{proof}

We apply \zcref{l:contour-stokes} to the BB and FF blocks of $N$ supermatrices, with $d = 8k^2 N$. We use the real coordinates of \zcref{d:rep-flat} on $\R^d$. On $\C^d$, we use at each site the complex coordinates introduced after \zcref{d:rep-cycle}, in which a BB block takes the form $P^{\C} - \iu \Lambda_{\mathrm B} X^{\C}$, with $P^{\C}$ complex block diagonal and $X^{\C}$ complex block off-diagonal, and an FF block the form $\iu M^{\C}$. These complex coordinates restrict to the real coordinates on the flat cycle $\mathcal S_\flat$, so that $j_t = 1$ there. A superfunction whose coefficients are entire functions of the BB and FF entries is integrated over a deformed cycle coefficientwise, and the integral of \zcref{d:rep-cycle} is the case in which $\mathbf z_t$ parametrizes $\mathcal S$.

\section{Gaussian integrals on the flat cycle}\label{s:contour-flat}

We first prove \eqref{e:rep-hs-flat}. This is the Gaussian Fourier transform on flat superspace, and we include the short proof because \zcref{p:rep-exact} and \zcref{p:rep-anchored} use the fact that its constant equals one. For families $R = (R_i)_{i \in [N]}$ and $B = (B_i)_{i \in [N]}$ of even $(2k|2k)$ supermatrices and a real symmetric $N \times N$ matrix $\mathbf M$, we set
\[
\langle R, \mathbf M B \rangle = \sum_{i, j = 1}^N \mathbf M_{ij} \Str(R_i B_j) .
\]
This form is symmetric, since $\Str(R_i B_j) = \Str(B_j R_i)$ for even supermatrices by \zcref{l:grass-sdet}\ref{i:grass-sdet-trace}.

\begin{lemma}\label{l:contour-flat}
Let $\mathbf M$ be a positive definite real symmetric $N \times N$ matrix, and let $B_1, \dots, B_N$ be even supermatrices whose entries lie in auxiliary generators, with complex bodies. Then
\[
\int_{\mathcal S_\flat^N} e^{-\langle R, \mathbf M R \rangle/2 - \iu \langle R, B \rangle} \dd R = e^{-\langle B, \mathbf M^{-1} B \rangle/2} .
\]
The same identity holds with $\mathcal S_\flat^N$ replaced by its translate by any family $(C_i)_{i \in [N]}$ of supermatrices with complex entries and vanishing odd blocks.
\end{lemma}

\begin{proof}
We first take $B = 0$. For $\mathbf M = \Id$, the identity is the $N$-fold product of \eqref{e:rep-flat-normalization}.

For general $\mathbf M$, we substitute
\[
R = (\mathbf M^{-1/2} \otimes \Id) R' ,
\]
which is a real linear change of the even coordinates and a linear change of the odd coordinates, given in each case by the matrix $\mathbf M^{-1/2}$ acting on the site index. This substitution turns $\langle R, \mathbf M R \rangle$ into $\langle R', R' \rangle$. Since each site carries $8k^2$ even and $8k^2$ odd coordinates, the Lebesgue integral over the even coordinates gains the factor $\det(\mathbf M)^{-4k^2}$, and by \zcref{l:grass-berezin}\ref{i:grass-berezin-linear} the Berezin integral over the odd coordinates gains the factor $\det(\mathbf M)^{4k^2}$. The two factors cancel, as in the direct computation after \eqref{e:rep-hs-flat}.

For general $B$, we complete the square,
\[
-\frac12 \langle R, \mathbf M R \rangle - \iu \langle R, B \rangle = -\frac12 \big\langle R + \iu \mathbf M^{-1} B, \mathbf M (R + \iu \mathbf M^{-1} B) \big\rangle - \frac12 \langle B, \mathbf M^{-1} B \rangle .
\]
The shift $R \mapsto R + \iu \mathbf M^{-1} B$ adds odd elements to the odd entries, even nilpotent elements to the even entries, and complex numbers to the even entries. Recall that on $\mathcal S_\flat$,
\[
\langle R, \mathbf M R \rangle_\emptyset = \sum_{i, j} \mathbf M_{ij} \big( \Tr(R_i^{\mathrm{BB}} R_j^{\mathrm{BB}}) + \Tr(M_i M_j) \big) ,
\]
so that the coefficients of the Gaussian are polynomials times $e^{-\langle R, \mathbf M R \rangle_\emptyset/2}$, and all their derivatives are integrable. By \zcref{l:grass-berezin}\ref{i:grass-berezin-odd-shift} and \ref{i:grass-berezin-even-shift}, the first two shifts do not change the integral.

For the shift by a family $c$ of complex constants, we apply \zcref{l:contour-stokes} with $\mathbf z_t(x) = x + t c$, coefficientwise. Here $j_t = 1$ and $|\partial_t \mathbf z_t| = |c|$. In the notation $| \cdot |_\emptyset$ of the proof of \zcref{p:rep-exact} for the modulus of the body, we have
\[
\Big| e^{-\langle R, \mathbf M R \rangle/2} \Big|_\emptyset \le e^{-\lambda_{\min}(\mathbf M) |x|^2/2 + C |x| + C} \qquad \text{at } R = x + t c ,
\]
uniformly in $t \in [0, 1]$. The other coefficients are bounded by polynomials times the right side, and \eqref{e:contour-stokes-majorant} holds. Together with the case $B = 0$, this implies the identity. The last assertion follows in the same way from the shift by $C$.
\end{proof}

\section{Real parts along the deformation}\label{s:contour-real}

We now compute the real parts of the Gaussian terms along the deformations. For $c \ge 0$, $t \in [0, 1]$, $P \in \mathrm{Herm}_{\rm d}(2k)$, and $X \in \mathrm{Herm}_{\rm o}(2k)$, we set
\begin{equation}\label{e:contour-homotopy}
R_t(P, X) = P - \iu \Lambda_{\mathrm B} \big( (1 - t)(f_0(X) - c) + X \big) .
\end{equation}
Thus $R_1(P, X) = P - \iu \Lambda_{\mathrm B} X$ is Hermitian, and $R_0(P, X)$ is the point of $\mathcal S^{\mathrm B} + \iu c \Lambda_{\mathrm B}$ with coordinates $(P, X)$. With $c = \eta$, the matrices $R_t(P, X)$ are the BB blocks of the cycle $\mathcal S_{\Gamma, t}$ of \zcref{s:rep-cycle}.

\begin{lemma}\label{l:contour-real}
Let $c \ge 0$ and $\eta' \ge 0$, let $\zeta \in \C^{2k}$, and let $t \in [0, 1]$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:contour-real-vertex} For $m \ge 0$ and $R = R_t(P, X)$,
\begin{equation}\label{e:contour-vertex}
\begin{aligned}
& \Re \Big[ -\frac{m}{2} \Tr R^2 - \iu \Tr\big( R (\zeta \zeta^* + \eta') \Lambda_{\mathrm B} \big) \Big] \\
& \qquad = -\frac{m}{2} \Big( \|P\|^2 + \|X\|^2 - (1 - t)^2 \|f_0(X) - c\|^2 \Big) \\
& \qquad\quad - \zeta^* \big( (1 - t)(f_0(X) - c) + X \big) \zeta - (1 - t) \eta' \big( \Tr f_0(X) - 2kc \big) .
\end{aligned}
\end{equation}
If $m > 0$ and $\bar\eta = m c + \eta' > 0$, the right side is at most
\begin{equation}\label{e:contour-vertex-bound}
-\frac{m}{2} \|P\|^2 - \frac{\bar\eta}{4} \Tr f_0(X) + m k (s_E^2 + c^2) + 2k \eta' c + (s_E + c) |\zeta|^2 + \frac{|\zeta|^4}{m} + \frac{k \bar\eta s_E}{2} + \frac{k \bar\eta^2}{6m} .
\end{equation}
\item\label{i:contour-real-edge} For $w \ge 0$, $R = R_t(P, X)$, and $R' = R_t(P', X')$,
\begin{equation}\label{e:contour-edge}
\Re \Big[ -\frac{w}{2} \Tr(R - R')^2 \Big] = -\frac{w}{2} \Big( \|P - P'\|^2 + \|X - X'\|^2 - (1 - t)^2 \|f_0(X) - f_0(X')\|^2 \Big) ,
\end{equation}
and
\begin{equation}\label{e:contour-edge-identity}
\|X - X'\|^2 - \|f_0(X) - f_0(X')\|^2 = s_E^2 \Tr\big( \mathsf g_X^2 \mathsf g_{X'}^{-2} + \mathsf g_X^{-2} \mathsf g_{X'}^2 - 2 \Id \big) \ge 0 .
\end{equation}
In particular, the right side of \eqref{e:contour-edge} is at most $-\frac{w}{2} \|P - P'\|^2$, and $\|f_0(X) - f_0(X')\| \le \|X - X'\|$.
\item\label{i:contour-real-pinned} Let $c = 0$, $w \ge 0$, and $R_* = P_* - \iu \Lambda_{\mathrm B}(f_0(X_*) + X_*) \in \mathcal S^{\mathrm B}$. Then for $R = R_t(P, X)$,
\[
\Re \Big[ -\frac{w}{2} \Tr(R - R_*)^2 \Big] \le -\frac{w}{2} \|P - P_*\|^2 - \frac{w s_E}{2 \|\mathsf g_{X_*}^2\|_{\rm op}} \Tr f_0(X) + 2 w k s_E^2 + \frac{w}{2} \|f_0(X_*)\|^2 .
\]
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \eqref{e:contour-vertex}. We set
\[
Z = (1 - t)(f_0(X) - c) + X ,
\]
which is Hermitian, so that $R = P - \iu \Lambda_{\mathrm B} Z$. As in the proof of \zcref{l:rep-cycle}\ref{i:rep-cycle-gauss}, the trace $\Tr(P \Lambda_{\mathrm B} Z)$ is real, and
\[
\Lambda_{\mathrm B} Z \Lambda_{\mathrm B} = (1 - t)(f_0(X) - c) - X ,
\]
since $f_0(X)$ is block diagonal and $X$ block off-diagonal. We obtain
\[
\Re \Tr R^2 = \|P\|^2 - \Tr(\Lambda_{\mathrm B} Z \Lambda_{\mathrm B} Z) = \|P\|^2 + \|X\|^2 - (1 - t)^2 \|f_0(X) - c\|^2 .
\]
Moreover, $\Tr(P (\zeta \zeta^* + \eta') \Lambda_{\mathrm B})$ is real, since $\Lambda_{\mathrm B} P$ and $\zeta \zeta^* + \eta'$ are Hermitian, and
\[
\Tr\big( \Lambda_{\mathrm B} Z (\zeta \zeta^* + \eta') \Lambda_{\mathrm B} \big) = \zeta^* Z \zeta + \eta' \Tr Z, \qquad \Tr Z = (1 - t)(\Tr f_0(X) - 2kc) .
\]
Combining the last two displays yields \eqref{e:contour-vertex}.

We next prove \eqref{e:contour-vertex-bound}, and we begin with the Gaussian term. Since $f_0(X)^2 = s_E^2 + X^2$, we have
\[
\|f_0(X) - c\|^2 = 2k s_E^2 + \|X\|^2 - 2c \Tr f_0(X) + 2k c^2 ,
\]
and the Gaussian term in \eqref{e:contour-vertex} becomes
\[
-\frac{m}{2} \|P\|^2 - \frac{m}{2} t (2 - t) \|X\|^2 + (1 - t)^2 m k (s_E^2 + c^2) - (1 - t)^2 m c \Tr f_0(X) .
\]

We now bound the source term. We write
\[
Z = \big( f_0(X) + X \big) - t f_0(X) - (1 - t) c .
\]
Using $f_0(X) + X > 0$, $\|f_0(X)\|_{\rm op} \le s_E + \|X\|$, and Young's inequality, we find
\[
-\zeta^* Z \zeta \le t (s_E + \|X\|) |\zeta|^2 + c |\zeta|^2 \le (s_E + c) |\zeta|^2 + \frac{m t}{4} \|X\|^2 + \frac{|\zeta|^4}{m} .
\]

We now combine the two bounds. Since $t \le t(2 - t)$, the sum of the terms with $\|X\|^2$ and $\Tr f_0(X)$ is bounded by
\[
-\frac{m}{4} t (2 - t) \|X\|^2 - \big( (1 - t)^2 m c + (1 - t) \eta' \big) \Tr f_0(X) .
\]
When $t \le 1/2$, the second term is bounded by $-\frac{\bar\eta}{4} \Tr f_0(X)$. When $t \ge 1/2$, we have $t(2 - t) \ge 3/4$. Using $\Tr f_0(X) \le 2k s_E + \sqrt{2k} \|X\|$ and maximizing over $\|X\|$, we find
\[
\frac{\bar\eta}{4} \Tr f_0(X) - \frac{3m}{16} \|X\|^2 \le \frac{k \bar\eta s_E}{2} + \frac{k \bar\eta^2}{6m} .
\]
In each case, the last display but one is bounded by
\[
-\frac{\bar\eta}{4} \Tr f_0(X) + \frac{k \bar\eta s_E}{2} + \frac{k \bar\eta^2}{6m} .
\]
Collecting the terms gives \eqref{e:contour-vertex-bound}.

We now prove \ref{i:contour-real-edge}. The constant $c$ cancels in $R - R'$. The computation of \ref{i:contour-real-vertex} with $m = w$, $\zeta = 0$, and $\eta' = 0$, and with $f_0(X) - f_0(X')$ and $X - X'$ in place of $f_0(X) - c$ and $X$, gives \eqref{e:contour-edge}.

For \eqref{e:contour-edge-identity}, we use $s_E \mathsf g_X^2 = f_0(X) + X$ and $s_E \mathsf g_X^{-2} = f_0(X) - X$ to compute
\[
s_E^2 \Tr\big( \mathsf g_X^2 \mathsf g_{X'}^{-2} + \mathsf g_X^{-2} \mathsf g_{X'}^2 \big) = 2 \Tr\big( f_0(X) f_0(X') \big) - 2 \Tr(X X') .
\]
On the other hand, since $\|f_0(X)\|^2 = 2k s_E^2 + \|X\|^2$, we have
\[
\|X - X'\|^2 - \|f_0(X) - f_0(X')\|^2 = 2 \Tr\big( f_0(X) f_0(X') \big) - 2 \Tr(X X') - 4k s_E^2 ,
\]
and the identity follows. To see that its right side is nonnegative, note that for positive definite matrices $A$ and $A'$,
\[
\Tr\big( A A'^{-1} + A^{-1} A' - 2 \Id \big) = \sum_\lambda \big( \lambda + \lambda^{-1} - 2 \big) \ge 0 ,
\]
where $\lambda$ runs over the eigenvalues of $A'^{-1/2} A A'^{-1/2}$. Finally, the right side of \eqref{e:contour-edge} is nonincreasing in $t$, and \eqref{e:contour-edge-identity} bounds its value at $t = 0$ by $-\frac{w}{2} \|P - P'\|^2$.

Finally, we prove \ref{i:contour-real-pinned}. As in the computation of \ref{i:contour-real-edge}, we have
\[
\Re \Big[ -\frac{w}{2} \Tr(R - R_*)^2 \Big] = -\frac{w}{2} \Big( \|P - P_*\|^2 + \|X - X_*\|^2 - \|(1 - t) f_0(X) - f_0(X_*)\|^2 \Big) .
\]
Expanding the last norm, we see that this exceeds its value at $t = 0$ by
\[
\begin{aligned}
\frac{w}{2} \Big( -t(2 - t) \|f_0(X)\|^2 + 2t \Tr\big( f_0(X) f_0(X_*) \big) \Big) &\le \frac{w}{2} \sup_{y \ge 0} \big( -t y^2 + 2t y \|f_0(X_*)\| \big) \\
&= \frac{w}{2} t \|f_0(X_*)\|^2 .
\end{aligned}
\]
The identity \eqref{e:contour-edge-identity} shows that the value at $t = 0$ is
\begin{multline*}
-\frac{w}{2} \|P - P_*\|^2 - \frac{w s_E^2}{2} \Tr\big( \mathsf g_X^2 \mathsf g_{X_*}^{-2} + \mathsf g_X^{-2} \mathsf g_{X_*}^2 - 2 \Id \big) \\
\le -\frac{w}{2} \|P - P_*\|^2 - \frac{w s_E^2}{2} \Tr\big( \mathsf g_X^2 \mathsf g_{X_*}^{-2} \big) + 2 w k s_E^2 ,
\end{multline*}
and
\[
\Tr\big( \mathsf g_X^2 \mathsf g_{X_*}^{-2} \big) \ge \|\mathsf g_{X_*}^2\|_{\rm op}^{-1} \Tr \mathsf g_X^2 = \big( s_E \|\mathsf g_{X_*}^2\|_{\rm op} \big)^{-1} \Tr f_0(X) ,
\]
where we used $\Tr X = 0$.
\end{proof}

The main point of this appendix is the identity \eqref{e:contour-vertex} at $t = 0$, which explains the role of the translation discussed after \zcref{l:rep-cycle}. With $c = \eta' = 0$, its right side does not decay in $X$ at $t = 0$. On the untranslated cycle $\mathcal S^{\mathrm B}$, the Gaussian is flat in the noncompact directions, and the source $\zeta \zeta^*$ of rank one produces decay in only some of them. Either a translation $c > 0$ or a linear damping $\eta' > 0$ restores the decay, through the term $-\frac{\bar\eta}{4} \Tr f_0(X)$ in \eqref{e:contour-vertex-bound}. In the open chain, the damping enters as the translation $c = \eta$, and in the anchored chain as the linear term with $\eta' = \eta$.

\section{The deformation lemma}\label{s:contour-deform}

We now combine these computations for a finite graph. Let $n \ge 1$ and let $m_1, \dots, m_n \ge 0$ be masses. We fix edge weights $w_{ij} = w_{ji} \ge 0$ for $i \ne j$, and weights $w_{i *}, w_{i **} \ge 0$ of links to two fixed points $R_*, R_{**} \in \mathcal S^{\mathrm B}$. Fix also $c, \eta' \ge 0$ and $\zeta_1, \dots, \zeta_n \in \C^{2k}$. For $R = (R_1, \dots, R_n) \in (\C^{2k \times 2k})^n$, we set
\begin{equation}\label{e:contour-V}
\begin{aligned}
\mathcal E(R) = {} & - \sum_i \frac{m_i}{2} \Tr R_i^2 - \sum_{i < j} \frac{w_{ij}}{2} \Tr(R_i - R_j)^2 \\
& - \sum_i \Big( \frac{w_{i *}}{2} \Tr(R_i - R_*)^2 + \frac{w_{i **}}{2} \Tr(R_i - R_{**})^2 \Big) \\
& - \iu \sum_i \Tr\big( R_i (\zeta_i \zeta_i^* + \eta') \Lambda_{\mathrm B} \big) .
\end{aligned}
\end{equation}

\begin{lemma}\label{l:contour-deform}
Suppose that one of the following holds.
\begin{enumerate}[label=(\roman*)]
\item\label{i:contour-deform-massive} We have $m_i > 0$ for all $i$, $c + \eta' > 0$, and $c = 0$ if some $w_{i *}$ or $w_{i **}$ is positive.
\item\label{i:contour-deform-heat} We have $n = 1$, $m_1 = 0$, $\zeta_1 = 0$, $c = \eta' = 0$, and $w_{1 *} > 0$.
\end{enumerate}
Then there exist $C, c' > 0$, depending on all the data, such that for all $t \in [0, 1]$ and all $(P_i, X_i)_{i \in [n]}$,
\begin{equation}\label{e:contour-deform-bound}
\Re \mathcal E\big( (R_t(P_i, X_i))_{i \in [n]} \big) \le C - c' \sum_{i = 1}^n \big( \|P_i\|^2 + \Tr f_0(X_i) \big) .
\end{equation}
Consequently, for every polynomial $p$ in the entries of $R$, the integral $\int p(R_t) e^{\mathcal E(R_t)} \dd P \dd X$ over $(\mathrm{Herm}_{\rm d}(2k) \times \mathrm{Herm}_{\rm o}(2k))^n$, with $R_t = (R_t(P_i, X_i))_i$, converges absolutely and does not depend on $t \in [0, 1]$.
\end{lemma}

\begin{proof}
We begin with case \ref{i:contour-deform-massive}. We bound the vertex terms with \eqref{e:contour-vertex-bound}, where $\bar\eta_i = m_i c + \eta' > 0$, the edge terms with \zcref{l:contour-real}\ref{i:contour-real-edge}, and the links to $R_*$ and $R_{**}$, if present, with \zcref{l:contour-real}\ref{i:contour-real-pinned}, keeping only the constants. Summing these bounds, we obtain \eqref{e:contour-deform-bound} with $c' = \min_i \min(m_i/2, \bar\eta_i/4)$.

In case \ref{i:contour-deform-heat}, the vertex term vanishes. We use \zcref{l:contour-real}\ref{i:contour-real-pinned} for the link to $R_*$, together with
\[
\|P - P_*\|^2 \ge \frac12 \|P\|^2 - \|P_*\|^2 ,
\]
and we note that the link to $R_{**}$ contributes at most a constant.

For the second assertion, we apply \zcref{l:contour-stokes} with $\mathbf z_t(P, X) = R_t$ and with the entire function $F = p \, e^{\mathcal E}$. In the complex coordinates of \zcref{s:contour-stokes}, the map $\mathbf z_t$ is
\[
(P, X) \mapsto \big( P - \iu (1 - t) \Lambda_{\mathrm B} (f_0(X) - c), \, X \big) ,
\]
at each site. The Jacobian matrix of $\mathbf z_t$ is block triangular with identity blocks on the diagonal, and therefore $j_t = 1$. The bound $\|f_0(X) - f_0(X')\| \le \|X - X'\|$ of \zcref{l:contour-real}\ref{i:contour-real-edge} implies that $\|D \mathbf z_t\|_{\rm op} \le 2$. Moreover, there exist $C$ and $K$ such that for all $t \in [0, 1]$,
\[
\begin{gathered}
|\partial_t \mathbf z_t| = \Big( \sum_i \|f_0(X_i) - c\|^2 \Big)^{1/2} \le C \Big( 1 + \sum_i \|X_i\| \Big) , \\
|p(R_t)| \le C \Big( 1 + \sum_i \big( \|P_i\| + \|X_i\| \big) \Big)^K .
\end{gathered}
\]
Using \eqref{e:contour-deform-bound} and $\Tr f_0(X) \ge \|X\|$, we conclude that \eqref{e:contour-stokes-majorant} holds with an integrable $F^\sharp$.
\end{proof}

\section{Proof of \texorpdfstring{\zcref{p:rep-contour}}{the product-contour identity}}\label{s:contour-proof}

Recall the cycles $\mathcal S_{\Gamma, t}$ of \zcref{s:rep-cycle} and the integrand
\[
F_\psi(R) = \exp\Big( -\frac12 \langle R, J^{-1} R \rangle - \iu \sum_{i = 1}^N \Str\big( R_i \psi_i \psi_i^* \Lambda \big) \Big) ,
\]
and fix $\zeta_1, \dots, \zeta_N \in \C^{2k}$. We expand $p F_\psi$ in the odd entries of the $R_i$ and the odd coordinates $\chi_i$, $\bar \chi_i$. By \eqref{e:rep-str-vector}, we have $\Str(R_i \psi_i \psi_i^* \Lambda) = \psi_i^* \Lambda R_i \psi_i$, whose body $\zeta_i^* \Lambda_{\mathrm B} R_i^{\mathrm{BB}} \zeta_i$ involves only the BB block, and the odd entries appear in the exponent only through nilpotent terms. Each coefficient of $p F_\psi$ is therefore a polynomial in the BB and FF entries of the $R_i$, times
\[
e^{\mathcal E(R^{\mathrm{BB}})} \exp\Big( \frac12 \sum_{i, j} (J^{-1})_{ij} \Tr\big( R_i^{\mathrm{FF}} R_j^{\mathrm{FF}} \big) \Big) .
\]
Here $\mathcal E$ is \eqref{e:contour-V} with $n = N$, $m_i = 1$, $w_{i, i + 1} = W^2$, all other weights zero, and $\eta' = 0$, with the given $\zeta_i$. We used \eqref{e:rep-precision} and the identity $\zeta_i^* \Lambda_{\mathrm B} R_i^{\mathrm{BB}} \zeta_i = \Tr(R_i^{\mathrm{BB}} \zeta_i \zeta_i^* \Lambda_{\mathrm B})$, and the plus sign in the FF factor comes from the minus sign of $\Str$ on the FF block.

The BB blocks of $\mathcal S_{\Gamma, t}$ are the matrices $R_t(P, X)$ of \eqref{e:contour-homotopy} with $c = \eta$, so case \ref{i:contour-deform-massive} of \zcref{l:contour-deform} applies, with $\bar\eta_i = \eta$. Summing the bound \eqref{e:contour-vertex-bound} over the sites and the bound of \zcref{l:contour-real}\ref{i:contour-real-edge} over the links yields \eqref{e:rep-homotopy}, with any constant $C_{k, \eta}$ such that
\[
C_{k, \eta} \big( 1 + |\zeta_i|^4 \big) \ge k(s_E^2 + \eta^2) + (s_E + \eta) |\zeta_i|^2 + |\zeta_i|^4 + \frac{k \eta s_E}{2} + \frac{k \eta^2}{6} .
\]
Since $s_E \le 1$, we may choose $C_{k, \eta}$ depending only on $k$ and $\eta$. This completes the proof of \zcref{p:rep-contour}\ref{i:rep-contour-bound}.

Recall that the FF blocks of $\mathcal S_{\Gamma, t}$ have the form $\iu M_i + (1 - t) c_{\mathrm F}$, where $c_{\mathrm F} = \kappa_E \Id + \iu \eta_0 \Gamma_{\mathrm F}$. Since $J^{-1} \ge \Id$, the real part of the exponent of the FF factor is at most $-\frac12 \sum_i \|M_i\|^2 + C \sum_i \|M_i\| + C$ on $\mathcal S_{\Gamma, t}$, uniformly in $t$. We now apply \zcref{l:contour-stokes} to each coefficient, with $d = 8k^2 N$ and the joint deformation $\mathbf z_t$ of the BB and FF blocks that parametrizes $\mathcal S_{\Gamma, t}^N$. The Jacobian of $\mathbf z_t$ is one, as in the proof of \zcref{l:contour-deform}, since the FF blocks are translated by a constant. The derivative $\partial_t \mathbf z_t$ and the polynomial factors grow at most polynomially. By \eqref{e:contour-deform-bound} and the Gaussian bound on the FF factor, \eqref{e:contour-stokes-majorant} holds with an integrable $F^\sharp$. We conclude that each coefficient of $\int_{\mathcal S_{\Gamma, t}^N} p F_\psi \dd R$ converges absolutely and is independent of $t$, which is \zcref{p:rep-contour}\ref{i:rep-contour-deform}.
\hfill\qed

\section{The heat kernel on the cycle}\label{s:contour-heat}

We now prove \zcref{l:rep-heat}. Let $t, t' > 0$ and $S, S'' \in \mathcal S$. To avoid a clash with the heat time, we write $\tau \in [0, 1]$ for the deformation parameter in \eqref{e:contour-homotopy}, with $c = 0$.

The symmetry $k_t(S, S') = k_t(S', S)$ is clear. Let $(P, X, M)$ and $(P', X', M')$ be the coordinates of $S$ and $S'$. Since $S - S'$ has the FF block $\iu (M - M')$, the body of $\Str(S - S')^2$ has real part
\[
\|P - P'\|^2 + \|X - X'\|^2 - \|f_0(X) - f_0(X')\|^2 + \|M - M'\|^2 \ge \|P - P'\|^2 + \|M - M'\|^2 ,
\]
where we used \eqref{e:contour-edge} with $\tau = 0$ and \eqref{e:contour-edge-identity}. This implies the bound on the body of $k_t(S, S')$.

For the first identity, consider the coefficients of $k_t(S, S')$ in the odd entries of $S$ and $S'$ as functions of $S'^{\mathrm{BB}}$ and $S'^{\mathrm{FF}}$. They are polynomials times $e^{\mathcal E(S'^{\mathrm{BB}})} e^{\Tr(S'^{\mathrm{FF}} - S^{\mathrm{FF}})^2/(4t)}$, where $\mathcal E$ is \eqref{e:contour-V} with $n = 1$, $m_1 = 0$, $\zeta_1 = 0$, $\eta' = 0$, $w_{1 *} = (2t)^{-1}$, $w_{1 **} = 0$, and $R_* = S^{\mathrm{BB}}$. The FF factor is a Gaussian in $M'$, since $\Tr(\iu (M' - M))^2 = -\|M' - M\|^2$. Case \ref{i:contour-deform-heat} of \zcref{l:contour-deform} shows that $k_t(S, \cdot)$ is integrable on $\mathcal S$. Together with \zcref{l:contour-stokes}, applied as in \zcref{s:contour-proof}, it also shows that the integral does not change if the BB block of $S'$ is deformed from $\mathcal S^{\mathrm B}$ to the Hermitian matrices. The resulting cycle is the translate of $\mathcal S_\flat$ by $\diag(0 \mid \kappa_E \Id)$, and on it we write
\[
-\frac{1}{4t} \Str(S' - S)^2 = -\frac{1}{4t} \Str S'^2 - \iu \Str\Big( S' \cdot \frac{\iu S}{2t} \Big) - \frac{1}{4t} \Str S^2 .
\]
We apply \zcref{l:contour-flat} with $N = 1$, $\mathbf M = (2t)^{-1}$, and $B = \iu S/(2t)$, and we obtain
\[
\int_{\mathcal S} k_t(S, S') \dd S' = e^{\Str(S^2)/(4t)} e^{-\Str(S^2)/(4t)} = 1 .
\]

For the second identity, we argue in the same way with $w_{1 *} = (2t)^{-1}$, $w_{1 **} = (2t')^{-1}$, $R_* = S^{\mathrm{BB}}$, and $R_{**} = S''^{\mathrm{BB}}$. We then apply \zcref{l:contour-flat} on the flat cycle, with $\mathbf M = (t + t')/(2 t t')$ and $B = \iu (S/(2t) + S''/(2t'))$, and we find
\[
\int_{\mathcal S} k_t(S, S') k_{t'}(S', S'') \dd S' = \exp\Big( \frac{t t'}{t + t'} \Str\Big( \frac{S}{2t} + \frac{S''}{2t'} \Big)^2 - \frac{\Str S^2}{4t} - \frac{\Str S''^2}{4t'} \Big) .
\]
Expanding the square and using $\Str(S S'') = \Str(S'' S)$, we find
\[
\frac{t t'}{t + t'} \Str\Big( \frac{S}{2t} + \frac{S''}{2t'} \Big)^2 - \frac{\Str S^2}{4t} - \frac{\Str S''^2}{4t'} = -\frac{\Str(S - S'')^2}{4(t + t')} .
\]
This completes the proof of \zcref{l:rep-heat}.
\hfill\qed

\section{The deformation in the anchored chain}\label{s:contour-anchored}

In the proof of \zcref{p:rep-anchored}, we deform the variables $S_1, \dots, S_n$ of the anchored chain from $\mathcal S$ to the cycle with Hermitian BB blocks, for fixed $\psi$. We now explain why \zcref{l:contour-deform} applies. The BB part of the exponent of $\ell_n(S; \psi)$ is \eqref{e:contour-V} with unit masses, $w_{i, i + 1} = W^2$, the link $w_{1 *} = W^2$ to $R_* = S^{\mathrm{BB}}$, the given $\zeta_i$, and $\eta' = \eta$. Indeed, since the BB block of $\Gamma$ is $\eps \Lambda_{\mathrm B}$, the BB part of $-\iu \eta_0 \Str(S_i \Gamma)$ is the linear term $-\iu \eta \Tr(S_i^{\mathrm{BB}} \Lambda_{\mathrm B})$. The cycle $\mathcal S$ is the case $\tau = 0$ of \eqref{e:contour-homotopy} with $c = 0$, so case \ref{i:contour-deform-massive} applies. The FF blocks stay on $\kappa_E \Id + \iu \mathrm{Herm}(2k)$, where the FF part of the exponent is a Gaussian in the $M_i$, and the odd variables appear polynomially. As in \zcref{s:contour-proof}, each coefficient of the integrand is integrable on the deformed cycles, uniformly in $\tau$, and its integral is independent of $\tau$. At $\tau = 1$, the cycle is the translate of $\mathcal S_\flat^n$ by $\diag(0 \mid \kappa_E \Id)$ at each site, and applying \zcref{l:contour-flat} with $\mathbf M = J_n^{-1}$ and $B_i = \iu b_i$, we obtain the Gaussian integral used there.

For the marked anchored chains, the extra factor $\prod_i p_{\alpha_i}(\psi_i)$ involves only the $\psi_i$, and the same argument applies. The deformation of this section, with the inverse superdeterminants written as $\psi$-integrals and the order of the integrals exchanged, proves \zcref{p:rep-anchored} directly in the direction from the chain to the Gaussian matrix, without an auxiliary regularization. The damping of the onsite weights supplies the decay in the noncompact directions, through the term $-\frac{\bar\eta}{4} \Tr f_0(X)$ of \eqref{e:contour-vertex-bound} with $\bar\eta = \eta' = \eta$.

\section{The source factor on the saddle manifold}\label{s:contour-kick}

We now bound the BB part of the source factor $w_\Gamma/w_0 = e^{-\iu \eta_0 \Str(S \Gamma)}$ of \zcref{s:rep-symmetry}, together with its derivatives, at the points $S$ with $S^{\mathrm{BB}} = \kappa_E - \iu s_E Q$ and $Q \in H_k$. These bounds are used in Part IV, and they are stated in the geometry of \zcref{s:sigma}. Recall the space $H_k$, the weight $v$ of \eqref{e:sigma-weight}, and the Levi-Civita connection $\nabla$ of \zcref{s:sigma}.

\begin{lemma}\label{l:rep-kick}
Let $\Gamma$ be a source supermatrix, and for $Q \in H_k$ set $\mathbf S(Q) = e^{-\iu \eta_0 \Tr((\kappa_E - \iu s_E Q) \eps \Lambda_{\mathrm B})}$, which is the BB part of the source factor at a point $S$ with $S^{\mathrm{BB}} = \kappa_E - \iu s_E Q$. Then
\begin{equation}\label{e:rep-kick}
\mathbf S(Q) = e^{-2 \eta s_E (v(Q) - 1)} \in \big( 0, e^{-2k \eta s_E} \big] .
\end{equation}
In particular $0 < \mathbf S \le 1$. Moreover, for $m \in \{1, 2\}$ there exists $C$, depending only on $k$ and $s_E$, such that $|\nabla^m \mathbf S| \le C \eta v e^{-s_E \eta v}$ on $H_k$ whenever $\eta \le 1$. For a point $S \in \mathcal S$ with coordinates $(\kappa_E \Id, X, M)$, we have $\mathbf S(Q_X) = e^{-\eta \Tr f_0(X)}$, where $Q_X$ is as in \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}.
\end{lemma}

\begin{proof}
Using $\Tr \Lambda_{\mathrm B} = 0$, we compute the exponent,
\[
-\iu \eta \Tr\big( \kappa_E \Lambda_{\mathrm B} - \iu s_E Q \Lambda_{\mathrm B} \big) = -\eta s_E \Tr(\Lambda_{\mathrm B} Q) ,
\]
and the definition \eqref{e:sigma-weight} of $v$ gives \eqref{e:rep-kick}. The bound follows from $\frac12 \Tr(\Lambda_{\mathrm B} Q) \ge k$, which is \zcref{l:sigma-body}\ref{i:sigma-body-trace}.

For the derivatives, we write $\mathbf S = e^{-2 \eta s_E (v - 1)}$, so that
\[
\nabla \mathbf S = -2 \eta s_E (\nabla v) \mathbf S, \qquad \nabla^2 \mathbf S = \big( 4 \eta^2 s_E^2 \, \nabla v \otimes \nabla v - 2 \eta s_E \nabla^2 v \big) \mathbf S .
\]
Recall from \zcref{l:sigma-body}\ref{i:sigma-body-derivatives} that
\[
|\nabla v| + |\nabla^2 v| \le C v .
\]
For the term with $\eta^2 v^2$, we use
\[
\eta^2 v^2 e^{-2 \eta s_E v} \le \big( \eta v e^{-\eta s_E v} \big) \sup_{x \ge 0} x e^{-s_E x} = (e s_E)^{-1} \eta v e^{-\eta s_E v} ,
\]
and together with $e^{2 \eta s_E} \le e^2$, this implies the bound.

Finally, \zcref{l:rep-cycle}\ref{i:rep-cycle-g} implies that $\Lambda_{\mathrm B} \mathsf g_X^{-1} \Lambda_{\mathrm B} = \mathsf g_X$, and therefore
\[
s_E \Tr(\Lambda_{\mathrm B} Q_X) = s_E \Tr\big( \Lambda_{\mathrm B} \mathsf g_X^{-1} \Lambda_{\mathrm B} \mathsf g_X \big) = s_E \Tr \mathsf g_X^2 = \Tr\big( f_0(X) + X \big) = \Tr f_0(X) ,
\]
where we used $\Tr X = 0$ in the last step. Inserting this into the exponent above, we find $\mathbf S(Q_X) = e^{-\eta \Tr f_0(X)}$.
\end{proof}

In the critical regime, $\eta = \eps/(N \rho(E))$ is of order $\delta_W = W^{-2}$, so that the bound of \zcref{l:rep-kick} reads $|\nabla^m \mathbf S| \le C \delta_W v e^{-c \delta_W v}$, with constants depending on $\eps$, $E$, and the range of $X_N(E)$.