Some changes.

1 files changed, 41 insertions, 187 deletions

diff --git a/topology.tex b/topology.tex
index f8eec7f..3e8ce9b 100644
--- a/topology.tex
+++ b/topology.tex
@@ -341,11 +341,44 @@ We can treat the integral over $\sigma_0$ immediately. It gives
 This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well
 as setting everything depending on $\bar H$ and $H$ to zero.
 
+\begin{equation}
+  \begin{aligned}
+    \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
+    &=1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+    -\frac6{R^4}\bar H_0H_0\bar{\hat H}\hat H
+    \\
+    &\qquad+\frac2{R^3}\left[
+      (mR-\hat m)(\bar{\hat H}H_0+\bar H_0\hat H)
+      -(D+R^2)\bar H_0H_0
+      +(1-m^2)\bar{\hat H}\hat H
+    \right]
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \operatorname{sdet}f(\mathbb Q)
+  =1+\frac{Df(1)}{R^2f'(1)}
+  +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H
+\end{equation}
+\begin{equation}
+  \int d1\,d2\,F(\mathbb Q)^{-1}(1,2)
+  =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1}
+  +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H
+\end{equation}
 
 \subsection{Behavior with extensively many constraints}
 
 \subsection{Behavior with finitely many constraints}
 
+The correct scaling to find a nontrivial answer with finite $M$ is to scale
+both the covariance functions and fixed constants with $N$ like
+$v_0=\frac1NV_0$, $f(q)=\frac1NF(q)$, so that $v_0$ and $f(q)$ are finite at
+large $N$. With these scalings and $M=1$, this problem reduces to examining the
+levels sets of the spherical spin glasses at energy density $E=v_0$.
+
+$v_0^{\chi>2}=\sqrt{2f(1)}$
+
+$v_0^{m=0}=2\sqrt{f(1)-\frac{f(1)^2}{f'(1)}}$
+
 \subsection{What does the average Euler characteristic tell us?}
 
 It is not straightforward to directly use the average Euler characteristic to
@@ -393,86 +426,6 @@ Thus we find the average Euler characteristic in this simple example is 2
 despite the fact that the possible manifolds resulting from the constraints
 have characteristics of either 0 or 4.
 
-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}
-  \bar\chi\propto\int dm\,e^{N\mathcal S_\mathrm a(m)}
-\end{equation}
-where the annealed action $\mathcal S_a$ is given by
-\begin{equation} \label{eq:ann.action}
-  \begin{aligned}
-    &\mathcal S_\mathrm a(m)
-    =\frac12\Bigg[
-      \log\left(
-        \frac{\frac{f'(1)}{f(1)}(1-m^2)-1}{\alpha-1}
-      \right) \\
-    &\hspace{4em} -\alpha\log\left(
-        \frac{\alpha}{\alpha-1}\left(
-          1-\frac1{\frac{f'(1)}{f(1)}(1-m^2)}
-        \right)
-      \right)
-    \Bigg]
-  \end{aligned}
-\end{equation}
-and must be evaluated at a maximum with respect to $m$. This function is
-plotted for a specific covariance function $f$ in Fig.~\ref{fig:action}, where
-several distinct regimes can be seen.
-
-\begin{figure}
-  \includegraphics{figs/action.pdf}
-
-  \caption{
-    The annealed action $\mathcal S_\mathrm a$ of \eqref{eq:ann.action} plotted
-    as a function of $m$ at several values of $\alpha$. Here, the covariance
-    function is $f(q)=\frac12q^2$ and $\alpha_\text{\textsc{sat}}=2$. When
-    $\alpha<1$, the action is maximized for $m^2>0$ and its value is zero. When
-    $1\leq\alpha<\alpha_\text{\textsc{sat}}$, the action is maximized at
-    $m=0$ and is positive. When $\alpha>\alpha_\text{\textsc{sat}}$ there is no
-    maximum.
-  } \label{fig:action}
-\end{figure}
-
-First, when $\alpha<1$ the action $\mathcal S_\mathrm a$ is strictly negative
-and has maxima at some $m^2>0$. At these maxima, $\mathcal S_\mathrm a(m)=0$.
-When $\alpha>1$, the action flips over and becomes strictly positive. In the
-regime $1<\alpha<\alpha_\text{\textsc{sat}}$, there is a single maximum at
-$m=0$ where the action is positive. When $\alpha\geq\alpha_\text{\textsc{sat}}$
-the maximum in the action vanishes.
-
-This results in distinctive regimes for $\overline\chi$, with an example plotted in Fig.~\ref{fig:characteristic}. If $m^*$ is the maximum of $\mathcal S_\mathrm a$, then
-\begin{equation}
-  \frac1N\log\overline\chi=\mathcal S_\mathrm a(m^*)
-\end{equation}
-When $\alpha<1$, the action evaluates to zero, and therefore $\overline\chi$ is
-positive and subexponential in $N$. When $1<\alpha<\alpha_\text{\textsc{sat}}$, the action
-is positive, and $\overline\chi$ is exponentially large in $N$. Finally, when
-$\alpha\geq\alpha_\text{\textsc{sat}}$ the action and $\overline\chi$ are ill-defined.
-
-\begin{figure}
-  \includegraphics{figs/quenched.pdf}
-
-  \caption{
-    The logarithm of the average Euler characteristic $\overline\chi$ as a
-    function of $\alpha$. The covariance function is $f(q)=\frac12+\frac12q^3$ and
-    $\alpha_\text{\textsc{sat}}=\frac32$. The dashed line shows the average of
-    $\log\chi$, the so-called quenched average, whose value differs in the
-    region $1<\alpha<\alpha_\text{\textsc{sat}}$ but whose transition points
-    are the same.
-  } \label{fig:characteristic}
-\end{figure}
-
-We can interpret this by reasoning about topology of $\Omega$ consistent with
-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
-$(m^*)^2=1-\alpha/\alpha_\text{\textsc{sat}}$ 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
-solution manifold.
-
 \begin{figure}
   \includegraphics[width=0.32\columnwidth]{figs/connected.pdf}
   \hfill
@@ -495,49 +448,6 @@ solution manifold.
   } \label{fig:cartoons}
 \end{figure}
 
-When $1<\alpha<\alpha_\text{\textsc{sat}}$, $\overline\chi$ is positive and
-very large. This is consistent with a solution manifold made up of
-exponentially many disconnected components, each with the topology of a
-hypersphere. If this interpretation is correct, our calculation effectively
-counts these components. This is a realization of a shattering transition in
-the solution manifold. Here $m^*$ is zero because for any choice of height
-axis, the vast majority of stationary points that contribute to the Euler
-characteristic are found near the equator. Finally, for
-$\alpha\geq\alpha_\text{\textsc{sat}}$, there are no longer solutions that
-satisfy the constraints. The Euler characteristic is not defined for an empty
-set, and in this regime the calculation yields no solution.
-
-We have made the above discussion assuming that $\alpha_\text{\textsc{sat}}>1$.
-However, this isn't necessary, and it is straightforward to produce covariance
-functions $f$ where $\alpha_\text{\textsc{sat}}<1$. In this case, the picture
-changes somewhat. When $\alpha_\text{\textsc{sat}}<\alpha<1$, the action
-$\mathcal S_\mathrm a$ has a single maximum at $m^*=0$, where it is negative.
-This corresponds to an average Euler characteristic $\overline\chi$ which is
-exponentially small in $N$. Such a situation is consistent with typical
-constraints leading to no solutions and a zero characteristic, but rare and
-atypical configurations having some solutions.
-
-In the regime where $\log\overline\chi$ is positive, it is possible that our
-calculation yields a value which is not characteristic of typical sets of
-constraints. This motivates computing $\overline{\log\chi}$, the average of
-the logarithm, which should produce something characteristic of typical
-samples, the so-called quenched calculation. In an appendix to this paper we
-sketch the quenched calculation and report its result in the replica symmetric
-approximation. This differs from the annealed calculation above only when
-$f(0)>0$. The replica symmetric calculation produces the same transitions at
-$\alpha=1$ and $\alpha=\alpha_\text{\textsc{sat}}$, but modifies the value
-$m^*$ in the connected phase and predicts
-$\frac1N\overline{\log\chi}<\frac1N\log\overline\chi$ in the shattered phase.
-The fact that $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ is the same in the annealed and
-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 $\alpha_\text{\textsc{sat}}$.
 
 \cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
 
@@ -547,75 +457,19 @@ eigenvalues changes \cite{Kent-Dobias_2023_When}. At the $\alpha_\text{\textsc{r
 
 \subsection{Behavior with finitely many constraints}
 
-\section{Interpretation of our results}
-
-\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\}
+  Mv_0^2=\frac4{f''(1)}\left[
+    f'(1)-f(1)-2\frac{f(1)}{f'(1)}f''(1)
+  \right]\left[
+    \frac{f(1)}{f'(1)}f''(1)-2\big(f'(1)-f(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}
+  Mv_0^2=\frac{f'(1)^2}{f''(1)}
 \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
-$\hat\omega_0$. 
+\section{Interpretation of our results}
 
 \paragraph{Quenched average of the Euler characteristic.}