\chapter{The supersymmetric representation of the band matrix}\label{s:rep}

In this section, we write the source ratios $T_N^{(k)}$ of \eqref{e:intro-ratio} as the integral of a one-dimensional chain. The chain has one supermatrix-valued variable at each site of $[N]$, a weight attached to each site, and a Gaussian link between neighboring sites. By \zcref{p:rep-exact}, the chain integral is a bilinear pairing in which a power of an explicit transfer operator $T_\Gamma$ appears. In Part IV, we start from this representation and from the identity for partial chains in \zcref{p:rep-anchored}, and we prove that the chain converges to the sigma model of \zcref{s:sigma}. In Part III, we use \zcref{l:rep-dictionary}, which relates resolvent entries of $H$ to derivatives of source ratios and to insertions in the chain.

The derivation is the supersymmetric method, which \zcref{s:grass-example} carries out for the GUE, applied to the band matrix of \zcref{d:model}. The supersymmetric approach to band matrices goes back to Fyodorov and Mirlin \cite{fyodorov1991scaling}, who used it to derive, at the level of physics, the one-dimensional sigma model. Our derivation has three steps. We write each determinant in $T_N^{(k)}$ as a Gaussian integral, bosonic for the inverse determinants and fermionic for the others, we average over $H$, and we decouple the resulting quartic interaction by a Hubbard--Stratonovich transformation, with one supermatrix for each site. For the density of states, which involves only the upper half plane, these steps are carried out for the same variance profile in \cite[Proposition~2.1]{disertori2021density}. For block band matrices, M.~Shcherbina and T.~Shcherbina derived an integral representation of the two-point function, which involves both half planes, with the noncompact hyperbolic sector and the anticommuting variables \cite[Proposition~2.1]{shcherbina2018universality} (see also \cite{shcherbina2021universality}).

Two points need care. The first is the choice of the variance profile, which makes the decoupled integrand a chain with nearest-neighbor coupling (see \zcref{s:rep-decoupling}). The second is that $T_N^{(k)}$ contains resolvents in the upper and the lower half plane. In this case, the commuting variables of the Hubbard--Stratonovich transformation cannot be integrated over Hermitian matrices. In \cite{shcherbina2018universality}, the commuting variables of each block are instead integrated out by a bosonization formula (\cite[Proposition~7.1]{shcherbina2018universality}, taken there from \cite{fyodorov2002negative}). Such a formula replaces an integral over several commuting vectors, of a function of the matrix of their inner products, by an integral over this matrix. It applies there because each block carries as many vector variables as its size, which is at least two. In our model, each site carries a single vector variable, and we integrate the commuting variables of the Hubbard--Stratonovich transformation over a cycle introduced by Sch\"afer and Wegner \cite{schafer1980disordered}. We prove the identity that justifies this in \zcref{s:contour}. After the exact representation, \zcref{s:rep-heuristics} describes in elementary terms the saddle manifold, near which Part IV analyzes the chain. It is a union of $2k + 1$ conjugation orbits, each consisting of critical points of the onsite weight. In \zcref{s:sigma}, we describe its balanced component as a homogeneous superspace.

We use the Grassmann algebras, supermatrices, the supertrace $\Str$, the superdeterminant $\Sdet$, and the Berezin integral of \zcref{s:grassmann}. Throughout this section, the integers $k \ge 1$ and $N \ge 2$, the parameters $\eps > 0$ and $W > 0$, and the energy $E \in (-2, 2)$ are fixed. Recall $s_E$, $\rho(E)$, and $a_N$ from \eqref{e:scaling}, and recall that $\kappa_E = E/2$, so that $\kappa_E^2 + s_E^2 = 1$. We let $\|A\|$ denote the Hilbert--Schmidt norm of a complex square matrix $A$ and $\|A\|_{\rm op}$ its operator norm, and we set $\Re A = (A + A^*)/2$ and $\Im A = (A - A^*)/(2\iu)$. We write $A > 0$ if $A$ is Hermitian and positive definite.

\section{Source ratios as superdeterminants}\label{s:rep-sources}

We first collect the spectral parameters of $T_N^{(k)}$ in one diagonal supermatrix. The supermatrices of this section have size $(2k|2k)$. We index the rows of the BB block, and those of the FF block, by $2k$ \emph{flavors}, and we call the flavors $1, \dots, k$ \emph{retarded} and the flavors $k + 1, \dots, 2k$ \emph{advanced}. We let $\diag(A \mid D)$ denote the supermatrix with BB block $A$, FF block $D$, and vanishing odd blocks.

\begin{definition}\label{d:rep-sources}
We set $\Lambda_{\mathrm B} = \Lambda_{\mathrm F} = \diag(\Id_k, -\Id_k)$ and $\Lambda = \diag(\Lambda_{\mathrm B} \mid \Lambda_{\mathrm F})$. We set $\eta_0 = a_N^{-1}$ and $\eta = \eps \eta_0$. For $\Gamma_{\mathrm F} \in \C^{2k \times 2k}$, we call
\[
\Gamma = \diag(\eps \Lambda_{\mathrm B} \mid \Gamma_{\mathrm F})
\]
a \emph{source supermatrix}. For $u \in \R^k$ and $b \in (0, \infty)^k$, we let $\Gamma(u, b)$ denote the source supermatrix with
\[
\Gamma_{\mathrm F} = \diag(b_1 - \iu u_1, \dots, b_k - \iu u_k, -b_1 - \iu u_1, \dots, -b_k - \iu u_k),
\]
and we set $\Gamma_0 = \Gamma(0, \eps \1) = \eps \Lambda$.
\end{definition}

The diagonal entries of $E + \iu \eta_0 \Gamma(u, b)$ are the spectral parameters of the determinants in $T_N^{(k)}$. Recall from \zcref{s:intro-results} that $z_a = E + (u_a + \iu b_a)/a_N$ for $a \in [k]$ and $z_0 = E + \iu \eps/a_N$. Then
\begin{equation}\label{e:rep-spectral}
E + \iu \eta_0 \Gamma(u, b) = \diag\big( z_0 \Id_k, \bar z_0 \Id_k \mid z_1, \dots, z_k, \bar z_1, \dots, \bar z_k \big) .
\end{equation}

We will also perturb the spectral parameter at single sites, which produces eigenvector observables (see \zcref{l:rep-dictionary}). We call a family $Y = (Y_i)_{i \in [N]}$ of supermatrices $Y_i = \diag(Y_i^{\mathrm B} \mid Y_i^{\mathrm F})$ with complex $2k \times 2k$ blocks a family of \emph{local sources}, and we set $\|Y^{\mathrm B}\|_{\rm op} = \max_i \|Y_i^{\mathrm B}\|_{\rm op}$. Given a family $(A_i)_{i \in [N]}$ of $(2k|2k)$ supermatrices, we let $A - H$ denote the supermatrix of size $(2kN|2kN)$ with blocks
\[
(A - H)_{ij} = \delta_{ij} A_i - H_{ij} \Id, \qquad i, j \in [N],
\]
where $\Id$ is the identity supermatrix of size $(2k|2k)$. We write $A - H$ as well when $A_i = A$ for all $i$.

\begin{definition}\label{d:rep-TY}
For a source supermatrix $\Gamma$ and local sources $Y$ with $\|Y^{\mathrm B}\|_{\rm op} < \eta$, we define
\begin{equation}\label{e:rep-TY}
T_N^{(k)}(\Gamma; Y) = \E \Sdet\big( E + \iu \eta_0 \Gamma + Y - H \big)^{-1} ,
\end{equation}
and we set $T_N^{(k)}(\Gamma) = T_N^{(k)}(\Gamma; 0)$.
\end{definition}

We call the sites $i_1 < \dots < i_m$ of \zcref{d:iface-local} the \emph{marked sites}. When $Y_i = 0$ outside the marked sites, $T_N^{(k)}(\Gamma(u, b); Y)$ is the function defined there, through which Part III uses local sources. Since the supermatrix in \eqref{e:rep-TY} has vanishing odd blocks, its superdeterminant is the ratio of the determinants of its BB and FF blocks.

\begin{lemma}\label{l:rep-sdet}
Let $\Gamma$ be a source supermatrix and $Y$ local sources with $\|Y^{\mathrm B}\|_{\rm op} < \eta$. Then the BB block of $E + \iu \eta_0 \Gamma + Y - H$ is invertible for every Hermitian $H$, and $T_N^{(k)}(\Gamma; Y)$ is finite and holomorphic in $\Gamma_{\mathrm F}$ and in the entries of $Y$. Moreover, for all $u \in \R^k$ and $b \in (0, \infty)^k$,
\begin{equation}\label{e:rep-T-identification}
T_N^{(k)}(\Gamma(u, b)) = T_N^{(k)}(u, b), \qquad T_N^{(k)}(\Gamma_0) = 1, \qquad \Str \Gamma_0^2 = 0 .
\end{equation}
\end{lemma}

\begin{proof}
Let $D_{\mathrm B}$ denote the BB block of $E + \iu \eta_0 \Gamma + Y - H$, which has the blocks
\[
(D_{\mathrm B})_{ij} = \delta_{ij} \big( E + \iu \eta \Lambda_{\mathrm B} + Y_i^{\mathrm B} \big) - H_{ij} \Id_{2k}, \qquad i, j \in [N] .
\]
Since the matrices $E \Lambda_{\mathrm B}$ and $H \otimes \Lambda_{\mathrm B}$ are Hermitian, we have
\[
\Im \big( (\Id_N \otimes \Lambda_{\mathrm B}) D_{\mathrm B} \big) = \eta \Id + \diag_{i \in [N]} \big( \Im(\Lambda_{\mathrm B} Y_i^{\mathrm B}) \big) \ge (\eta - \|Y^{\mathrm B}\|_{\rm op}) \Id > 0 .
\]
Recall that a matrix $A$ with $\Im A > 0$ is invertible and satisfies $|\det A| \ge \det \Im A$. Indeed, the matrix $K = (\Im A)^{-1/2} (\Re A) (\Im A)^{-1/2}$ is Hermitian, and $A = (\Im A)^{1/2} (K + \iu \Id) (\Im A)^{1/2}$. Taking determinants, we find
\[
|\det A| = \det(\Im A) \prod_{\mu} |\mu + \iu| \ge \det \Im A ,
\]
where $\mu$ runs over the eigenvalues of $K$, which are real. Together with the previous display, this inequality implies that
\[
|\det D_{\mathrm B}|^{-1} \le (\eta - \|Y^{\mathrm B}\|_{\rm op})^{-2kN}
\]
for every Hermitian $H$. The determinant of the FF block is a polynomial in the entries of $H$, $\Gamma_{\mathrm F}$, and $Y^{\mathrm F}$, and the entries of $H$ have Gaussian tails. Dominated convergence now shows that the expectation is finite and holomorphic.

We now prove \eqref{e:rep-T-identification}. The two blocks of $E + \iu \eta_0 \Gamma(u, b) - H$ are block diagonal over the flavors by \eqref{e:rep-spectral}, so that
\[
\Sdet \big( E + \iu \eta_0 \Gamma(u, b) - H \big)^{-1} = \prod_{a = 1}^k \frac{\det(z_a - H) \det(\bar z_a - H)}{\det(z_0 - H) \det(\bar z_0 - H)} = \prod_{\alpha = 1}^N \prod_{a = 1}^k \frac{(x_\alpha - u_a)^2 + b_a^2}{x_\alpha^2 + \eps^2} .
\]
In the second step, we used that for $x = a_N (\lambda - E)$,
\[
(z_a - \lambda)(\bar z_a - \lambda) = \frac{(x - u_a)^2 + b_a^2}{a_N^2}, \qquad (z_0 - \lambda)(\bar z_0 - \lambda) = \frac{x^2 + \eps^2}{a_N^2} .
\]
Taking expectations yields the first identity. For $\Gamma_0 = \eps \Lambda$, the BB and FF blocks coincide, which implies that $T_N^{(k)}(\Gamma_0) = 1$, and we have
\[
\Str \Gamma_0^2 = \eps^2 (\Tr \Id_{2k} - \Tr \Id_{2k}) = 0 . \qedhere
\]
\end{proof}

By \zcref{t:process}, the limits of the source ratios $T_N^{(k)}(u, b)$ determine the limit of the point process $\Xi_N$, and the local sources give access to eigenvector observables. The rest of the section rewrites $T_N^{(k)}(\Gamma; Y)$.

\section{Gaussian representation and the average over \texorpdfstring{$H$}{H}}\label{s:rep-gaussian}

We now write the superdeterminant in \eqref{e:rep-TY} as a Gaussian integral, as in \zcref{l:grass-resolvent}, and compute the average over $H$. For each site $i \in [N]$, we take $2k$ complex coordinates $\zeta_i = (\zeta_{i, 1}, \dots, \zeta_{i, 2k})$ and $4k$ odd coordinates $\chi_{i, a}, \bar \chi_{i, a}$, $a \in [2k]$, and we let $\psi_i = (\zeta_i, \chi_i)$ denote the supervector at the site $i$. As in \zcref{d:grass-flat}, the integral over all these coordinates is denoted $\int \cdot \dd \psi$. For a $(2k|2k)$ supermatrix $M$, we set $\psi_i^* M \psi_j = (\zeta_i, \chi_i)^* M (\zeta_j, \chi_j)$, in the notation preceding \zcref{p:grass-supergauss}, and we let $\psi_i \psi_j^*$ denote the even supermatrix with entries $(\psi_i)_a (\psi_j^*)_b$. One checks on the four blocks that
\begin{equation}\label{e:rep-str-vector}
\Str(\psi_i \psi_j^* M) = \psi_j^* M \psi_i
\end{equation}
for every even supermatrix $M$; in the FF block, we used $-\chi_{i, a} \bar \chi_{j, b} = \bar \chi_{j, b} \chi_{i, a}$.

\begin{lemma}\label{l:rep-spinor}
Let $\Gamma$ be a source supermatrix and $Y$ local sources with $\|Y^{\mathrm B}\|_{\rm op} < \eta$. Then
\begin{equation}\label{e:rep-quartic}
T_N^{(k)}(\Gamma; Y) = \int \exp\Big( \iu \sum_{i = 1}^N \psi_i^* \Lambda (E + \iu \eta_0 \Gamma + Y_i) \psi_i - \frac12 \sum_{i, j = 1}^N J_{ij} \Str\big( \psi_i \psi_i^* \Lambda \, \psi_j \psi_j^* \Lambda \big) \Big) \dd \psi ,
\end{equation}
where the integrand is integrable.
\end{lemma}

\begin{proof}
Fix a Hermitian matrix $H$, and set
\[
M = -\iu (\Id_N \otimes \Lambda)(E + \iu \eta_0 \Gamma + Y - H) ,
\]
which has vanishing odd blocks. The proof of \zcref{l:rep-sdet} shows that
\[
\Re M^{\mathrm{BB}} = \Im \big( (\Id_N \otimes \Lambda_{\mathrm B}) (E + \iu \eta_0 \Gamma + Y - H)^{\mathrm{BB}} \big) \ge (\eta - \|Y^{\mathrm B}\|_{\rm op}) \Id .
\]
Note that $\Sdet(-\iu \, \Id_N \otimes \Lambda) = 1$, since its BB and FF blocks coincide. By \zcref{p:grass-supergauss}, we have
\[
\Sdet \big( E + \iu \eta_0 \Gamma + Y - H \big)^{-1} = \int \exp\Big( \iu \sum_{i} \psi_i^* \Lambda (E + \iu \eta_0 \Gamma + Y_i) \psi_i - \iu \sum_{i, j} H_{ij} \, \psi_i^* \Lambda \psi_j \Big) \dd \psi .
\]
The body of this integrand has modulus
\[
\exp\Big( -\sum_i \zeta_i^* \Im\big( \Lambda_{\mathrm B} (\iu \eta \Lambda_{\mathrm B} + Y_i^{\mathrm B}) \big) \zeta_i \Big) \le e^{-(\eta - \|Y^{\mathrm B}\|_{\rm op}) |\zeta|^2} ,
\]
which is independent of $H$, and the other coefficients are this function times polynomials in $\zeta$, $\bar \zeta$, and the entries of $H$. Using Fubini's theorem, we may take the expectation inside the integral.

We now compute the expectation. Let $c = (c_{ij})$ be a complex matrix. The variable $\sum_{i, j} H_{ij} c_{ij}$ is a complex linear combination of centered jointly Gaussian variables. By \eqref{e:intro-covariance}, we have
\[
\E \Big( \sum_{i, j} H_{ij} c_{ij} \Big)^2 = \sum_{i, j} J_{ij} c_{ij} c_{ji}, \qquad \E \exp\Big( -\iu \sum_{i, j} H_{ij} c_{ij} \Big) = \exp\Big( -\frac12 \sum_{i, j} J_{ij} c_{ij} c_{ji} \Big) .
\]
Each side of the second identity is entire in $c$. Using \zcref{l:grass-calculus}\ref{i:grass-calculus-identities}, we conclude that the identity holds for the even elements $c_{ij} = \psi_i^* \Lambda \psi_j$. Finally, since $\psi_i^* \Lambda \psi_j$ is even, we have
\[
\Str\big( \psi_i \psi_i^* \Lambda \psi_j \psi_j^* \Lambda \big) = (\psi_i^* \Lambda \psi_j) \Str\big( \psi_i \psi_j^* \Lambda \big) = (\psi_i^* \Lambda \psi_j)(\psi_j^* \Lambda \psi_i) ,
\]
where we used \eqref{e:rep-str-vector} in the last step.
\end{proof}

The quadratic term in \eqref{e:rep-quartic} contains the spectral parameters and is a sum of terms attached to single sites. The quartic term comes from the randomness of $H$. It couples every pair of sites with the weight $J_{ij}$, and by \eqref{e:intro-images} the weight is of order $W^{-1}$ for pairs at distance up to order $W$. The rest of the derivation removes this coupling.

\section{Decoupling and the nearest-neighbor chain}\label{s:rep-decoupling}

The Hubbard--Stratonovich transformation decouples a quadratic form in the exponent by a Gaussian integral. In one variable, it is the identity $e^{-a^2/2} = (2\pi)^{-1/2} \int_\R e^{-r^2/2 - \iu a r} \dd r$. Given a positive definite matrix $J$ and a vector $a$, one uses instead a centered Gaussian vector $g$ with covariance $J$,
\[
e^{-\langle a, J a \rangle/2} = \E e^{-\iu \langle a, g \rangle}, \qquad \text{where $g$ has the density } \frac{e^{-\langle r, J^{-1} r \rangle/2}}{(2\pi)^{N/2} (\det J)^{1/2}} .
\]
We apply this with the supermatrix $\psi_i \psi_i^* \Lambda$ in place of $a_i$, and with one supermatrix-valued variable $R_i$ for each site in place of $g_i$. Each $R_i$ is a $(2k|2k)$ supermatrix, integrated over a cycle. We use this word as in contour integration. A \emph{cycle} is a set of $(2k|2k)$ supermatrices whose odd entries are independent odd coordinates and whose pairs of BB and FF blocks range over a real submanifold of $\C^{2k \times 2k} \times \C^{2k \times 2k}$ of real dimension $8k^2$, which is the complex dimension of this space. The integrands below are holomorphic in the BB and FF entries, and in \zcref{s:rep-cycle} we move the cycle in complex space, as one moves a contour of integration in one complex variable. We begin with the flat cycle.

\begin{definition}\label{d:rep-flat}
We let $\mathcal S_\flat$ denote the set of $(2k|2k)$ supermatrices whose BB block is Hermitian, whose FF block is $\iu M$ with $M$ Hermitian, and whose $8k^2$ odd entries are independent odd coordinates. We identify $\mathcal S_\flat$ with the superdomain $\R^{8k^2|8k^2}$, through the real coordinates of the BB block and of $M$, orthonormal for the Hilbert--Schmidt inner product, and the odd entries, ordered so that $R^{\mathrm{BF}}_{ab}$ is immediately followed by $R^{\mathrm{FB}}_{ba}$ for all $a, b \in [2k]$. We let $\dd R$ denote the Berezin integral of \zcref{d:grass-berezin} on this superdomain, multiplied by the constant $c_k \in \C \setminus \{0\}$ for which
\begin{equation}\label{e:rep-flat-normalization}
\int_{\mathcal S_\flat} e^{-\Str(R^2)/2} \dd R = 1 .
\end{equation}
On $\mathcal S_\flat^N$, we use the product of $N$ copies of $\dd R$, with independent odd coordinates.
\end{definition}

Note that on $\mathcal S_\flat$,
\[
\Str R^2 = \|R^{\mathrm{BB}}\|^2 + \|M\|^2 + 2 \Tr\big( R^{\mathrm{BF}} R^{\mathrm{FB}} \big) ,
\]
where we used $\Tr(R^{\mathrm{FB}} R^{\mathrm{BF}}) = -\Tr(R^{\mathrm{BF}} R^{\mathrm{FB}})$, since the entries of the odd blocks anticommute. We now compute $c_k$. The even coordinates contribute the Gaussian integral $(2\pi)^{4k^2}$ over $\R^{8k^2}$. The products $R^{\mathrm{BF}}_{ab} R^{\mathrm{FB}}_{ba}$ are even and commute, and each pair of their factors is consecutive in our order of the odd entries. Applying \zcref{l:grass-berezin}\ref{i:grass-berezin-fubini} pair by pair, we find
\[
\int e^{-\Tr(R^{\mathrm{BF}} R^{\mathrm{FB}})} \dd \xi = \prod_{a, b = 1}^{2k} \int \big( 1 - R^{\mathrm{BF}}_{ab} R^{\mathrm{FB}}_{ba} \big) \dd R^{\mathrm{BF}}_{ab} \dd R^{\mathrm{FB}}_{ba} = (-1)^{4k^2} = 1 ,
\]
where $\xi$ denotes the $8k^2$ odd entries of $R$ in this order. Therefore $c_k = (2\pi)^{-4k^2}$. In another order of the odd entries, this Berezin integral may be $-1$; for instance, reading the two odd blocks row by row gives $(-1)^k$.

The flat cycle has the three features of the GUE computation listed after the proof of \zcref{p:grass-gue}: the Gaussian measure in \eqref{e:rep-hs-flat} below has total integral one; the FF block is rotated by $\iu$, since $\Str$ carries a minus sign on the FF block; and the BB block is Hermitian, which suffices only when one half plane appears (see \zcref{s:rep-cycle}). We prove in \zcref{l:contour-flat} that for every positive definite real symmetric $N \times N$ matrix $\mathbf M$ and all even supermatrices $B_1, \dots, B_N$ with entries in auxiliary generators,
\begin{equation}\label{e:rep-hs-flat}
\exp\Big( -\frac12 \sum_{i, j} (\mathbf M^{-1})_{ij} \Str(B_i B_j) \Big) = \int_{\mathcal S_\flat^N} \exp\Big( -\frac12 \sum_{i, j} \mathbf M_{ij} \Str(R_i R_j) - \iu \sum_i \Str(R_i B_i) \Big) \dd R .
\end{equation}
The right side carries the constant one for all $\mathbf M$, and this is the basic supersymmetric cancellation. We spell it out for $B = 0$, to which the general case reduces by completing the square. Let $x_i \in \R^{8k^2}$ denote the even coordinates of $R_i$. Since $\mathbf M$ is symmetric, we find as in the computation of $\Str R^2$ above that
\[
\frac12 \sum_{i, j = 1}^N \mathbf M_{ij} \Str(R_i R_j) = \frac12 \sum_{i, j = 1}^N \mathbf M_{ij} \langle x_i, x_j \rangle + \sum_{a, b = 1}^{2k} \sum_{i, j = 1}^N (R_i^{\mathrm{BF}})_{ab} \, \mathbf M_{ij} \, (R_j^{\mathrm{FB}})_{ba} .
\]
For the first term, the Gaussian integral over the even coordinates is
\[
\begin{aligned}
\int_{\R^{8k^2 N}} \exp\Big( -\frac12 \sum_{i, j = 1}^N \mathbf M_{ij} \langle x_i, x_j \rangle \Big) \dd x &= (2\pi)^{4k^2 N} \det(\mathbf M \otimes \Id_{8k^2})^{-1/2} \\
&= (2\pi)^{4k^2 N} \det(\mathbf M)^{-4k^2} .
\end{aligned}
\]
For each pair $(a, b)$, the second term is a fermionic Gaussian in the $2N$ odd entries $(R_i^{\mathrm{BF}})_{ab}$ and $(R_i^{\mathrm{FB}})_{ba}$, and since the odd entries of each $R_i$ come in consecutive pairs, the Berezin integral factors over $(a, b)$ without signs. By \zcref{p:grass-gauss}\ref{i:grass-gauss-fermion}, applied with $\chi_i = (R_i^{\mathrm{BF}})_{ab}$, $\bar \chi_i = (R_i^{\mathrm{FB}})_{ba}$, and the matrix $-\mathbf M$, each pair contributes the factor $\det(-\mathbf M) = (-1)^N \det \mathbf M$. Over the $4k^2$ pairs, we obtain
\[
\int_{\mathcal S_\flat^N} e^{-\frac12 \sum_{i, j} \mathbf M_{ij} \Str(R_i R_j)} \dd R = c_k^N \, (2\pi)^{4k^2 N} \det(\mathbf M)^{-4k^2} \big( (-1)^N \det \mathbf M \big)^{4k^2} = 1 .
\]
The even coordinates contribute $\det(\mathbf M)^{-4k^2}$ and the odd coordinates $\det(\mathbf M)^{4k^2}$, and the two factors cancel because $\mathcal S_\flat$ has as many even as odd dimensions, $8k^2$ of each.

To decouple \eqref{e:rep-quartic}, we take $\mathbf M = J^{-1}$ and $B_i = \psi_i \psi_i^* \Lambda$. This is where the choice of the variance profile matters. Recall that $J^{-1} = \Id - W^2 \Delta$, and note that
\[
\sum_{i, j = 1}^N (-\Delta)_{ij} f(i) f(j) = \sum_{i = 1}^{N - 1} \big( f(i + 1) - f(i) \big)^2 .
\]
Using this identity in each entry and $\Str(A B) = \Str(B A)$ for even supermatrices, we obtain
\begin{equation}\label{e:rep-precision}
\sum_{i, j = 1}^N (J^{-1})_{ij} \Str(R_i R_j) = \sum_{i = 1}^N \Str R_i^2 + W^2 \sum_{i = 1}^{N - 1} \Str(R_{i + 1} - R_i)^2 .
\end{equation}
Thus the precision of the Hubbard--Stratonovich field is a nearest-neighbor operator, although $J$ itself couples all pairs of sites. This is the reason for the choice of $J$ in \zcref{d:model}. The same profile is used in \cite{disertori2021density}, where it is observed that the dual supersymmetric model has nearest-neighbor coupling, and in three dimensions in \cite{disertori2002density}.

We now insert \eqref{e:rep-hs-flat} into \eqref{e:rep-quartic} and, for the moment formally, integrate out $\psi$. By \eqref{e:rep-str-vector}, we have $\Str(R_i \psi_i \psi_i^* \Lambda) = \psi_i^* \Lambda R_i \psi_i$. The exponent becomes $\iu \sum_i \psi_i^* \Lambda (E + \iu \eta_0 \Gamma + Y_i - R_i) \psi_i$, and the $\psi$-integral factors over the sites. Applying \zcref{p:grass-supergauss} at each site, with $Y = 0$, we would obtain
\begin{multline}\label{e:rep-formal}
T_N^{(k)}(\Gamma) \overset{?}{=} \int_{\mathcal S_\flat^N} \exp\Big( - \sum_{i = 1}^N \Big[ \frac12 \Str R_i^2 + \Str \log\big( E + \iu \eta_0 \Gamma - R_i \big) \Big] \\
- \frac{W^2}{2} \sum_{i = 1}^{N - 1} \Str(R_{i + 1} - R_i)^2 \Big) \dd R ,
\end{multline}
where we wrote $\Sdet(A)^{-1} = e^{-\Str \log A}$, which follows from \zcref{l:grass-sdet}\ref{i:grass-sdet-exp} when the logarithm is defined. The source appears only in the onsite terms. We translate $R_i = S_i + \iu \eta_0 \Gamma$, which leaves the links unchanged, and we expand
\[
\frac12 \Str(S_i + \iu \eta_0 \Gamma)^2 = \frac12 \Str S_i^2 + \iu \eta_0 \Str(S_i \Gamma) - \frac12 \eta_0^2 \Str \Gamma^2 .
\]
The right side of \eqref{e:rep-formal} then becomes
\begin{equation}\label{e:rep-chain-formal}
e^{N \eta_0^2 \Str(\Gamma^2)/2} \int \prod_{i = 1}^N e^{-\Str(S_i^2)/2 - \iu \eta_0 \Str(S_i \Gamma)} \Sdet(E - S_i)^{-1} \prod_{i = 1}^{N - 1} e^{-W^2 \Str(S_{i + 1} - S_i)^2/2} \dd S .
\end{equation}

The integrand in \eqref{e:rep-chain-formal} is a product of $N$ factors attached to single sites and $N - 1$ Gaussian links with variance of order $W^{-2}$ in $S_{i + 1} - S_i$, so that the integral is a matrix element of the $(N - 1)$st power of an integral operator in one supermatrix variable, which we define in \zcref{s:rep-chain} (see \zcref{f:rep-chain}). Recall from the discussion after \eqref{e:ov-chain} that the resemblance of such an integral to a Feynman--Kac formula is only formal, since the weights are complex and contain odd variables.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1.4cm, y=1cm, site/.style={circle, draw, fill=white, inner sep=1.6pt}]
\foreach \x/\lab in {0/S_1, 1/S_2, 3/S_m, 4/S_{m + 1}, 5/S_{m + 2}, 7/S_{N - m}, 8/S_{N - m + 1}, 10/S_N} {
  \node[site] (n\x) at (\x, 0) {};
  \node[below=3pt] at (\x, 0) {\small $\lab$};
  \node[above=3pt] at (\x, 0) {\scriptsize $w_\Gamma$};
}
\foreach \a/\b in {0/1, 3/4, 4/5, 7/8} { \draw[thick] (n\a) -- (n\b); }
\foreach \a/\b in {1/3, 5/7, 8/10} { \draw[thick, dotted] (n\a) -- (n\b); }
\node[below=2pt] at (0.5, 0) {\scriptsize $k_{\tau_{\rm phys}}$};
\draw[decorate, decoration={brace, mirror, amplitude=5pt}] (-0.2, -0.75) -- (3.8, -0.75) node[midway, below=6pt] {$\ell_m$};
\draw[decorate, decoration={brace, mirror, amplitude=5pt}] (3.85, -0.75) -- (7.2, -0.75) node[midway, below=6pt] {$T_\Gamma^{N - 2m}$};
\draw[decorate, decoration={brace, mirror, amplitude=5pt}] (7.35, -0.75) -- (10.2, -0.75) node[midway, below=6pt] {$f_m$};
\end{tikzpicture}
\caption{The open chain of \zcref{p:rep-exact}. Each site carries the weight $w_\Gamma$ of \eqref{e:rep-weight}, and each pair of neighboring sites the link kernel $k_{\tau_{\rm phys}}$ of \zcref{d:rep-chain}. In the pairing \eqref{e:rep-pairing}, the partial chain $f_m$ collects the last $m$ sites, the power $T_\Gamma^{N - 2m}$ the middle $N - 2m$ sites, and $\ell_m$ the first $m$ sites together with the link from $S_m$ to $S_{m + 1}$.}
\label{f:rep-chain}
\end{figure}

For other variance profiles, the first two steps are unchanged, since \zcref{l:rep-spinor} uses only the variances \eqref{e:intro-covariance}. However, the decoupling by one supermatrix per site needs a Gaussian field with covariance $J$, and the precision $J^{-1}$ of this field appears in the action. For the sharp profile $J_{ij} = (2W + 1)^{-1} \1_{|i - j| \le W}$, such a field does not exist once $N$ is large compared with $W$, since $J$ is then not positive semidefinite. Indeed, its symbol, the normalized Dirichlet kernel, takes negative values, and for $W = 8$ and $N = 200$ the smallest eigenvalue of $J$ is about $-0.22$. A positive definite profile whose inverse is not sparse leads to a decoupled integrand that contains $\Str(R_i R_j)$ for all pairs of sites and does not factor into nearest-neighbor terms. We do not know how to write the chain for such profiles with an integral operator in one supermatrix variable, and our proof uses the nearest-neighbor structure \eqref{e:rep-precision} in an essential way.

\section{The noncompact sector and the Sch\"afer--Wegner cycle}\label{s:rep-cycle}

The right side of \eqref{e:rep-formal} is not defined as written. We now explain why, and how the derivation is repaired. The identity \eqref{e:rep-hs-flat} holds for each fixed $\psi$. To integrate out $\psi$ afterward, we must exchange the $\psi$-integral with the integral over $R$, and for this the integrand must be integrable on the product. The odd variables appear polynomially, and the FF blocks through convergent Gaussians. The difficulty is in the BB blocks. For $R_i \in \mathcal S_\flat$, the commuting part of the exponent at the site $i$ has the real part
\[
\Re \big[ \iu \zeta_i^* \Lambda_{\mathrm B} (E + \iu \eta \Lambda_{\mathrm B} - R_i^{\mathrm{BB}}) \zeta_i \big] = -\eta |\zeta_i|^2 + \zeta_i^* \Im\big( \Lambda_{\mathrm B} R_i^{\mathrm{BB}} \big) \zeta_i .
\]
Since $R_i^{\mathrm{BB}}$ is Hermitian, $\Im(\Lambda_{\mathrm B} R_i^{\mathrm{BB}})$ is a Hermitian matrix with trace zero, whose eigenvalues are of the order of the off-diagonal blocks of $R_i^{\mathrm{BB}}$. The $\zeta_i$-integral therefore diverges whenever these blocks are large compared with $\eta$. Equivalently, the inverse superdeterminant in \eqref{e:rep-formal} has poles on $\mathcal S_\flat$. For instance, for $k = 1$ and the Hermitian matrix $R^{\mathrm{BB}} = \big(\begin{smallmatrix} E & r \\ \bar r & E \end{smallmatrix}\big)$, we have
\[
\det\big( E + \iu \eta \Lambda_{\mathrm B} - R^{\mathrm{BB}} \big) = \eta^2 - |r|^2 ,
\]
which vanishes for $|r| = \eta$.

The source of the difficulty is that the quadratic form $\zeta^* \Lambda_{\mathrm B} \zeta$ is indefinite, of signature $(k, k)$, so that the symmetry group of the commuting variables is the noncompact group $U(k, k)$. This hyperbolic symmetry was first observed by Wegner \cite{wegner1979mobility} (see also \cite[p.~2]{fyodorov2008hyperbolic}), and the cure is due to Sch\"afer and Wegner \cite{schafer1980disordered}. One deforms the BB blocks of the Hubbard--Stratonovich variables into the complex domain, in such a way that $\Im(\Lambda_{\mathrm B} (E + \iu \eta \Lambda_{\mathrm B} - R_i^{\mathrm{BB}}))$ becomes positive definite. We use the following form of their cycle. We let $\mathrm{Herm}(2k)$ denote the space of Hermitian $2k \times 2k$ matrices, and we let $\mathrm{Herm}_{\rm d}(2k)$ and $\mathrm{Herm}_{\rm o}(2k)$ denote its subspaces of block-diagonal and block off-diagonal matrices, for the decomposition into $k \times k$ blocks given by the retarded and advanced flavors.

From here on, in this section, in \zcref{s:hist,s:end}, and in \zcref{s:contour}, the letter $X$ denotes a matrix in $\mathrm{Herm}_{\rm o}(2k)$, which will be a coordinate of the cycle. The parameter $X$ of \zcref{d:intro-critical} appears there only as $X_N(E)$.

For $X \in \mathrm{Herm}_{\rm o}(2k)$, we set
\begin{equation}\label{e:rep-f0}
f_0(X) = \big( s_E^2 \Id + X^2 \big)^{1/2}, \qquad \mathsf g_X = \big( s_E^{-1} (f_0(X) + X) \big)^{1/2} .
\end{equation}

\begin{definition}\label{d:rep-cycle}
We let the \emph{Sch\"afer--Wegner cycle} $\mathcal S$ be the set of $(2k|2k)$ supermatrices $S$ with
\begin{equation}\label{e:rep-cycle}
S^{\mathrm{BB}} = P - \iu \Lambda_{\mathrm B} \big( f_0(X) + X \big), \qquad S^{\mathrm{FF}} = \kappa_E \Id + \iu M ,
\end{equation}
where $P \in \mathrm{Herm}_{\rm d}(2k)$, $X \in \mathrm{Herm}_{\rm o}(2k)$, and $M \in \mathrm{Herm}(2k)$, and whose odd entries are independent odd coordinates. We call $(P, X, M)$ the coordinates of $S \in \mathcal S$. For $R \in \mathcal S_\flat$ with BB block $P - \iu \Lambda_{\mathrm B} X$ and FF block $\iu M$, we let $S(R)$ denote the point of $\mathcal S$ with coordinates $(P, X, M)$ and the same odd entries as $R$. For a superfunction $F$ of $S$ whose coefficients are defined on $\mathcal S$, we call $F$ \emph{integrable on $\mathcal S$} if $R \mapsto F(S(R))$ is integrable in the sense of \zcref{d:grass-berezin}, and we define
\[
\int_{\mathcal S} F(S) \dd S = \int_{\mathcal S_\flat} F(S(R)) \dd R .
\]
We let $\mathcal S^N$ denote the product of $N$ copies of $\mathcal S$, with independent odd coordinates, and the product integral.
\end{definition}

The map $X \mapsto -\iu \Lambda_{\mathrm B} X$ is a linear isometry of $\mathrm{Herm}_{\rm o}(2k)$ onto itself. Every Hermitian matrix therefore has the form $P - \iu \Lambda_{\mathrm B} X$ for unique $P$ and $X$, and the flat cycle has BB block $P - \iu \Lambda_{\mathrm B} X$. The cycle $\mathcal S$ is thus a graph over $\mathcal S_\flat$: its BB block differs from that of $\mathcal S_\flat$ by the block-diagonal matrix $-\iu \Lambda_{\mathrm B} f_0(X)$, and its FF block by the constant $\kappa_E \Id$. \zcref{f:rep-cycle} shows this graph for $k = 1$.

The definition of $\int_{\mathcal S} F \dd S$ contains no Jacobian factor, because it is a contour integral of a holomorphic integrand. To see this, we write the BB block of a complex supermatrix as $P^{\C} - \iu \Lambda_{\mathrm B} X^{\C}$, with $P^{\C}$ complex block diagonal and $X^{\C}$ complex block off-diagonal, and its FF block as $\iu M^{\C}$. We use the entries of $P^{\C}$, $X^{\C}$, and $M^{\C}$ in the complexifications of orthonormal bases of the corresponding real spaces as complex coordinates $z \in \C^{8k^2}$. These coordinates are real exactly on $\mathcal S_\flat$, where they are the coordinates of \zcref{d:rep-flat}. The form $\dd R$ then extends to the \emph{holomorphic Berezin form} $c_k \dd z_1 \wedge \dots \wedge \dd z_{8k^2} \dd \xi$, the product of a holomorphic volume form in the even entries and the Berezin integral in the odd entries. As for a contour integral $\int_\gamma f(z) \dd z = \int f(\gamma(t)) \gamma'(t) \dd t$, integrating this form over a cycle parametrized by $x \mapsto z(x)$ means integrating $F(z(x)) \det z'(x)$ against $c_k \dd x \dd \xi$, where $z'(x)$ denotes the complex $8k^2 \times 8k^2$ matrix of partial derivatives of $z$ in $x$. By \zcref{l:contour-stokes}, the result does not change when the cycle is deformed, provided the integrand decays. The cycle $\mathcal S$ is parametrized by
\[
(P, X, M) \mapsto (P^{\C}, X^{\C}, M^{\C}) = \big( P - \iu \Lambda_{\mathrm B} f_0(X), \, X, \, M - \iu \kappa_E \Id \big) ,
\]
whose Jacobian matrix is triangular with unit diagonal. Hence $\det z' = 1$, and $\int_{\mathcal S} F \dd S$ is the integral of the holomorphic Berezin form over $\mathcal S$.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\begin{axis}[
  width=0.74\linewidth, height=0.58\linewidth,
  view={32}{24},
  xmin=-1.8, xmax=1.8, ymin=-1.8, ymax=1.8, zmin=0, zmax=2.1,
  xlabel={$\Re X_{12}$}, ylabel={$\Im X_{12}$}, zlabel={$f_0(X)_{11}$},
  xtick={-1, 0, 1}, ytick={-1, 0, 1}, ztick={0, 1}, zticklabels={$0$, $s_E$},
  tick label style={font=\small}, label style={font=\small},
  colormap={rep-cycle}{gray(0cm)=(0.62); gray(1cm)=(0.84)},
  axis lines=left, clip=false]
\addplot3[surf, shader=flat, draw=gray!45, line width=0.1pt, domain=-1.8:1.8, y domain=-1.8:1.8, samples=7, samples y=7, point meta=0, colormap={rep-flat}{gray(0cm)=(0.93); gray(1cm)=(0.93)}] {0};
\addplot3[thick, dashed, domain=0:360, samples=73, samples y=0] ({0.3*cos(x)}, {0.3*sin(x)}, {0});
\addplot3[densely dotted, thick] coordinates {(0, 0, 0) (0, 0, 1)};
\addplot3[densely dotted, thick] coordinates {(1.2, -0.6, 0) (1.2, -0.6, {sqrt(1 + 1.8)})};
\addplot3[only marks, mark=*, mark size=1.1pt] coordinates {(1.2, -0.6, 0)};
\addplot3[surf, z buffer=sort, shader=faceted, colormap name=rep-cycle, faceted color=black!50, line width=0.1pt, fill opacity=0.8, domain=0:1.6, y domain=0:360, samples=9, samples y=33, point meta=z] ({x*cos(y)}, {x*sin(y)}, {sqrt(1 + x^2)});
\addplot3[only marks, mark=*, mark size=1.1pt] coordinates {(0, 0, 1) (1.2, -0.6, {sqrt(1 + 1.8)})};
\node[font=\small, anchor=north west] at (axis cs:1.2, -0.6, 0) {$R$};
\node[font=\small, anchor=west] at (axis cs:1.2, -0.6, {sqrt(1 + 1.8)}) {$S(R)$};
\node[font=\small] at (axis cs:-1.45, -1.3, 0) {$\mathcal S_\flat$};
\node[font=\small] at (axis cs:-1.2, -1.2, 2.05) {$\mathcal S$};
\end{axis}
\end{tikzpicture}
\caption{The Sch\"afer--Wegner cycle for $k = 1$, drawn for $E = 0$ and $P = 0$, so that $s_E = 1$ and $P = E \Id = \kappa_E \Id$. Here $X$ has the single entry $X_{12} = \bar X_{21}$. The plane of $X_{12}$ at height zero carries $\mathcal S_\flat$, and $\mathcal S$ is the graph of $f_0(X)_{11} = -\Im S^{\mathrm{BB}}_{11}$ over it, the upper sheet of the hyperboloid $f_0(X)_{11}^2 - |X_{12}|^2 = s_E^2$. The two dotted segments join points $R$ of $\mathcal S_\flat$ to the points $S(R)$ above them, one at $X = 0$, where $S(R)$ is the vertex at height $s_E$, and one at the labeled point $R$. The dashed circle $|X_{12}| = \eta$, drawn with $\eta = 0.3$, carries the poles on $\mathcal S_\flat$ of the example in \zcref{s:rep-cycle}, with $|r| = |X_{12}|$, and $\mathcal S$ stays away from it. The points of $\mathcal S$ shown have the BB blocks $\kappa_E - \iu s_E Q_X$ of \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}, the BB blocks of critical points of the weight of the chain, and the vertex gives $\kappa_E - \iu s_E \Lambda_{\mathrm B}$.}
\label{f:rep-cycle}
\end{figure}

Since $P$ ranges over all of $\mathrm{Herm}_{\rm d}(2k)$, the cycle $\mathcal S$ is also the set of supermatrices with BB block $\kappa_E \Id + P - \iu \Lambda_{\mathrm B}(f_0(X) + X)$, which is the form used in Part IV. We will need the following properties of $\mathcal S$.

\begin{lemma}\label{l:rep-cycle}
Let $X \in \mathrm{Herm}_{\rm o}(2k)$ and $S \in \mathcal S$ with coordinates $(P, X, M)$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:rep-cycle-g} The matrix $f_0(X)$ is block diagonal and commutes with $X$, the matrix $\mathsf g_X^2 = s_E^{-1}(f_0(X) + X)$ is positive definite with $\det \mathsf g_X = 1$ and $\mathsf g_X \Lambda_{\mathrm B} \mathsf g_X = \Lambda_{\mathrm B}$, and $\Tr f_0(X) \ge \|X\|$.
\item\label{i:rep-cycle-inverse} We have $\Im\big( \Lambda_{\mathrm B} (E - S^{\mathrm{BB}}) \big) = s_E \mathsf g_X^2$. In particular $E - S^{\mathrm{BB}}$ is invertible, with
\[
|\det(E - S^{\mathrm{BB}})|^{-1} \le s_E^{-2k}, \qquad \|(E - S^{\mathrm{BB}})^{-1}\|_{\rm op} \le s_E^{-2} \big( s_E + 2 \|X\|_{\rm op} \big) .
\]
\item\label{i:rep-cycle-gauss} The body of $\Str S^2$ has real part $\|P\|^2 - 2k s_E^2 + \|M\|^2 - 2k \kappa_E^2$, and for $\theta \in [0, 1]$,
\begin{equation}\label{e:rep-translation}
\Re \Big[ -\frac12 \Tr\big( S^{\mathrm{BB}} + \iu \theta \eta \Lambda_{\mathrm B} \big)^2 \Big] = -\frac12 \|P\|^2 - \theta \eta \Tr f_0(X) + k \big( s_E^2 + \theta^2 \eta^2 \big) .
\end{equation}
\item\label{i:rep-cycle-saddle} For $Q_X = \mathsf g_X^{-1} \Lambda_{\mathrm B} \mathsf g_X$, we have $Q_X^2 = \Id$ and $-\iu \Lambda_{\mathrm B}(f_0(X) + X) = -\iu s_E Q_X$. In particular, the point of $\mathcal S$ with $P = \kappa_E \Id$ has BB block $\kappa_E \Id - \iu s_E Q_X$.
\end{enumerate}
\end{lemma}

\begin{proof}
We begin with \ref{i:rep-cycle-g}. The matrices $X^2$ and $f_0(X)$ are block diagonal, since $X$ is block off-diagonal. The identity $\Lambda_{\mathrm B} X \Lambda_{\mathrm B} = -X$ shows that the eigenvalues of $X$ come in pairs $\pm x$, and $f_0(X) + X$ has the eigenvalues $(s_E^2 + x^2)^{1/2} + x > 0$. Using
\[
\big( (s_E^2 + x^2)^{1/2} + x \big) \big( (s_E^2 + x^2)^{1/2} - x \big) = s_E^2 ,
\]
we find that $\det \mathsf g_X^2 = 1$. Moreover
\[
\Lambda_{\mathrm B} \mathsf g_X^2 \Lambda_{\mathrm B} = s_E^{-1} \big( f_0(X) - X \big) = \mathsf g_X^{-2} ,
\]
and taking positive square roots, we obtain $\Lambda_{\mathrm B} \mathsf g_X \Lambda_{\mathrm B} = \mathsf g_X^{-1}$, which is equivalent to $\mathsf g_X \Lambda_{\mathrm B} \mathsf g_X = \Lambda_{\mathrm B}$. Finally,
\[
\Tr f_0(X) = \sum_x (s_E^2 + x^2)^{1/2} \ge \Big( \sum_x x^2 \Big)^{1/2} = \|X\| .
\]

For \ref{i:rep-cycle-inverse}, we write
\[
\Lambda_{\mathrm B}(E - S^{\mathrm{BB}}) = \Lambda_{\mathrm B}(E - P) + \iu \big( f_0(X) + X \big) = \Lambda_{\mathrm B}(E - P) + \iu s_E \mathsf g_X^2 .
\]
Since $P$ commutes with $\Lambda_{\mathrm B}$, the matrix $\Lambda_{\mathrm B}(E - P)$ is Hermitian, and the formula for the imaginary part follows. The bound on the determinant follows from the inequality $|\det A| \ge \det \Im A$ of the proof of \zcref{l:rep-sdet} and $\det(s_E \mathsf g_X^2) = s_E^{2k}$.

To bound the inverse, let $A$ be a matrix with $\Im A > 0$. For all unit vectors $v$, we have
\[
\|A v\| \ge |v^* A v| \ge v^* (\Im A) v ,
\]
which implies that
\[
\|A^{-1}\|_{\rm op} \le \|(\Im A)^{-1}\|_{\rm op} .
\]
We apply this with $A = \Lambda_{\mathrm B}(E - S^{\mathrm{BB}})$, and we obtain
\[
\big\| (E - S^{\mathrm{BB}})^{-1} \big\|_{\rm op} \le s_E^{-1} \|\mathsf g_X^{-2}\|_{\rm op} = s_E^{-2} \|f_0(X) - X\|_{\rm op} \le s_E^{-2} \big( s_E + 2 \|X\|_{\rm op} \big) ,
\]
where we used $\mathsf g_X^{-2} = s_E^{-1}(f_0(X) - X)$ and $\|f_0(X)\|_{\rm op} \le s_E + \|X\|_{\rm op}$.

For \ref{i:rep-cycle-gauss}, we set $Z = f_0(X) + X - \theta \eta$, which is Hermitian, so that
\[
S^{\mathrm{BB}} + \iu \theta \eta \Lambda_{\mathrm B} = P - \iu \Lambda_{\mathrm B} Z .
\]
The trace $\Tr(P \Lambda_{\mathrm B} Z) = \Tr((\Lambda_{\mathrm B} P) Z)$ is real, since $P$ and $\Lambda_{\mathrm B}$ commute. The cross terms between the block-diagonal and the block off-diagonal components of $Z$ have trace zero, and
\[
\Lambda_{\mathrm B} Z \Lambda_{\mathrm B} = f_0(X) - \theta \eta - X .
\]
Using $\|f_0(X)\|^2 = 2k s_E^2 + \|X\|^2$, we find
\[
\begin{aligned}
\Re \Tr \big( P - \iu \Lambda_{\mathrm B} Z \big)^2 &= \|P\|^2 + \|X\|^2 - \|f_0(X) - \theta \eta\|^2 \\
&= \|P\|^2 - 2k s_E^2 + 2 \theta \eta \Tr f_0(X) - 2k \theta^2 \eta^2 .
\end{aligned}
\]
This is \eqref{e:rep-translation}, and the case $\theta = 0$ gives the BB part of the first claim. The FF block contributes
\[
-\Re \Tr(\kappa_E + \iu M)^2 = \|M\|^2 - 2k \kappa_E^2 ,
\]
and the odd entries contribute only nilpotent terms.

For \ref{i:rep-cycle-saddle}, note that part \ref{i:rep-cycle-g} implies $\Lambda_{\mathrm B} \mathsf g_X = \mathsf g_X^{-1} \Lambda_{\mathrm B}$, and therefore
\[
\Lambda_{\mathrm B} \big( f_0(X) + X \big) = s_E \Lambda_{\mathrm B} \mathsf g_X^2 = s_E \mathsf g_X^{-1} \Lambda_{\mathrm B} \mathsf g_X . \qedhere
\]
\end{proof}

The cycle is designed for \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}. For $R_i \in \mathcal S + \iu \eta_0 \Gamma$, the BB block of $\Lambda (E + \iu \eta_0 \Gamma - R_i)$ has imaginary part $s_E \mathsf g_{X_i}^2 > 0$. This makes the $\psi_i$-integral converge and shows that the inverse superdeterminant in \eqref{e:rep-chain-formal} is a smooth function on $\mathcal S$. Next, \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle} shows that the BB blocks of $\mathcal S$ include the matrices $\kappa_E - \iu s_E g \Lambda_{\mathrm B} g^{-1}$ with $g > 0$ and $g \Lambda_{\mathrm B} g = \Lambda_{\mathrm B}$, and the FF blocks of $\mathcal S$ include the matrices $\kappa_E - \iu s_E Q_{\mathrm F}$ with $Q_{\mathrm F}$ Hermitian and $Q_{\mathrm F}^2 = \Id$. These are the blocks of the critical points of the weight of the chain near which the chain concentrates, which we describe in \zcref{s:rep-heuristics}. This is the reason for the choice $s = s_E$ in $f_0$.

The integrability on the deformed cycle is delicate, and \eqref{e:rep-translation} shows why. Consider first the cycle $\mathcal S$ itself, which is the case $\theta = 0$. There the modulus of the BB part of the Gaussian factor $e^{-\Str(R^2)/2}$ is constant in $X$, so that this factor does not decay in the noncompact directions. The decay comes from the translation by $\iu \eta_0 \Gamma$, which is the case $\theta = 1$ and produces the factor $e^{-\eta \Tr f_0(X)}$. In the variables $S_i = R_i - \iu \eta_0 \Gamma$, this factor is the BB part of $|e^{-\iu \eta_0 \Str(S \Gamma)}|$, the damping factor of the weight that each site of the chain carries (\zcref{d:rep-chain}). For this reason, we deform the flat cycle directly onto the translated cycle $\mathcal S + \iu \eta_0 \Gamma$, through the cycles $\mathcal S_{\Gamma, t}$, $t \in [0, 1]$, with BB blocks
\[
P - \iu \Lambda_{\mathrm B} \big( (1 - t)(f_0(X) - \eta) + X \big), \qquad P \in \mathrm{Herm}_{\rm d}(2k), \ X \in \mathrm{Herm}_{\rm o}(2k),
\]
FF blocks $\iu M + (1 - t)(\kappa_E \Id + \iu \eta_0 \Gamma_{\mathrm F})$ with $M \in \mathrm{Herm}(2k)$, and independent odd entries. Thus $\mathcal S_{\Gamma, 1} = \mathcal S_\flat$ and $\mathcal S_{\Gamma, 0} = \mathcal S + \iu \eta_0 \Gamma$. The integral over $\mathcal S_{\Gamma, t}$ is defined as in \zcref{d:rep-cycle}, and $\mathcal S_{\Gamma, t}^N$ denotes the product cycle. We will need the following identity.

\begin{proposition}\label{p:rep-contour}
Let $\Gamma$ be a source supermatrix, and fix $\zeta_1, \dots, \zeta_N \in \C^{2k}$. Let $F_\psi(R)$ denote the integrand on the right side of \eqref{e:rep-hs-flat} with $\mathbf M = J^{-1}$ and $B_i = \psi_i \psi_i^* \Lambda$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:rep-contour-deform} For every polynomial $p$ in the entries of $R_1, \dots, R_N$, the superfunction $p F_\psi$ is integrable on $\mathcal S_{\Gamma, t}^N$ for all $t \in [0, 1]$, and its integral does not depend on $t$. In particular,
\[
\int_{\mathcal S_\flat^N} p F_\psi \dd R = \int_{(\mathcal S + \iu \eta_0 \Gamma)^N} p F_\psi \dd R .
\]
\item\label{i:rep-contour-bound} For $t \in [0, 1]$ and $R_i \in \mathcal S_{\Gamma, t}$ with BB coordinates $(P_i, X_i)$, the body of the BB part of the exponent of $F_\psi$ satisfies
\begin{multline}\label{e:rep-homotopy}
\Re \Big[ - \sum_{i, j} \frac{(J^{-1})_{ij}}{2} \Tr\big( R_i^{\mathrm{BB}} R_j^{\mathrm{BB}} \big) - \iu \sum_i \zeta_i^* \Lambda_{\mathrm B} R_i^{\mathrm{BB}} \zeta_i \Big] \\
\le \sum_{i = 1}^N \Big( -\frac12 \|P_i\|^2 - \frac{\eta}{4} \Tr f_0(X_i) + C_{k, \eta} \big( 1 + |\zeta_i|^4 \big) \Big) ,
\end{multline}
where $C_{k, \eta}$ depends only on $k$ and $\eta$.
\end{enumerate}
\end{proposition}

We prove \zcref{p:rep-contour} in \zcref{s:contour}, and we now explain the two computations on which it rests. First, at each site, the real part in \eqref{e:rep-homotopy} is
\begin{multline*}
-\frac12 \|P\|^2 - \frac{t(2 - t)}{2} \|X\|^2 - (1 - t)^2 \eta \Tr f_0(X) \\
+ (1 - t)^2 k (s_E^2 + \eta^2) - \zeta^* \big( (1 - t)(f_0(X) - \eta) + X \big) \zeta .
\end{multline*}
Here the Gaussian term $-\frac12 t(2 - t)\|X\|^2$ and the damping term $-(1 - t)^2 \eta \Tr f_0(X)$ together give decay along $X$ for all $t$. The source term is bounded by $t \zeta^* f_0(X) \zeta + \eta |\zeta|^2$, since $f_0(X) + X > 0$. Second, the link terms do not grow along the deformation. The translation by $\eta$ cancels in $R_{i + 1} - R_i$, and the identity
\[
\|X - X'\|^2 - \|f_0(X) - f_0(X')\|^2 = s_E^2 \Tr\big( \mathsf g_X^2 \mathsf g_{X'}^{-2} + \mathsf g_X^{-2} \mathsf g_{X'}^2 - 2 \Id \big) \ge 0
\]
shows that $f_0$ is a contraction for the Hilbert--Schmidt norm. It follows that the real part of the link terms is bounded by $-\frac12 W^2 \|P_{i + 1} - P_i\|^2$ on every $\mathcal S_{\Gamma, t}$. The supplementary script \texttt{rep\_cycle\_check.py} checks the identities of \zcref{l:rep-cycle} and the last identity at random points, and \texttt{rep\_homotopy\_check.py} checks the per-site identity above, the bound \eqref{e:rep-homotopy} for each site, and the bounds of \zcref{l:contour-real}.

We now compare \zcref{p:rep-contour} with earlier results. Sch\"afer and Wegner \cite{schafer1980disordered} introduced the parametrization $P - \iu \lambda T s T^{-1}$ of the bosonic Hubbard--Stratonovich variable, with $T$ in the noncompact group, which makes the integral converge while keeping the integrand bounded (see the description in \cite[p.~2]{fyodorov2008hyperbolic}). The BB part of $\mathcal S$ has this form with $\lambda = s_E$, by \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}. A rigorous version of their prescription was given by Zirnbauer \cite{zirnbauer1996riemannian}. Following \cite[Section~III.B]{zirnbauer1996riemannian}, we say that an ensemble of Hermitian matrices is of \emph{class C} if its matrices have the form $H = \big(\begin{smallmatrix} a & b \\ b^* & -a^T \end{smallmatrix}\big)$ with $a$ Hermitian and $b$ symmetric, so that $\iu H$ lies in the Lie algebra of the compact symplectic group. For the Gaussian ensemble of class C, he shows in \cite[Lemma~3.2 and Theorem~3.3]{zirnbauer1996riemannian} that the image of the Sch\"afer--Wegner parametrization is a manifold without boundary on which the Gaussian factor and the coupling to the vector variables are bounded, and he expresses the ensemble average of a generating function as a superintegral over a single supermatrix. The class of the GUE, with the noncompact group $U(n_A, n_R)$, is treated in the same way in \cite[Section~IV.G]{zirnbauer1996riemannian}. The corresponding identity for a different domain, due to Pruisken and Sch\"afer, was established by Fyodorov for $U(p, q)$ \cite{fyodorov2005hubbard} (see \cite[p.~4]{fyodorov2008hyperbolic}), by Fyodorov, Wei, and Zirnbauer for $O(p, q)$ \cite[Theorem~1]{fyodorov2008hyperbolic}, where the integral becomes a sum of integrals over subdomains with alternating signs, and by M\"uller-Hill and Zirnbauer for general noncompact groups \cite[Theorem~2.1]{mullerhill2011equivalence}. In \cite[Appendix~B]{mullerhill2011equivalence}, they also sketch an explicit deformation of the Sch\"afer--Wegner domain into the flat domain, for a single matrix variable and without odd variables, along a homotopy of a different form.

All these results concern a single matrix variable. In \cite{zirnbauer1996riemannian}, the convergence in the noncompact directions comes from the coupling to the vector variables, and \cite{fyodorov2008hyperbolic, mullerhill2011equivalence} assume that the source $A$ satisfies $A s > 0$, in the notation of \cite{fyodorov2008hyperbolic}. (In the chains of \cite{shcherbina2018universality}, the commuting variables are bosonized instead, as explained at the beginning of this section.) In contrast, \zcref{p:rep-contour} concerns $N$ matrix variables coupled by the nearest-neighbor precision \eqref{e:rep-precision}, with a source $\zeta_i \zeta_i^* \Lambda_{\mathrm B}$ at each site whose product with $\Lambda_{\mathrm B}$ has rank one. The source alone then produces decay in only some of the noncompact directions, and the decay must come from the damping $\eta$ instead. What is new in \zcref{s:contour} is the control of the link terms, through the contraction property of $f_0$, and an explicit deformation onto the translated cycle with a majorant that is uniform along the deformation. The argument is elementary, and we give it in full.

\section{The open chain}\label{s:rep-chain}

We now define the objects of the chain and state the exact representation. Recall the cycle $\mathcal S$ of \zcref{d:rep-cycle}. The superfunctions on $\mathcal S$ below are even, with values in the Grassmann algebra generated by the odd entries of their arguments, and integrals over $\mathcal S$ are taken in the sense of \zcref{d:rep-cycle}.

\begin{definition}\label{d:rep-chain}
Let $\Gamma$ be a source supermatrix.
\begin{enumerate}[label=(\alph*)]
\item\label{i:rep-chain-weight} For $S \in \mathcal S$ and a $(2k|2k)$ supermatrix $Y = \diag(Y^{\mathrm B} \mid Y^{\mathrm F})$, we define the \emph{onsite weight} and the \emph{marked onsite weight}
\begin{equation}\label{e:rep-weight}
\begin{gathered}
 w_\Gamma(S) = e^{-\Str(S^2)/2 - \iu \eta_0 \Str(S \Gamma)} \Sdet(E - S)^{-1} , \\
 w_{\Gamma, Y}(S) = e^{-\Str(S^2)/2 - \iu \eta_0 \Str(S \Gamma)} \Sdet(E - S + Y)^{-1} ,
\end{gathered}
\end{equation}
the second for $Y$ in a neighborhood of zero that may depend on $S$. We let $w_0$ denote $w_\Gamma$ with $\Gamma = 0$, so that $w_\Gamma = w_0 e^{-\iu \eta_0 \Str(S \Gamma)}$.
\item\label{i:rep-chain-heat} For $t > 0$ and $S, S' \in \mathcal S$, we define the \emph{heat kernel} $k_t(S, S') = e^{-\Str(S - S')^2/(4t)}$, and we let $K_t$ denote the operator
\[
(K_t f)(S) = \int_{\mathcal S} k_t(S, S') f(S') \dd S' ,
\]
where the Berezin integral is over the odd entries of $S'$. We set $\tau_{\rm phys} = (2W^2)^{-1}$, so that $k_{\tau_{\rm phys}}(S, S') = e^{-W^2 \Str(S - S')^2/2}$ is the link factor of \eqref{e:rep-chain-formal}.
\item\label{i:rep-chain-transfer} We define the \emph{transfer operator} $T_\Gamma = M_{w_\Gamma} K_{\tau_{\rm phys}}$, where $M_w$ denotes multiplication by $w$, so that
\begin{equation}\label{e:rep-transfer}
(T_\Gamma f)(S) = w_\Gamma(S) \int_{\mathcal S} k_{\tau_{\rm phys}}(S, S') f(S') \dd S' ,
\end{equation}
so that the heat link acts first and the onsite weight is applied after it. We define its \emph{analytic transpose} $T_\Gamma^{\rm tr} = K_{\tau_{\rm phys}} M_{w_\Gamma}$, which applies the weight first. For integers $m \ge 1$, we define the \emph{partial chains}
\[
f_m = T_\Gamma^m 1, \qquad \ell_m = (T_\Gamma^{\rm tr})^m 1 .
\]
We say that $\ell_m$, and more generally a product of steps $T_\Gamma^{\rm tr}$ with insertions, is built in the \emph{analytic transpose order}.
\item\label{i:rep-chain-pairing} For superfunctions $f$ and $g$ on $\mathcal S$ such that $f g$ is integrable on $\mathcal S$, we define the bilinear pairing $\mathcal B_{\mathcal S}(f, g) = \int_{\mathcal S} f(S) g(S) \dd S$.
\item\label{i:rep-chain-partition} For a superfunction $F$ on $\mathcal S^N$, we define the \emph{chain integral}
\begin{equation}\label{e:rep-partition}
\mathcal Z_N(\Gamma; F) = \int_{\mathcal S^N} F(S_1, \dots, S_N) \prod_{i = 1}^N w_\Gamma(S_i) \prod_{i = 1}^{N - 1} k_{\tau_{\rm phys}}(S_i, S_{i + 1}) \dd S ,
\end{equation}
whenever the integrand is integrable, and we write $\mathcal Z_N(\Gamma) = \mathcal Z_N(\Gamma; 1)$ for the chain integral of the open chain without insertion.
\end{enumerate}
\end{definition}

The superdeterminant in \eqref{e:rep-weight} is defined on all of $\mathcal S$. Indeed, let $A, B, C, D$ denote the blocks of $E - S$. By \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}, the block $A$ is invertible. Using \zcref{l:grass-sdet}\ref{i:grass-sdet-block}, we obtain
\[
\Sdet(E - S)^{-1} = \frac{\det(D - C A^{-1} B)}{\det A} .
\]
The right side is a polynomial in the FF and odd entries of $S$ and in the entries of $A^{-1}$, divided by $\det A$. The same holds for $w_{\Gamma, Y}$ when $\|Y^{\mathrm B}\|_{\rm op}$ is smaller than $s_E$ times the smallest eigenvalue of $\mathsf g_X^2$, and the derivatives of $w_{\Gamma, Y}$ in $Y$ at $Y = 0$ are defined at every point of $\mathcal S$. We will need three facts about the heat kernel, which we prove in \zcref{s:contour}.

\begin{lemma}\label{l:rep-heat}
Let $t, t' > 0$. For all $S, S'' \in \mathcal S$, the superfunctions $k_t(S, \cdot)$ and $k_t(S, \cdot) k_{t'}(\cdot, S'')$ are integrable on $\mathcal S$, and
\[
(K_t 1)(S) = 1, \qquad \int_{\mathcal S} k_t(S, S') k_{t'}(S', S'') \dd S' = k_{t + t'}(S, S'') .
\]
Moreover, $k_t(S, S') = k_t(S', S)$, and the body of $k_t(S, S')$ has modulus at most $e^{-(\|P - P'\|^2 + \|M - M'\|^2)/(4t)}$, where $(P, X, M)$ and $(P', X', M')$ are the coordinates of $S$ and $S'$.
\end{lemma}

Since $\dd S$ is normalized by \eqref{e:rep-flat-normalization}, the identity $K_t 1 = 1$ holds without a constant factor, for all $t > 0$ and all $W$; the reason is the cancellation of determinants after \eqref{e:rep-hs-flat}, with $N = 1$ and $\mathbf M = (2t)^{-1}$.

We can now state the exact representation.

\begin{proposition}\label{p:rep-exact}
Let $\Gamma$ be a source supermatrix.
\begin{enumerate}[label=(\alph*)]
\item\label{i:rep-exact-chain} The integrand of $\mathcal Z_N(\Gamma)$ is integrable on $\mathcal S^N$, and
\begin{equation}\label{e:rep-exact}
T_N^{(k)}(\Gamma) = e^{N \eta_0^2 \Str(\Gamma^2)/2} \, \mathcal Z_N(\Gamma) .
\end{equation}
\item\label{i:rep-exact-transfer} The partial chains $f_m$ and $\ell_m$ are defined by absolutely convergent integrals, $\ell_m = K_{\tau_{\rm phys}} f_m$, and for all integers $1 \le m \le N/2$,
\begin{equation}\label{e:rep-pairing}
\mathcal Z_N(\Gamma) = \mathcal B_{\mathcal S}(1, T_\Gamma^N 1) = \mathcal B_{\mathcal S}\big( \ell_m, T_\Gamma^{N - 2m} f_m \big) .
\end{equation}
Moreover, $\mathcal B_{\mathcal S}(T_\Gamma^{\rm tr} f, g) = \mathcal B_{\mathcal S}(f, T_\Gamma g)$ whenever both sides are given by absolutely convergent iterated integrals.
\item\label{i:rep-exact-marks} Let $Y$ be local sources. Then $T_N^{(k)}(\Gamma; Y)$ is holomorphic in the entries of $Y$ for $\|Y^{\mathrm B}\|_{\rm op} < \eta$, and for every multi-index $\alpha = (\alpha_1, \dots, \alpha_N)$, where $\alpha_i$ is a multi-index in the entries of $Y_i$,
\begin{equation}\label{e:rep-exact-marks}
\partial_Y^\alpha T_N^{(k)}(\Gamma; Y) \big|_{Y = 0} = e^{N \eta_0^2 \Str(\Gamma^2)/2} \int_{\mathcal S^N} \prod_{i = 1}^N \big( \partial_{Y_i}^{\alpha_i} w_{\Gamma, Y_i}(S_i) \big|_{Y_i = 0} \big) \prod_{i = 1}^{N - 1} k_{\tau_{\rm phys}}(S_i, S_{i + 1}) \dd S ,
\end{equation}
with an integrable integrand. If $Y^{\mathrm B} = 0$, then $T_N^{(k)}(\Gamma; Y)$ equals the right side of \eqref{e:rep-exact-marks} with $\alpha = 0$ and $w_{\Gamma, Y_i}(S_i)$ in place of the derivatives, for all FF blocks $Y_i^{\mathrm F}$.
\end{enumerate}
In particular,
\begin{equation}\label{e:rep-normalization}
\mathcal Z_N(\Gamma_0) = 1, \qquad T_N^{(k)}(\Gamma) = e^{N \eta_0^2 \Str(\Gamma^2)/2} \, \frac{\mathcal Z_N(\Gamma)}{\mathcal Z_N(\Gamma_0)} .
\end{equation}
\end{proposition}

Later sections use only the ratio form in \eqref{e:rep-normalization}, which holds for every normalization of $\dd S$, and its prefactor tends to one together with its derivatives in the sources, since $N \eta_0^2 = (N \rho(E)^2)^{-1}$.

\begin{proof}
We begin with \ref{i:rep-exact-marks}, which contains \ref{i:rep-exact-chain} as the case $\alpha = 0$, $Y = 0$. Holomorphy was shown in \zcref{l:rep-sdet}. In \eqref{e:rep-quartic}, the local sources appear only through the factor $e^{\iu \sum_i \psi_i^* \Lambda Y_i \psi_i}$. For $\|Y^{\mathrm B}\|_{\rm op} \le \eta/2$, the body of the integrand is bounded by $e^{-\eta |\zeta|^2/2}$ times polynomials in $\zeta$, uniformly in $Y$, so we may differentiate under the integral. We obtain
\begin{equation}\label{e:rep-marks-psi}
\partial_Y^\alpha T_N^{(k)}(\Gamma; Y) \big|_{Y = 0} = \int p_\alpha(\psi) \exp\Big( \iu \sum_{i} \psi_i^* \Lambda (E + \iu \eta_0 \Gamma) \psi_i - \frac12 \sum_{i, j} J_{ij} \Str\big( \psi_i \psi_i^* \Lambda \, \psi_j \psi_j^* \Lambda \big) \Big) \dd \psi ,
\end{equation}
where
\[
p_\alpha(\psi) = \partial_Y^\alpha e^{\iu \sum_i \psi_i^* \Lambda Y_i \psi_i} \big|_{Y = 0}
\]
is a polynomial in $\psi$.

For fixed $\psi$, we first apply \eqref{e:rep-hs-flat} with $\mathbf M = J^{-1}$ and $B_i = \psi_i \psi_i^* \Lambda$, and we then deform the cycle by \zcref{p:rep-contour}\ref{i:rep-contour-deform} with $p = 1$. Using \eqref{e:rep-str-vector}, we find
\begin{equation}\label{e:rep-marks-double}
\partial_Y^\alpha T_N^{(k)}(\Gamma; Y) \big|_{Y = 0} = \int \dd \psi \int_{(\mathcal S + \iu \eta_0 \Gamma)^N} p_\alpha(\psi) \, e^{\iu \sum_i \psi_i^* \Lambda (E + \iu \eta_0 \Gamma - R_i) \psi_i} \, e^{-\frac12 \sum_{i, j} (J^{-1})_{ij} \Str(R_i R_j)} \dd R .
\end{equation}

We next show that the integrand in \eqref{e:rep-marks-double} is integrable on the product, so that the two integrals may be exchanged. Let $R_i = S_i + \iu \eta_0 \Gamma$ with $S_i \in \mathcal S$. The body of the BB part of the $\psi_i$-exponent has real part
\[
-\zeta_i^* \Im\big( \Lambda_{\mathrm B}(E - S_i^{\mathrm{BB}}) \big) \zeta_i = -s_E \, \zeta_i^* \mathsf g_{X_i}^2 \zeta_i ,
\]
by \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}. The coefficients of the integrand are bounded by polynomials in $\zeta$, in the coordinates of the $S_i$ and in the entries of $\eta_0 \Gamma$, times
\[
\exp\Big( -s_E \sum_i \zeta_i^* \mathsf g_{X_i}^2 \zeta_i \Big) \, \Big| e^{-\frac12 \sum_{i, j} (J^{-1})_{ij} \Str(R_i R_j)} \Big|_\emptyset ,
\]
where $| \cdot |_\emptyset$ denotes the modulus of the body. For fixed $X_i$, the integral of the first factor times a polynomial of degree $2q$ in $\zeta_i$ is at most
\[
C_q s_E^{-q} \|\mathsf g_{X_i}^{-2}\|_{\rm op}^q \det(s_E \mathsf g_{X_i}^2)^{-1} \le C_q s_E^{-2q - 2k} \big( s_E + 2\|X_i\|_{\rm op} \big)^q ,
\]
where we used \zcref{l:rep-cycle}.

We now bound the second factor. The case $t = 0$, $\zeta = 0$ of \eqref{e:rep-homotopy} implies that its BB part is bounded by
\[
\prod_{i = 1}^N \exp\Big( -\frac12 \|P_i\|^2 - \frac{\eta}{4} \Tr f_0(X_i) + C_{k, \eta} \Big) .
\]
Its FF part is a Gaussian in the $M_i$ times $e^{C |M|}$, since $J^{-1} \ge \Id$. The polynomial factors in $X_i$ of the previous paragraph are dominated by $e^{-\eta \Tr f_0(X_i)/4}$, since $\Tr f_0(X_i) \ge \|X_i\|$. Altogether, the integrand is integrable on the product.

Since the supervectors $\psi_i$ have $4kN$ odd coordinates in all and the supermatrices $R_i$ have $8k^2 N$, both even numbers, \zcref{l:grass-berezin}\ref{i:grass-berezin-fubini} shows that the two Berezin integrals may be taken in either order without a sign, and Fubini's theorem allows us to integrate over $\psi$ first.

We now integrate over $\psi$. At the site $i$, \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse} yields
\[
\Re \big[ -\iu \Lambda_{\mathrm B} \big( E + \iu \eta_0 \Gamma + Y_i - R_i \big)^{\mathrm{BB}} \big] = s_E \mathsf g_{X_i}^2 + \Im\big( \Lambda_{\mathrm B} Y_i^{\mathrm B} \big) ,
\]
which is positive definite for $Y_i$ near zero. Applying \zcref{p:grass-supergauss} and differentiating under the $\psi$-integral, we obtain
\[
\int p_\alpha(\psi) \, e^{\iu \sum_i \psi_i^* \Lambda (E + \iu \eta_0 \Gamma - R_i) \psi_i} \dd \psi = \prod_{i = 1}^N \partial_{Y_i}^{\alpha_i} \Sdet\big( E + \iu \eta_0 \Gamma + Y_i - R_i \big)^{-1} \big|_{Y_i = 0} .
\]

It remains to substitute $R_i = S_i + \iu \eta_0 \Gamma$, under which $\int_{(\mathcal S + \iu \eta_0 \Gamma)^N} \dd R$ becomes $\int_{\mathcal S^N} \dd S$ by \zcref{d:rep-cycle}. The links are unchanged by \eqref{e:rep-precision}, and
\[
\frac12 \sum_{i = 1}^N \Str(S_i + \iu \eta_0 \Gamma)^2 = \frac12 \sum_{i = 1}^N \Str S_i^2 + \iu \eta_0 \sum_{i = 1}^N \Str(S_i \Gamma) - \frac{N \eta_0^2}{2} \Str \Gamma^2 ,
\]
where we used $\Str(S_i \Gamma) = \Str(\Gamma S_i)$. Collecting the factors yields \eqref{e:rep-exact-marks}, whose integrand is integrable by the bounds above.

Now let $Y^{\mathrm B} = 0$. The FF block of $E + \iu \eta_0 \Gamma + Y - H$ appears in the superdeterminant only through the determinant of a Schur complement. Each side of the claimed identity is therefore a polynomial in the entries of the $Y_i^{\mathrm F}$, and their Taylor coefficients at zero agree by \eqref{e:rep-exact-marks}. This proves \ref{i:rep-exact-marks} and \ref{i:rep-exact-chain}.

We next prove \ref{i:rep-exact-transfer}. Recall that $K_{\tau_{\rm phys}} 1 = 1$ by \zcref{l:rep-heat}, so that $T_\Gamma 1 = w_\Gamma$. Using \eqref{e:rep-transfer} and induction, we find that
\[
f_m(S_1) = w_\Gamma(S_1) \int_{\mathcal S^{m - 1}} \prod_{i = 2}^m w_\Gamma(S_i) \prod_{i = 1}^{m - 1} k_{\tau_{\rm phys}}(S_i, S_{i + 1}) \dd S_2 \cdots \dd S_m
\]
has $m$ weights and $m - 1$ links. Moreover,
\[
(T_\Gamma^{\rm tr})^m 1 = K_{\tau_{\rm phys}} (M_{w_\Gamma} K_{\tau_{\rm phys}})^{m - 1} w_\Gamma = K_{\tau_{\rm phys}} T_\Gamma^m 1 ,
\]
which shows that $\ell_m = K_{\tau_{\rm phys}} f_m$ has one more link.

The integrand of each of these iterated integrals is a factor of the integrand of $\mathcal Z_N(\Gamma)$, possibly multiplied by one more link, whose body has modulus at most one by \zcref{l:rep-heat}. The bounds in the proof of \ref{i:rep-exact-chain}, with $\zeta = 0$, show that it is integrable for each value of the free variable. The function $T_\Gamma^{N - 2m} f_m$ has $N - m$ weights and $N - m - 1$ links, so its pairing with $\ell_m$ has $N$ weights and $N - 1$ links, arranged as in \zcref{f:rep-chain}. Each site carries $8k^2$ odd coordinates, an even number, so that the Berezin integrals over different sites may be taken in any order, and Fubini's theorem identifies both pairings in \eqref{e:rep-pairing} with \eqref{e:rep-partition} for $F = 1$.

The transpose identity follows from Fubini's theorem and the symmetry $k_{\tau_{\rm phys}}(S, S') = k_{\tau_{\rm phys}}(S', S)$, without signs, since all superfunctions involved are even. Finally, recall from \zcref{l:rep-sdet} that $T_N^{(k)}(\Gamma_0) = 1$ and $\Str \Gamma_0^2 = 0$. Inserting these into \eqref{e:rep-exact} at $\Gamma = \Gamma_0$, we conclude that $\mathcal Z_N(\Gamma_0) = 1$, and \eqref{e:rep-normalization} follows.
\end{proof}

\section{The saddle manifold}\label{s:rep-heuristics}

We now describe the critical points of the onsite weight, around which Part IV analyzes the chain. Recall the onsite weight $w_\Gamma$ of \eqref{e:rep-weight}, and write $w_\Gamma(S) = e^{-\mathcal F(S) - \iu \eta_0 \Str(S \Gamma)}$ with $\mathcal F(S) = \frac12 \Str S^2 + \Str \log(E - S)$. The critical points of $\mathcal F$ solve $S = (E - S)^{-1}$, or equivalently $S^2 - E S + 1 = 0$. Let $Q$ be an even supermatrix of size $(2k|2k)$ with $Q^2 = \Id$, and let $S = \kappa_E - \iu s_E Q$. Then $E - S = \kappa_E + \iu s_E Q$, and $(\kappa_E + \iu s_E Q)(\kappa_E - \iu s_E Q) = \kappa_E^2 + s_E^2 = 1$, so that
\begin{equation}\label{e:rep-saddle-inverse}
(E - S)^{-1} = \kappa_E - \iu s_E Q = S .
\end{equation}
Every such supermatrix $S$ is therefore a critical point of $\mathcal F$.

For $p \in \{0, \dots, 2k\}$, let $R_*^{(p)}$ denote the diagonal supermatrix with BB block $\kappa_E - \iu s_E \Lambda_{\mathrm B}$ and FF block $\kappa_E - \iu s_E \diag(\Id_p, -\Id_{2k - p})$, which is of this form. The supermatrices $V R_*^{(p)} V^{-1}$, where $V$ is an even invertible supermatrix of size $(2k|2k)$ with entries in a Grassmann algebra, form $2k + 1$ complex orbits, indexed by the \emph{signature} $p$, and we call their union the \emph{saddle manifold} (see \zcref{s:tube-geometry}, where $p$ is fixed and $R_*^{(p)}$ is written $R_*$). The orbits of signature $p \ne k$ have more commuting than anticommuting normal directions, and their contributions to the chain become exponentially small in $W$ after a macroscopic number of steps, as explained in \zcref{s:ov-averaging} and proved in \zcref{l:asm-unbalanced}. The limit of the chain is therefore carried by the complex orbit of $\kappa_E - \iu s_E \Lambda$, the orbit of signature $p = k$.

We call the \emph{balanced component} of the saddle manifold the real form of this orbit whose body is the set of matrices
\[
 \kappa_E - \iu s_E \diag\big( g \Lambda_{\mathrm B} g^{-1} \mid u \Lambda_{\mathrm F} u^{-1} \big) ,
\]
where $g$ ranges over the complex $2k \times 2k$ matrices with $g^* \Lambda_{\mathrm B} g = \Lambda_{\mathrm B}$ and $u$ over the unitary $2k \times 2k$ matrices. Here the body of the orbit is the complex manifold of the matrices of this form with $g, u \in \mathrm{GL}(2k, \C)$, since the body of $V R_*^{(k)} V^{-1}$ is the conjugate of $R_*^{(k)}$ by the body of $V$. By a \emph{real form} of a complex manifold we mean, as for $\R^n$ in $\C^n$ or the unitary group in $\mathrm{GL}(n, \C)$, a real submanifold whose tangent space at each point spans the complex tangent space over $\C$ and has half its real dimension. The matrices with the above restrictions on $g$ and $u$ form such a submanifold. At the point $\kappa_E - \iu s_E \Lambda$, for instance, the tangent space of the orbit consists of the matrices $-\iu s_E \diag(B \mid C)$ with $B$ and $C$ complex and block off-diagonal, and that of the submanifold of those with $B$ anti-Hermitian and $C$ Hermitian; at the other points, this follows by conjugation.

The cycle $\mathcal S$ passes through this body, by \zcref{l:rep-cycle}\ref{i:rep-cycle-saddle}. In \zcref{s:sigma}, we describe the balanced component, together with its odd directions, as a homogeneous superspace: its points are the supermatrices $\kappa_E - \iu s_E Q$, where $Q$ is the field on the target space $\mathcal M_k$ of the sigma model, and the odd elements of $\mathfrak{gl}(2k|2k)$ act on it infinitesimally. Since the supertrace and the superdeterminant are invariant under conjugation, by \zcref{l:grass-sdet}, the function $\mathcal F$ is constant on each component of the saddle manifold, and on the balanced component the onsite weight varies only through the small source term.

The links $e^{-W^2 \Str(S_{i + 1} - S_i)^2/2}$ force neighboring variables to be close. Heuristically, the chain therefore stays near the saddle manifold, its normal fluctuations relax over about $W$ sites, and its tangent variables on the balanced component perform a random walk whose scaling limit is the heat semigroup on $\mathcal M_k$ of \zcref{s:sigma}. This heuristic is explained in \zcref{s:ov-scales,s:sigma-heuristic}, and Part IV makes it rigorous.

\section{Symmetry and the source factor}\label{s:rep-symmetry}

We record the symmetries of the weights and the size of the source factor, which Part IV uses in the tube around the saddle manifold.

\begin{remark}\label{r:rep-symmetry}
We let $\mathrm{GL}(2k|2k)$ denote the group of even invertible $(2k|2k)$ supermatrices whose entries lie in a Grassmann algebra, and for an even supermatrix $A$ we let the stabilizer of $A$ be the subgroup of $V \in \mathrm{GL}(2k|2k)$ with $V A = A V$. For $V \in \mathrm{GL}(2k|2k)$ and even supermatrices $S$ and $S'$, \zcref{l:grass-sdet}\ref{i:grass-sdet-mult} and \ref{i:grass-sdet-trace} give
\[
\begin{gathered}
 \Str (V S V^{-1})^2 = \Str S^2 , \\
 \Sdet\big( E - V S V^{-1} \big) = \Sdet(E - S) , \\
 \Str\big( V (S - S') V^{-1} \big)^2 = \Str(S - S')^2 .
\end{gathered}
\]
It follows that $w_0(V S V^{-1}) = w_0(S)$ and $k_t(V S V^{-1}, V S' V^{-1}) = k_t(S, S')$, wherever both sides are defined, and $\Str(V S V^{-1} \Gamma) = \Str(S \Gamma)$ if $V$ commutes with $\Gamma$. By \eqref{e:rep-saddle-inverse}, the critical points of $w_0$ include the supermatrices $\kappa_E - \iu s_E Q$ with $Q^2 = \Id$, and the stabilizer of $\kappa_E - \iu s_E \Lambda$ consists of the $V \in \mathrm{GL}(2k|2k)$ that commute with $\Lambda$. These are algebraic identities. Conjugation does not preserve the cycle $\mathcal S$, and invariance statements for integrals over $\mathcal S$ need a separate argument. Conjugation by an element of $\mathrm{GL}(2k|2k)$ with body $\diag(g \mid u)$, where $g^* \Lambda_{\mathrm B} g = \Lambda_{\mathrm B}$ and $u$ is unitary, preserves the body of the balanced component, defined in \zcref{s:rep-heuristics}, and such an element fixes the point $\kappa_E - \iu s_E \Lambda$ if and only if it commutes with $\Lambda$.
\end{remark}

The BB part of the source factor $w_\Gamma/w_0 = e^{-\iu \eta_0 \Str(S \Gamma)}$ is real and at most one on the body of the balanced component of the saddle manifold. Part IV also uses bounds on its derivatives there. Since these are stated in the geometry of \zcref{s:sigma}, we give them in \zcref{l:rep-kick} of \zcref{s:contour-kick}.

\section{The anchored chain}\label{s:rep-anchored}

In Part IV, we need a priori bounds on the partial chains $\ell_n = K_{\tau_{\rm phys}} T_\Gamma^n 1$, as functions of their free variable (see \zcref{s:hist}). The partial chain $\ell_n$ is a chain of $n$ sites whose first site is linked to a fixed point $S$ of the cycle, and we say that it is \emph{anchored} at $S$. The Gaussian part of a partial chain can be integrated as in \zcref{s:rep-decoupling}, read backward. The result is an average over an $n \times n$ Gaussian matrix of an inverse superdeterminant in which $S$ appears linearly. To state it, we let $\Delta_n$ denote the Neumann Laplacian on $[n]$, and we set
\begin{equation}\label{e:rep-anchored-cov}
J_n = \big( \Id_n - W^2 \Delta_n + W^2 e_1 e_1^* \big)^{-1}, \qquad \varsigma = W^2 J_n e_1 \in \R^n, \qquad \sigma_n = \sum_{i = 1}^n \varsigma_i .
\end{equation}
The matrix $J_n^{-1}$ is the precision of the chain on $[n]$ with an additional link of weight $W^2$ from the site $1$ to the fixed point. It has positive diagonal, nonpositive off-diagonal entries, and positive row sums, so that all entries of $J_n$ are positive. Since $J_n^{-1} \1 = \1 + W^2 e_1$, we have
\begin{equation}\label{e:rep-anchored-identities}
J_n \1 = \1 - \varsigma, \qquad \sigma_n = W^2 (1 - \varsigma_1), \qquad \1^T J_n \1 = n - \sigma_n, \qquad W^2 - W^4 (J_n)_{11} = \sigma_n ,
\end{equation}
and in particular $0 < \varsigma_i < 1$ for all $i$. We let $H^{(n)}$ denote the $n \times n$ complex Hermitian Gaussian matrix with $\E H^{(n)}_{ij} H^{(n)}_{kl} = \delta_{il} \delta_{jk} (J_n)_{ij}$, defined as in \zcref{d:model}.

\begin{proposition}\label{p:rep-anchored}
Let $\Gamma$ be a source supermatrix, let $n \ge 1$, and set $\ell_n = K_{\tau_{\rm phys}} T_\Gamma^n 1$. Then for all $S \in \mathcal S$,
\begin{multline}\label{e:rep-anchored}
\ell_n(S) = \exp\Big( -\frac{\sigma_n}{2} \Str S^2 - \iu \eta_0 \sigma_n \Str(S \Gamma) - \frac{\eta_0^2 (n - \sigma_n)}{2} \Str \Gamma^2 \Big) \\
\times \E \Sdet\Big( E - H^{(n)} - \varsigma \otimes S + \iu \eta_0 (\1 - \varsigma) \otimes \Gamma \Big)^{-1} ,
\end{multline}
where $\varsigma \otimes S$ denotes the family $(\varsigma_i S)_{i \in [n]}$, in the notation of \zcref{s:rep-sources}. More generally, let $\ell_n^{(\alpha)}$ denote the function defined by the same iterated integral as $\ell_n$, with the weights $w_\Gamma(S_i)$ at the sites of the chain replaced by the derivatives $\partial_{Y_i}^{\alpha_i} w_{\Gamma, Y_i}(S_i)|_{Y_i = 0}$, where $S_1$ is the site linked to $S$. Then $\ell_n^{(\alpha)}(S)$ is the derivative $\partial_Y^\alpha$ at $Y = 0$ of the right side of \eqref{e:rep-anchored} with $Y_i$ added to the $i$th diagonal block of the supermatrix in the superdeterminant.
\end{proposition}

The supermatrix in \eqref{e:rep-anchored} has the blocks
\[
\delta_{ij} \big( E - \varsigma_i S + \iu \eta_0 (1 - \varsigma_i) \Gamma \big) - H^{(n)}_{ij} \Id, \qquad i, j \in [n] .
\]
The BB block of this supermatrix is invertible. Indeed, after multiplication by $\Id_n \otimes \Lambda_{\mathrm B}$, its body has imaginary part $\diag_i(\varsigma_i s_E \mathsf g_X^2 + (1 - \varsigma_i) \eta) > 0$ by \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}, where $X$ is the coordinate of $S$. The same argument as in \zcref{l:rep-sdet} shows that the expectation is finite.

\begin{proof}
Recall from \zcref{d:rep-chain} that $\ell_n(S)$ is the integral over $(S_1, \dots, S_n) \in \mathcal S^n$ of
\[
k_{\tau_{\rm phys}}(S, S_1) \prod_{i = 1}^n w_\Gamma(S_i) \prod_{i = 1}^{n - 1} k_{\tau_{\rm phys}}(S_i, S_{i + 1}) .
\]
We first write the superdeterminants as Gaussian integrals. Using \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse} and \zcref{p:grass-supergauss}, we have
\[
\Sdet(E - S_i)^{-1} = \int e^{\iu \psi_i^* \Lambda (E - S_i) \psi_i} \dd \psi_i
\]
on $\mathcal S$. The bounds in the proof of \zcref{p:rep-exact}, with the damping $e^{-\eta \Tr f_0(X_i)}$ of the weights in place of \eqref{e:rep-homotopy}, show that the resulting integrand is integrable on $\mathcal S^n$ times the space of the $\psi_i$. As in the proof of \zcref{p:rep-exact}, the numbers of odd coordinates involved are even, and using Fubini's theorem, we obtain
\[
\ell_n(S) = \int \ell_n(S; \psi) \dd \psi ,
\]
where
\begin{multline*}
\ell_n(S; \psi) = \int_{\mathcal S^n} \exp\Big( -\frac12 \sum_{i, j = 1}^n (J_n^{-1})_{ij} \Str(S_i S_j) + \sum_{i = 1}^n \Str(S_i b_i) - \frac{W^2}{2} \Str S^2 \Big) \dd S_1 \cdots \dd S_n \\
\times e^{\iu \sum_i \psi_i^* \Lambda E \psi_i}
\end{multline*}
and
\[
b_i = W^2 \delta_{i1} S - \iu \big( \psi_i \psi_i^* \Lambda + \eta_0 \Gamma \big) .
\]
Here we expanded $-\frac{W^2}{2} \Str(S_1 - S)^2$, and we used \eqref{e:rep-str-vector} and the analogue of \eqref{e:rep-precision} on $[n]$.

We next compute $\ell_n(S; \psi)$ for fixed $\psi$. The BB part of its exponent is of the form treated in \zcref{l:contour-deform}, with unit masses, the damping $\eta$ coming from $-\iu \eta_0 \Str(S_i \Gamma)$, and one link to the fixed point $S$. That lemma deforms each $S_i$ from $\mathcal S$ onto the cycle with Hermitian BB block and FF block in $\kappa_E \Id + \iu \mathrm{Herm}(2k)$, which is a translate of $\mathcal S_\flat$ (see \zcref{s:contour-anchored}). Completing the square on this cycle and applying \zcref{l:contour-flat}, we find
\[
\ell_n(S; \psi) = \exp\Big( \frac12 \sum_{i, j} (J_n)_{ij} \Str(b_i b_j) - \frac{W^2}{2} \Str S^2 \Big) e^{\iu \sum_i \psi_i^* \Lambda E \psi_i} .
\]

We now expand $\frac12 \sum_{i, j} (J_n)_{ij} \Str(b_i b_j)$. Together with the term $-\frac12 W^2 \Str S^2$ of $\ell_n(S; \psi)$, the terms with $S$ only, those with $S$ and $\Gamma$, and those with $\Gamma$ only give
\[
\begin{gathered}
\frac{W^4}{2} (J_n)_{11} \Str S^2 - \frac{W^2}{2} \Str S^2 = -\frac{\sigma_n}{2} \Str S^2 , \\
-\iu \eta_0 W^2 (J_n \1)_1 \Str(S \Gamma) = -\iu \eta_0 \sigma_n \Str(S \Gamma) , \\
-\frac{\eta_0^2}{2} \big( \1^T J_n \1 \big) \Str \Gamma^2 = -\frac{\eta_0^2 (n - \sigma_n)}{2} \Str \Gamma^2 ,
\end{gathered}
\]
where we used \eqref{e:rep-anchored-identities}. Using $(J_n \1)_i = 1 - \varsigma_i$ and \eqref{e:rep-str-vector}, we find that the remaining terms are
\[
-\frac12 \sum_{i, j} (J_n)_{ij} \Str\big( \psi_i \psi_i^* \Lambda \psi_j \psi_j^* \Lambda \big) - \iu \sum_i \varsigma_i \, \psi_i^* \Lambda S \psi_i - \eta_0 \sum_i (1 - \varsigma_i) \, \psi_i^* \Lambda \Gamma \psi_i .
\]
The computation in the proof of \zcref{l:rep-spinor}, read backward with $H^{(n)}$ in place of $H$, shows that
\[
\exp\Big( -\frac12 \sum_{i, j} (J_n)_{ij} \Str\big( \psi_i \psi_i^* \Lambda \psi_j \psi_j^* \Lambda \big) \Big) = \E \exp\Big( -\iu \sum_{i, j} H^{(n)}_{ij} \psi_i^* \Lambda \psi_j \Big) .
\]
Altogether, $\ell_n(S; \psi)$ is the Gaussian prefactor in \eqref{e:rep-anchored} times
\[
\E \exp\Big( \iu \sum_i \psi_i^* \Lambda \big( E - \varsigma_i S + \iu \eta_0 (1 - \varsigma_i) \Gamma \big) \psi_i - \iu \sum_{i, j} H^{(n)}_{ij} \psi_i^* \Lambda \psi_j \Big) .
\]
By \zcref{l:rep-cycle}\ref{i:rep-cycle-inverse}, the body of the exponent has the real part
\[
-\sum_{i = 1}^n \zeta_i^* \big( \varsigma_i s_E \mathsf g_X^2 + (1 - \varsigma_i) \eta \big) \zeta_i ,
\]
which is independent of $H^{(n)}$.

We may now exchange the expectation and the $\psi$-integral, and \eqref{e:rep-anchored} follows from \zcref{p:grass-supergauss}.

For the marked version, we argue as in the proof of \zcref{p:rep-exact}\ref{i:rep-exact-marks}. Differentiating under the integral at fixed $S_i \in \mathcal S$, we obtain
\[
\partial_{Y_i}^{\alpha_i} \Sdet(E - S_i + Y_i)^{-1} \big|_{Y_i = 0} = \int p_{\alpha_i}(\psi_i) \, e^{\iu \psi_i^* \Lambda (E - S_i) \psi_i} \dd \psi_i ,
\]
with the polynomial $p_{\alpha_i}$ of \eqref{e:rep-marks-psi}. This polynomial does not involve $S_1, \dots, S_n$, and the argument above applies with the extra factor $\prod_i p_{\alpha_i}(\psi_i)$. With $Y_i$ added to the $i$th diagonal block, the matrix $\varsigma_i s_E \mathsf g_X^2 + (1 - \varsigma_i) \eta$ in the real part of the exponent above becomes $\varsigma_i s_E \mathsf g_X^2 + (1 - \varsigma_i) \eta + \Im(\Lambda_{\mathrm B} Y_i^{\mathrm B})$, which is at least $((1 - \varsigma_i) \eta - \|Y_i^{\mathrm B}\|_{\rm op}) \Id$. For $\|Y_i^{\mathrm B}\|_{\rm op} \le (1 - \varsigma_i) \eta/2$, the coefficients of the integrand are therefore bounded, uniformly in $Y$, by polynomials in $\zeta$ and in the entries of $H^{(n)}$ times $e^{-\sum_i (1 - \varsigma_i) \eta |\zeta_i|^2/2}$. This allows us to differentiate under the $\psi$-integral, and the claim follows.
\end{proof}

\section{Resolvent entries and insertions}\label{s:rep-dictionary}

We now state the dictionary between the resolvent $G(z) = (H - z)^{-1}$ and the chain. Derivatives of $T_N^{(k)}(\Gamma; Y)$ in the local sources are expectations of polynomials in resolvent entries. In the chain, they are integrals with polynomials in the entries of $(E - S_i)^{-1}$ inserted at the marked sites.

Let $\Gamma$ be a source supermatrix with diagonal FF block. We let $z_1^{\mathrm F}, \dots, z_{2k}^{\mathrm F}$ denote the diagonal entries of $E + \iu \eta_0 \Gamma_{\mathrm F}$, and we set $z_a^{\mathrm B} = z_0$ for $a \le k$ and $z_a^{\mathrm B} = \bar z_0$ for $a > k$. We let $\mathbf G^{\mathrm B}$ and $\mathbf G^{\mathrm F}$ denote the $2kN \times 2kN$ matrices with entries
\[
(\mathbf G^{\mathrm B})_{(i, a), (j, b)} = \delta_{ab} \, G_{ij}(z_a^{\mathrm B}), \qquad (\mathbf G^{\mathrm F})_{(i, a), (j, b)} = \delta_{ab} \, G_{ij}(z_a^{\mathrm F}) ,
\]
indexed by sites $i, j \in [N]$ and flavors $a, b \in [2k]$. For local sources $Y$, we let $\mathbf Y^{\mathrm B}$ and $\mathbf Y^{\mathrm F}$ denote the block-diagonal matrices with $(i, i)$ blocks $Y_i^{\mathrm B}$ and $Y_i^{\mathrm F}$. We set $\mathcal R_N(\Gamma) = \Sdet(E + \iu \eta_0 \Gamma - H)^{-1}$, so that $\E \mathcal R_N(\Gamma) = T_N^{(k)}(\Gamma)$, and by the proof of \zcref{l:rep-sdet}, $\mathcal R_N(\Gamma(u, b))$ is the product inside the expectation in \eqref{e:intro-ratio}. For $a \in [k]$, we set $e_a^+ = e_a + e_{k + a} \in \C^{2k}$ and $e_a^- = (e_a - e_{k + a})/(2\iu)$. For $S \in \mathcal S$ such that $E - S$ is invertible, we let $[(E - S)^{-1}]^{\mathrm{BB}}$ and $[(E - S)^{-1}]^{\mathrm{FF}}$ denote the BB and FF blocks of the inverse supermatrix, and we define
\begin{equation}\label{e:rep-gamma}
\gamma_a(S) = -(e_a^-)^T \big[ (E - S)^{-1} \big]^{\mathrm{BB}} e_a^+ .
\end{equation}

\begin{lemma}\label{l:rep-dictionary}
Let $\Gamma$ be a source supermatrix with diagonal FF block, and let $Y$ be local sources with $\|Y^{\mathrm B}\|_{\rm op} < \eta$.
\begin{enumerate}[label=(\alph*)]
\item\label{i:rep-dictionary-matrix} We have
\begin{equation}\label{e:rep-dictionary-matrix}
T_N^{(k)}(\Gamma; Y) = \E \Big[ \mathcal R_N(\Gamma) \, \frac{\det(\Id - \mathbf G^{\mathrm F} \mathbf Y^{\mathrm F})}{\det(\Id - \mathbf G^{\mathrm B} \mathbf Y^{\mathrm B})} \Big] .
\end{equation}
\item\label{i:rep-dictionary-chain} At every point of $\mathcal S$ where $E - S$ is invertible, which is all of $\mathcal S$ if $E \ne 0$ and a set of full measure if $E = 0$, we have
\begin{equation}\label{e:rep-insertion}
w_{\Gamma, Y}(S) = w_\Gamma(S) \, \Sdet\big( \Id + (E - S)^{-1} Y \big)^{-1} .
\end{equation}
If $Y = \diag(Y^{\mathrm B} \mid 0)$, the last factor equals $\det(\Id + [(E - S)^{-1}]^{\mathrm{BB}} Y^{\mathrm B})^{-1}$, and if $Y = \diag(0 \mid Y^{\mathrm F})$, it equals $\det(\Id + [(E - S)^{-1}]^{\mathrm{FF}} Y^{\mathrm F})$.
\end{enumerate}
Consequently, in the sense of Taylor coefficients at $Y = 0$, the entries of $\mathbf G^{\mathrm B}$ and $\mathbf G^{\mathrm F}$ in \eqref{e:rep-dictionary-matrix} correspond to the entries of $-[(E - S_i)^{-1}]^{\mathrm{BB}}$ and $-[(E - S_i)^{-1}]^{\mathrm{FF}}$ in \eqref{e:rep-exact-marks}. In particular, let $\Gamma = \Gamma_0$, so that $z = z_0 = E + \iu \eta$, and let $i_1, \dots, i_m \in [N]$ be distinct sites. Then, in the notation \eqref{e:rep-partition},
\begin{align}
\E \operatorname{per}\big[ G_{i_a i_b}(z) \big]_{a, b = 1}^m &= \mathcal Z_N\Big( \Gamma_0; \prod_{a = 1}^m \big( -[(E - S_{i_a})^{-1}]^{\mathrm{BB}}_{11} \big) \Big), \label{e:rep-dict-perm} \\
\E \det\big[ G_{i_a i_b}(z) \big]_{a, b = 1}^m &= \mathcal Z_N\Big( \Gamma_0; \prod_{a = 1}^m \big( -[(E - S_{i_a})^{-1}]^{\mathrm{FF}}_{11} \big) \Big), \label{e:rep-dict-det} \\
\E \operatorname{per}\big[ \Im G_{i_a i_b}(z) \big]_{a, b = 1}^m &= \mathcal Z_N\Big( \Gamma_0; \prod_{a = 1}^m \gamma_1(S_{i_a}) \Big), \label{e:rep-dict-imperm}
\end{align}
where $\operatorname{per}$ denotes the permanent and $\Im G = (G(z) - G(z)^*)/(2\iu)$. If $m \le k$, then
\begin{equation}\label{e:rep-dict-ldos}
\E \prod_{a = 1}^m \Im G_{i_a i_a}(z) = \mathcal Z_N\Big( \Gamma_0; \prod_{a = 1}^m \gamma_a(S_{i_a}) \Big) .
\end{equation}
Finally, for all $i \in [N]$ and integers $q \ge 1$, writing $y = [(E - S_i)^{-1}]^{\mathrm{BB}}$,
\begin{equation}\label{e:rep-dict-moment}
\E |G_{ii}(z)|^{2q} = \mathcal Z_N\Big( \Gamma_0; \sum_{r = 0}^q \binom{q}{r}^2 (y_{11} \, y_{k + 1, k + 1})^{q - r} (y_{1, k + 1} \, y_{k + 1, 1})^r \Big) .
\end{equation}
All chain integrals in \eqref{e:rep-dict-perm}--\eqref{e:rep-dict-moment} have integrable integrands.
\end{lemma}

Recall from \zcref{s:intro-results} that $\ell^\eps_{i, N}(0) = s_E^{-1} \Im G_{ii}(z_0)$; for a center $u \neq 0$, \eqref{e:rep-dict-ldos} applies with $E$ replaced by $E + u/a_N$ and the same $\eta$. The local density $\ell^\eps_{i, N}$, a random function attached to the site $i$, is unrelated to the partial chains $\ell_m$ of \zcref{d:rep-chain}, which are superfunctions on $\mathcal S$ built from $m$ sites of the chain. The permanent \eqref{e:rep-dict-imperm} at distinct sites contains the products $\Im G_{i_a i_b} \Im G_{i_b i_a}$, and the product of diagonal entries is obtained from \eqref{e:rep-dict-ldos}, which uses one pair of flavors for each site. The sites in \eqref{e:rep-dict-ldos} must be distinct. At a single site $i$, the two pairs of flavors $1$ and $2$ give, with $y = [(E - S_i)^{-1}]^{\mathrm{BB}}$, the insertion
\[
\gamma_1(S_i) \gamma_2(S_i) + (e_1^-)^T y \, e_2^+ \, (e_2^-)^T y \, e_1^+
\]
for $\E (\Im G_{ii}(z))^2$, and not $\gamma_1(S_i) \gamma_2(S_i)$; moments at a single site are given by \eqref{e:rep-dict-moment}. In \eqref{e:rep-dict-moment}, the matrix side does not couple the retarded and advanced flavors, while the insertion contains the entries $y_{1, k + 1}$ and $y_{k + 1, 1}$, which couple them. Such terms distinguish coincident from distinct sites in \zcref{t:intensity}.

\begin{proof}
We begin with \ref{i:rep-dictionary-matrix}. Let $D = E + \iu \eta_0 \Gamma - H$. The BB and FF blocks of $D$ are block diagonal over the flavors, with $(a, a)$ blocks $z_a^{\mathrm B} - H$ and $z_a^{\mathrm F} - H$ on $\C^N$. Using $(z - H)^{-1} = -G(z)$ and reordering the indices, we find
\[
D^{-1} = -\diag\big( \mathbf G^{\mathrm B} \mid \mathbf G^{\mathrm F} \big), \qquad D + Y = D \big( \Id - \diag(\mathbf G^{\mathrm B} \mid \mathbf G^{\mathrm F}) \, Y \big) .
\]
By \zcref{l:grass-sdet}\ref{i:grass-sdet-mult}, we now have
\[
\Sdet(D + Y)^{-1} = \Sdet(D)^{-1} \frac{\det(\Id - \mathbf G^{\mathrm F} \mathbf Y^{\mathrm F})}{\det(\Id - \mathbf G^{\mathrm B} \mathbf Y^{\mathrm B})} ,
\]
and taking expectations, we obtain \eqref{e:rep-dictionary-matrix}.

For \ref{i:rep-dictionary-chain}, the multiplicativity of $\Sdet$ in \zcref{l:grass-sdet}\ref{i:grass-sdet-mult} implies that
\[
\Sdet(E - S + Y) = \Sdet(E - S) \Sdet\big( \Id + (E - S)^{-1} Y \big) ,
\]
and \eqref{e:rep-insertion} follows from the definition \eqref{e:rep-weight}. When $Y$ has only a BB block, the supermatrix $\Id + (E - S)^{-1} Y$ has FF block $\Id$ and BF block zero. When $Y$ has only an FF block, its BB block is $\Id$ and its FB block is zero. In these two cases, \zcref{l:grass-sdet}\ref{i:grass-sdet-block} and the definition of $\Sdet$ give, respectively,
\[
\begin{gathered}
\Sdet\big( \Id + (E - S)^{-1} Y \big) = \det\big( \Id + [(E - S)^{-1}]^{\mathrm{BB}} Y^{\mathrm B} \big) , \\
\Sdet\big( \Id + (E - S)^{-1} Y \big) = \det\big( \Id + [(E - S)^{-1}]^{\mathrm{FF}} Y^{\mathrm F} \big)^{-1} .
\end{gathered}
\]
Note that the FF block $\kappa_E - \iu M$ of $E - S$ is invertible for all $M$ if $E \ne 0$, and outside the null set where $M$ is singular if $E = 0$.

The correspondence follows by comparing \eqref{e:rep-dictionary-matrix}, at $\Gamma = \Gamma_0$ where $\mathcal R_N(\Gamma_0) = 1$, with \eqref{e:rep-exact-marks} and \eqref{e:rep-insertion}. We now verify the special cases. For \eqref{e:rep-dict-perm}, we set
\[
Y_{i_a} = \diag(t_a E_{11} \mid 0), \qquad A = \sum_{a = 1}^m t_a e_{i_a} e_{i_a}^* ,
\]
where $E_{11}$ is the matrix unit of the first flavor. The letter $E$ with two indices, such as $E_{11}$ here or $E_{i j}$ and $E_{\mathrm{RR}}$ in later sections, always denotes a matrix unit, and $E$ without indices is the energy. Then \eqref{e:rep-dictionary-matrix} reduces to
\[
T_N^{(k)}(\Gamma_0; Y) = \E \det\big( \Id_N - G(z) A \big)^{-1} .
\]
The coefficient of $t_1 \cdots t_m$ in $\det(\Id_N - G A)^{-1}$ is the permanent of the submatrix of $G$ with rows and columns $i_1, \dots, i_m$, by the expansion of $\det(\Id - A)^{-1}$ into permanents (MacMahon's master theorem). For the chain side, \eqref{e:rep-insertion} gives the factor
\[
\prod_{a = 1}^m \big( 1 + t_a [(E - S_{i_a})^{-1}]^{\mathrm{BB}}_{11} \big)^{-1} ,
\]
whose coefficient of $t_1 \cdots t_m$ is the insertion in \eqref{e:rep-dict-perm}.

For \eqref{e:rep-dict-det}, we use the FF sources $Y_{i_a} = \diag(0 \mid t_a E_{11})$ instead. The coefficients of $t_1 \cdots t_m$ on the matrix side and on the chain side are then, respectively,
\[
(-1)^m \, \E \det\big[ G_{i_a i_b}(z) \big]_{a, b = 1}^m \qquad \text{and} \qquad \prod_{a = 1}^m \big[ (E - S_{i_a})^{-1} \big]^{\mathrm{FF}}_{11} .
\]

For \eqref{e:rep-dict-imperm}, we use the sources
\[
Y_{i_a} = \diag\big( t_a e_1^+ (e_1^-)^T \mid 0 \big) .
\]
The matrix $e_1^+ (e_1^-)^T$ has rank one and $\mathbf G^{\mathrm B}$ is diagonal in the flavors, and therefore
\[
\det\big( \Id - \mathbf G^{\mathrm B} \mathbf Y^{\mathrm B} \big) = \det\Big( \Id_N - A \, \frac{G(z) - G(\bar z)}{2\iu} \Big) = \det\big( \Id_N - A \Im G(z) \big) ,
\]
where we used $G(\bar z) = G(z)^*$. As for \eqref{e:rep-dict-perm}, the expansion into permanents shows that $\operatorname{per}[\Im G_{i_a i_b}]$ is the coefficient of $t_1 \cdots t_m$ in the inverse of this determinant. For the chain side, we set $y = [(E - S_{i_a})^{-1}]^{\mathrm{BB}}$, and we have
\[
\det\big( \Id + t_a y e_1^+ (e_1^-)^T \big) = 1 + t_a (e_1^-)^T y e_1^+ = 1 - t_a \gamma_1(S_{i_a}) ,
\]
and the inverses of these factors give the insertion.

For \eqref{e:rep-dict-ldos}, we use the source $t_a e_a^+ (e_a^-)^T$ at the site $i_a$. Different sites now use different pairs of flavors. Since $\mathbf G^{\mathrm B}$ is diagonal in the flavors, the matrix side factors,
\[
\det\big( \Id - \mathbf G^{\mathrm B} \mathbf Y^{\mathrm B} \big)^{-1} = \prod_{a = 1}^m \big( 1 - t_a \Im G_{i_a i_a}(z) \big)^{-1} ,
\]
and the chain side is the product of the factors $(1 - t_a \gamma_a(S_{i_a}))^{-1}$, as above.

For \eqref{e:rep-dict-moment}, we use the source
\[
Y_i = \diag\big( t E_{11} + t' E_{k + 1, k + 1} \mid 0 \big) .
\]
On the matrix side, since $|G_{ii}(z)| \le \eta^{-1}$, we may expand both factors into geometric series for $|t|, |t'| < \eta$, and we obtain
\[
\E \big( 1 - t G_{ii}(z) \big)^{-1} \big( 1 - t' G_{ii}(\bar z) \big)^{-1} = \sum_{q, q' \ge 0} t^q t'^{q'} \, \E \big[ G_{ii}(z)^q G_{ii}(\bar z)^{q'} \big] .
\]
Since $G_{ii}(\bar z) = \overline{G_{ii}(z)}$, the coefficient of $t^q t'^q$ is the moment $\E |G_{ii}(z)|^{2q}$. The factor on the chain side is the inverse of
\[
(1 + t y_{11})(1 + t' y_{k + 1, k + 1}) - t t' y_{1, k + 1} y_{k + 1, 1} ,
\]
and expanding first in its last term gives the coefficient in \eqref{e:rep-dict-moment}.

The insertions are Taylor coefficients of $\Sdet(\Id + (E - S)^{-1} Y)^{-1}$, which are the derivatives of the marked weights in \eqref{e:rep-exact-marks} divided by $w_\Gamma$. Their integrability follows from \zcref{p:rep-exact}\ref{i:rep-exact-marks}.
\end{proof}

The supplementary script \texttt{rep\_dictionary\_check.py} checks the matrix and chain sides of \eqref{e:rep-dict-perm}--\eqref{e:rep-dict-moment} separately, before the expectation, together with the coincident-site insertion above and the saddle values below, for $k = 2$ and a random Hermitian matrix of size five, to a relative error of order $10^{-12}$.

The insertions take explicit values on the saddle manifold of \zcref{s:rep-heuristics}. Let $S = \kappa_E - \iu s_E Q$ with $Q^2 = \Id$. By \eqref{e:rep-saddle-inverse}, we have $-(E - S)^{-1} = -\kappa_E + \iu s_E Q$, the supermatrix that Part III uses under the name orbit resolvent, in \eqref{e:iface-orbit-resolvent}. At $Q = \Lambda$, its retarded and advanced diagonal entries $-\kappa_E \pm \iu s_E$ are the values of $G_{ii}(E \pm \iu 0)$ for the semicircle law. Inserting \eqref{e:rep-saddle-inverse} into \eqref{e:rep-gamma}, we find
\[
\gamma_a(S) = \frac{s_E}{2} \big( Q_{aa} + Q_{a, k + a} - Q_{k + a, a} - Q_{k + a, k + a} \big)^{\mathrm{BB}} ,
\]
which at $Q = \Lambda$ gives $\gamma_a(S) = s_E = \Im(-\kappa_E + \iu s_E)$.