From 43fc2fdf9e74351fdeb8abed136cbbd25ec69eb0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 19 Sep 2024 08:50:01 +0200
Subject: Reorganization of more things to the appendix.

---
 topology.tex | 169 +++++++++++++++++++++++++++++------------------------------
 1 file changed, 82 insertions(+), 87 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 127b15e..f56a2c7 100644
--- a/topology.tex
+++ b/topology.tex
@@ -271,71 +271,31 @@ The integral over the solution manifold $\Omega$ in \eqref{eq:kac-rice} becomes
 \end{equation}
 where $\partial=[\frac\partial{\partial\mathbf x},\frac\partial{\partial\pmb\omega}]$
 is the vector of partial derivatives with respect to all $N+M+1$ variables.
-This integral is now in a form where standard techniques from the mean-field
-theory of disordered systems can be applied to calculate it.
+This expression is now in a form where standard techniques from the mean-field
+theory of disordered systems can be applied to average over the random constraint functions and evaluate the integrals to leading order in large $N$.
 
-To evaluate the average of $\chi$ over the random constraints, we first translate the $\delta$-functions and determinant to integral form, with
-\begin{align}
-  \label{eq:delta.exp}
-  \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
-  &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
-  \,e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)}
-  \\
-  \label{eq:det.exp}
-  \det\partial\partial L(\mathbf x,\pmb\omega)
-  &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
-  e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]}
-\end{align}
-where $\hat{\mathbf x}$ and $\hat{\pmb\omega}$ are ordinary vectors and
-$\bar{\pmb\eta}$, $\pmb\eta$, $\bar{\pmb\gamma}$, and $\pmb\gamma$ are
-Grassmann vectors. With these expressions substituted into
-\eqref{eq:kac-rice.lagrange}, the result is an integral over an exponential
-whose argument is linear in the random functions $V_k$. These functions can
-therefore be averaged over, and the resulting expression treated with standard
-methods. Details of this calculation can be found in Appendix~\ref{sec:euler}.
-The result is the reduction of the average Euler characteristic to an expression of the
-form
-\begin{equation} \label{eq:pre-saddle.characteristic}
-  \overline{\chi(\Omega)}
-  =\left(\frac N{2\pi}\right)^2\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\chi(R,D,m,\hat m)}
+Details of this calculation can be found in Appendix~\ref{sec:euler}. The
+result is the reduction of the average Euler characteristic to an integral over
+a single order parameter $m=\frac1N\mathbf x\cdot\mathbf x_0$ of the form
+\begin{equation}
+  \overline{\chi(\Omega)}=\left(\frac{N}{2\pi}\right)^{\frac12}\int dm\,g(m)\,e^{N\mathcal S_\chi(m)}
 \end{equation}
-where $g$ is a prefactor of $o(N^0)$, $\mathcal S_\chi$ is an effective action defined by
-\begin{equation} \label{eq:euler.action}
-  \begin{aligned}
-    \mathcal S_\chi(R,D,m,\hat m)
-    &=-\hat m-\frac\alpha2\left[
-      \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
-      +\frac{V_0^2}{f(1)}\left(1+\frac{f'(1)R^2}{f(1)D}\right)^{-1}
-    \right] \\
-    &\hspace{7em}+\frac12\log\left(
-      1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+where $g(m)$ is a prefactor of order $N^0$ and $\mathcal S_\chi(m)$ is an
+effective action defined by
+\begin{equation} \label{eq:S.m}
+  \mathcal S_\chi(m)
+  =-\frac\alpha2\bigg[
+    \log\left(
+      1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R_*}}{1-m^2}
     \right)
-  \end{aligned}
+    +\frac{V_0^2}{f(1)}\left(
+      1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R_*}}
+    \right)^{-1}
+  \bigg]
+  +\frac12\log\left(-\frac m{R_*}\right)
 \end{equation}
-and where we have introduced the ratio $\alpha=M/N$.
-The remaining order parameters are defined by the scalar products
-\begin{align}
-  R=-i\frac1N\mathbf x\cdot\hat{\mathbf x}
-  &&
-  D=\frac1N\hat{\mathbf x}\cdot\hat{\mathbf x}
-  &&
-  m=\frac1N\mathbf x\cdot\mathbf x_0
-  &&
-  \hat m=-i\frac1N\hat {\mathbf x}\cdot\mathbf x_0
-\end{align}
-between the configurations $\mathbf x$, the auxiliary configurations
-$\hat{\mathbf x}$, and the height axis $\mathbf x_0$.
-The integral \eqref{eq:pre-saddle.characteristic} can be evaluated to leading
-order in $N$ by a saddle point approximation. First we extremize with respect
-to $R$, $D$, and $\hat m$, which take the saddle-point values
-\begin{align}
-  R=R^*
-  &&
-  D=-\frac{m+R_*}{1-m^2}R_*
-  &&
-  \hat m=0
-\end{align}
-where we have defined
+Here we have introduced the ratio $\alpha=M/N$ between the number of equations
+and the number of variables, and $R_*$ is a function of $m$ given by
 \begin{equation} \label{eq:rs}
   \begin{aligned}
     R_*
@@ -350,28 +310,15 @@ where we have defined
   \Bigg]
   \end{aligned}
 \end{equation}
-Upon substitution of these solutions into \eqref{eq:euler.action}, we find an
-effective action as a function of $m$ alone given by
-\begin{equation} \label{eq:S.m}
-  \mathcal S_\chi(m)
-  =-\frac\alpha2\bigg[
-    \log\left(
-      1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R_*}}{1-m^2}
-    \right)
-    +\frac{V_0^2}{f(1)}\left(
-      1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R_*}}
-    \right)^{-1}
-  \bigg]
-  +\frac12\log\left(-\frac m{R_*}\right)
-\end{equation}
-This function is plotted in Fig.~\ref{fig:action} for a
-selection of parameters.
-To finish evaluating the integral, this
-expression should be maximized with respect to $m$. If $m_*$ is such a maximum,
-then the resulting Euler characteristic is $\overline{\chi(\Omega)}\propto
-e^{N\mathcal S_\chi(m_*)}$.
-
-\begin{figure}[tbh]
+This function is plotted in Fig.~\ref{fig:action} for a selection of
+parameters. To finish evaluating the integral by the saddle-point
+approximation, the action should be maximized with respect to $m$. If $m_*$ is
+such a maximum, then the resulting average Euler characteristic is
+$\overline{\chi(\Omega)}\propto e^{N\mathcal S_\chi(m_*)}$. In the next
+subsection we examine the maxima of $\mathcal S_\chi$ their properties as the
+parameters are varied.
+
+\begin{figure}[tb]
   \includegraphics{figs/action_1.pdf}
   \hspace{-3.5em}
   \includegraphics{figs/action_3.pdf}
@@ -834,9 +781,27 @@ JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
 \section{Details of the calculation of the average Euler characteristic}
 \label{sec:euler}
 
-Our starting point is the expression \eqref{eq:kac-rice.lagrange} with the
-substitutions of the $\delta$-function and determinant \eqref{eq:delta.exp} and
-\eqref{eq:det.exp} made. To make the calculation compact, we introduce
+Our starting point is the expression \eqref{eq:kac-rice.lagrange}. To evaluate
+the average of $\chi$ over the random constraints, we first translate the
+$\delta$-function and determinant to integral form, with
+\begin{align}
+  \label{eq:delta.exp}
+  \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
+  &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
+  \,e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)}
+  \\
+  \label{eq:det.exp}
+  \det\partial\partial L(\mathbf x,\pmb\omega)
+  &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
+  e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]}
+\end{align}
+where $\hat{\mathbf x}$ and $\hat{\pmb\omega}$ are ordinary vectors and
+$\bar{\pmb\eta}$, $\pmb\eta$, $\bar{\pmb\gamma}$, and $\pmb\gamma$ are
+Grassmann vectors. With these expressions substituted into
+\eqref{eq:kac-rice.lagrange}, the result is an integral over an exponential
+whose argument is linear in the random functions $V_k$.
+
+To make the calculation compact, we introduce
 superspace coordinates \cite{DeWitt_1992_Supermanifolds, Kent-Dobias_2024_Conditioning}. Introducing the Grassmann indices $\bar\theta_1$
 and $\theta_1$, we define the supervectors
 \begin{align}
@@ -1007,7 +972,37 @@ as removing all dependence on $\bar H$ and $H$. With these solutions inserted, t
 \end{align}
 The Grassmann terms in these expressions do not contribute to the effective
 action, but will be important in our derivation of the prefactor for the
-exponential around the stationary points at $\pm m_*$. The substitution of these expressions into \eqref{eq:post.hubbard-strat} without the Grassmann terms yields \eqref{eq:euler.action} from the main text.
+exponential around the stationary points at $\pm m_*$. The substitution of these expressions into \eqref{eq:post.hubbard-strat} without the Grassmann terms yields
+\begin{equation} \label{eq:pre-saddle.characteristic}
+  \overline{\chi(\Omega)}
+  =\left(\frac N{2\pi}\right)^2\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\chi(R,D,m,\hat m)}
+\end{equation}
+where $g$ is a prefactor of $o(N^0)$ detailed in the following appendix, $\mathcal S_\chi$ is an effective action defined by
+\begin{equation} \label{eq:euler.action}
+  \begin{aligned}
+    \mathcal S_\chi(R,D,m,\hat m)
+    &=-\hat m-\frac\alpha2\left[
+      \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
+      +\frac{V_0^2}{f(1)}\left(1+\frac{f'(1)R^2}{f(1)D}\right)^{-1}
+    \right] \\
+    &\hspace{7em}+\frac12\log\left(
+      1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+    \right)
+  \end{aligned}
+\end{equation}
+and where we have introduced the ratio $\alpha=M/N$.
+The integral \eqref{eq:pre-saddle.characteristic} can be evaluated to leading
+order in $N$ by a saddle point approximation. To get the formula \eqref{eq:S.m}
+in the main text, we first extremize this expression with respect to $R$, $D$,
+and $\hat m$, which take the saddle-point values
+\begin{align}
+  R=R^*
+  &&
+  D=-\frac{m+R_*}{1-m^2}R_*
+  &&
+  \hat m=0
+\end{align}
+where $R^*$ is given by \eqref{eq:rs} from the main text.
 
 \section{Calculation of the prefactor of the average Euler characteristic}
 \label{sec:prefactor}
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