From ae9115876b609316ff02e0b596956d2459bc3448 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 2 Aug 2024 12:55:11 +0200
Subject: Added some more discussion and moved calculation details to an
 appendix.

---
 topology.tex | 160 ++++++++++++++++++++++++++++++++---------------------------
 1 file changed, 87 insertions(+), 73 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 097c5a4..decfa6f 100644
--- a/topology.tex
+++ b/topology.tex
@@ -122,6 +122,14 @@ When the Kac--Rice formula is used to \emph{count} stationary points, the sign
 of the determinant is a nuisance that one must take pains to preserve
 \cite{Fyodorov_2004_Complexity}. Here we are correct to exclude it.
 
+We need to choose a function $H$ for our calculation. Because $\chi$ is
+a topological invariant, any choice will work so long as it does not share some
+symmetry with the underlying manifold, i.e., that it $H$ satisfies the Smale condition. Because our manifold of random
+constraints has no symmetries, we can take a simple height function $H(\mathbf
+x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
+$\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is
+used as the polar axis, $H$ gives the height on the sphere.
+
 We treat the integral over the implicitly defined manifold $\Omega$ using the
 method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to
 enforce the spherical constraint and $M$ multipliers $\omega_k$ to enforce the vanishing of
@@ -139,79 +147,10 @@ The integral over $\Omega$ in \eqref{eq:kac-rice} then 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 mean-field theory
+can be applied to calculate it. Details of this calculation are reserved in an appendix.
 
-To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
-\begin{align}
-  \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)}
-  \\
-  \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 H[\pmb\eta,\pmb\gamma]}
-\end{align}
-for real variables $\hat{\mathbf x}$ and $\hat{\pmb\omega}$, and Grassmann
-variables $\bar{\pmb\eta}$, $\pmb\eta$, $\bar{\pmb\gamma}$, and $\pmb\gamma$.
-With these transformations in place, there is a compact way to express $\chi$
-using superspace notation. For a review of the superspace formalism for
-evaluating integrals of the form \eqref{eq:kac-rice.lagrange}, see Appendices A
-\& B of \cite{Kent-Dobias_2024_Conditioning}. Introducing the Grassmann indices
-$\bar\theta_1$ and $\theta_1$, we define superfields
-\begin{align}
-  \pmb\phi(1)
-  &=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\hat{\mathbf x}\bar\theta_1\theta_1 
-  \label{eq:superfield.phi} \\
-  \pmb\sigma(1)
-  &=\pmb\omega+\bar\theta_1\pmb\gamma+\bar{\pmb\gamma}\theta_1+\hat{\pmb\omega}\bar\theta_1\theta_1
-  \label{eq:superfield.sigma}
-\end{align}
-with which we can represent $\chi$ by
-\begin{equation}
-  \chi=\int d\pmb\phi\,d\pmb\sigma\,\exp\left\{
-    \int d1\,L\big(\pmb\phi(1),\pmb\sigma(1)\big)
-  \right\}
-\end{equation}
-We are now in a position to average over the distribution of constraints. Using
-standard manipulations, we find the average Euler characteristic is
-\begin{equation}
-  \begin{aligned}
-    \overline{\chi}&=\int d\pmb\phi\,d\sigma_0\,\exp\Bigg\{
-      -\frac M2\log\operatorname{sdet}f\left(\frac{\phi(1)^T\phi(2)}N\right) \\
-      &\qquad+\int d1\,\left[
-        H\big(\phi(1)\big)+\frac12\sigma_0(1)\big(\|\phi(1)\|^2-N\big)
-      \right]
-    \Bigg\}
-  \end{aligned}
-\end{equation}
-Now we are forced to make a decision about the function $H$. Because $\chi$ is
-a topological invariant, any choice will work so long as it does not share some
-symmetry with the underlying manifold, i.e., that it $H$ satisfies the Smale condition. Because our manifold of random
-constraints has no symmetries, we can take a simple height function $H(\mathbf
-x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
-$\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is
-used as the polar axis, $H$ gives the height on the sphere.
-
-With this choice made, we can integrate over the superfields $\pmb\phi$.
-Defining two order parameters $\mathbb Q(1,2)=\frac1N\phi(1)\cdot\phi(2)$ and
-$\mathbb M(1)=\frac1N\phi(1)\cdot\mathbf x_0$, the result is
-\begin{align}
-  \overline{\chi}
-  &=\int d\mathbb Q\,d\mathbb M\,d\sigma_0 \notag\\
-   &\quad\times\exp\Bigg\{
-    \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
-    -\frac M2\log\operatorname{sdet}f(\mathbb Q) \notag \\
-    &\qquad+N\int d1\,\left[
-      \mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
-    \right]
-  \Bigg\}
-\end{align}
-This expression is an integral of an exponential with a leading factor of $N$
-over several order parameters, and is therefore in a convenient position for
-evaluating at large $N$ with a saddle point. The order parameter $\mathbb Q$ is
-made up of scalar products of the original integration variables in our
-problem in \eqref{eq:superfield.phi}, while $\mathbb M$ contains their scalar
-project with $\mathbf x_0$, and $\pmb\sigma_0$ contains $\omega_0$ and
-$\hat\omega_0$. We can solve the saddle point equations in all of these
+We can solve the saddle point equations in all of these
 parameters save for $m=\frac1N\mathbf x_0\cdot\mathbf x$, the overlap with the
 height axis. The result reduces the average Euler characteristic to
 \begin{equation}
@@ -282,7 +221,7 @@ these results. Cartoons that depict this reasoning are shown in
 Fig.~\ref{fig:cartoons}. In the regime $\alpha<1$, $\overline\chi$ is positive but not
 very large. This is consistent with a solution manifold made up of few large
 components, each with the topology of a hypersphere. The saddle point value
-$m^*$ for the overlap with the height axis $\mathbf x_0$ corresponds to the
+$(m^*)^2=1-\alpha/\alpha_\text{\textsc{sat}}$ for the overlap with the height axis $\mathbf x_0$ corresponds to the
 latitude at which most stationary points that contribute to the Euler
 characteristic are found. This means we can interpret $1-m^*$ as the typical
 squared distance between a randomly selected point on the sphere and the
@@ -347,6 +286,13 @@ The fact that $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ is the same in the anneale
 replica symmetric calculations suggests that it may perhaps be exact. It is also
 consistent with the full RSB calculation of \cite{Urbani_2023_A}.
 
+We check the stability of the replica symmetric solution by calculating the
+eigenvalues of the Hessian of the effective action with respect to the order
+parameters. While for calculations of this kind the meaning of the sign of
+these eigenvalues is difficult to understand directly, in situations where
+there is a continuous \textsc{rsb} transition the sign of one of the
+eigenvalues changes \cite{Kent-Dobias_2023_When}. At the $\alpha_\text{\textsc{rsb}}$ predicted in \cite{Urbani_2023_A} we see no instability of this kind, and instead only observe such an instability at $\alpha_\text{\textsc{sat}}$.
+
 
 \begin{acknowledgements}
   JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
@@ -355,6 +301,74 @@ consistent with the full RSB calculation of \cite{Urbani_2023_A}.
 
 \bibliography{topology}
 
+\paragraph{Details of the annealed calculation.}
+
+To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
+\begin{align}
+  \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)}
+  \\
+  \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 H[\pmb\eta,\pmb\gamma]}
+\end{align}
+for real variables $\hat{\mathbf x}$ and $\hat{\pmb\omega}$, and Grassmann
+variables $\bar{\pmb\eta}$, $\pmb\eta$, $\bar{\pmb\gamma}$, and $\pmb\gamma$.
+With these transformations in place, there is a compact way to express $\chi$
+using superspace notation. For a review of the superspace formalism for
+evaluating integrals of the form \eqref{eq:kac-rice.lagrange}, see Appendices A
+\& B of \cite{Kent-Dobias_2024_Conditioning}. Introducing the Grassmann indices
+$\bar\theta_1$ and $\theta_1$, we define superfields
+\begin{align}
+  \pmb\phi(1)
+  &=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\hat{\mathbf x}\bar\theta_1\theta_1 
+  \label{eq:superfield.phi} \\
+  \pmb\sigma(1)
+  &=\pmb\omega+\bar\theta_1\pmb\gamma+\bar{\pmb\gamma}\theta_1+\hat{\pmb\omega}\bar\theta_1\theta_1
+  \label{eq:superfield.sigma}
+\end{align}
+with which we can represent $\chi$ by
+\begin{equation}
+  \chi=\int d\pmb\phi\,d\pmb\sigma\,\exp\left\{
+    \int d1\,L\big(\pmb\phi(1),\pmb\sigma(1)\big)
+  \right\}
+\end{equation}
+We are now in a position to average over the distribution of constraints. Using
+standard manipulations, we find the average Euler characteristic is
+\begin{equation}
+  \begin{aligned}
+    \overline{\chi}&=\int d\pmb\phi\,d\sigma_0\,\exp\Bigg\{
+      -\frac M2\log\operatorname{sdet}f\left(\frac{\phi(1)^T\phi(2)}N\right) \\
+      &\qquad+\int d1\,\left[
+        H\big(\phi(1)\big)+\frac12\sigma_0(1)\big(\|\phi(1)\|^2-N\big)
+      \right]
+    \Bigg\}
+  \end{aligned}
+\end{equation}
+
+With this choice made, we can integrate over the superfields $\pmb\phi$.
+Defining two order parameters $\mathbb Q(1,2)=\frac1N\phi(1)\cdot\phi(2)$ and
+$\mathbb M(1)=\frac1N\phi(1)\cdot\mathbf x_0$, the result is
+\begin{align}
+  \overline{\chi}
+  &=\int d\mathbb Q\,d\mathbb M\,d\sigma_0 \notag\\
+   &\quad\times\exp\Bigg\{
+    \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
+    -\frac M2\log\operatorname{sdet}f(\mathbb Q) \notag \\
+    &\qquad+N\int d1\,\left[
+      \mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
+    \right]
+  \Bigg\}
+\end{align}
+This expression is an integral of an exponential with a leading factor of $N$
+over several order parameters, and is therefore in a convenient position for
+evaluating at large $N$ with a saddle point. The order parameter $\mathbb Q$ is
+made up of scalar products of the original integration variables in our
+problem in \eqref{eq:superfield.phi}, while $\mathbb M$ contains their scalar
+project with $\mathbf x_0$, and $\pmb\sigma_0$ contains $\omega_0$ and
+$\hat\omega_0$. 
+
 \paragraph{Quenched average of the Euler characteristic.}
 
 \begin{equation}
-- 
cgit v1.2.3-70-g09d2

