From a809a6078ad7e052426270e689aef89efec380c4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 18 Sep 2024 10:09:54 +0200
Subject: Added a figure, rewrote the abstract, and some other fixes.

---
 topology.tex | 89 ++++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 53 insertions(+), 36 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index d6fb36f..8ff0de1 100644
--- a/topology.tex
+++ b/topology.tex
@@ -57,22 +57,18 @@ $\star$ \href{mailto:jaron.kent-dobias@roma1.infn.it}{\small jaron.kent-dobias@r
 \section*{\color{scipostdeepblue}{Abstract}}
 \textbf{\boldmath{%
 We consider the set of solutions to $M$ random polynomial equations whose $N$
-variables are restricted to the $(N-1)$-sphere. Each equation has independent Gaussian coefficients and a target
-value $V_0$. When
-solutions exist, they form a manifold. We compute the average Euler
-characteristic of this manifold in the limit of large $N$, and find different
-behavior depending on the target value $V_0$, the ratio $\alpha=M/N$, and the
-variances of the polynomial coefficients. We divide the behavior in four
-phases: a connected phase, an onset phase, a shattered phase, and an
-\textsc{unsat} phase. In the connected phase, the average characteristic is 2
-and there is a single extensive connected component, while in the onset phase
-the Euler characteristic is exponentially large in $N$. In the shattered phase
-the characteristic remains exponentially large but subextensive components
-appear, while in the \textsc{unsat} phase the manifold vanishes. When $M=1$
-there is a correspondence between this problem and
-level sets of the energy in the spherical spin glasses. We conjecture that
-the transition between the onset and shattered phases corresponds to the asymptotic
-limit of gradient descent from a random initial condition.
+variables are restricted to the $(N-1)$-sphere. Each equation has independent
+Gaussian coefficients and a target value $V_0$. When solutions exist, they form
+a manifold. We compute the average Euler characteristic of this manifold in the
+limit of large $N$, and find different behavior depending on the target value
+$V_0$, the ratio $\alpha=M/N$, and the variances of the coefficients. We divide
+this behavior into five phases with different implications for the topology of
+the solution manifold. When $M=1$ there is a correspondence between this
+problem and level sets of the energy in the spherical spin glasses. We
+conjecture that the transition energy dividing two of the topological phases
+corresponds to the asymptotic limit of gradient descent from a random initial
+condition, possibly resolving a recent open problem in out-of-equilibrium
+dynamics.
 }
 }
 
@@ -160,7 +156,7 @@ all possible terms of degree $p$ in $V_k$. In particular, one can explicitly con
 with the elements of the tensors $J^{(k,p)}$ as independently distributed
 unit normal random variables. The size of the
 series coefficients of $f$ therefore control the variances in the coefficients
-of random polynomial constraints. When $M=1$, this problem corresponds to the
+of the random polynomial constraints. When $M=1$, this problem corresponds to the
 level set of a spherical spin glass with energy density $E=V_0/\sqrt{N}$.
 
 
@@ -190,12 +186,11 @@ Because the constraints are all smooth functions, $\Omega$ is almost always a ma
   unlikely, requiring $NM+1$ independent equations to be simultaneously satisfied.
   This means that different connected components of the set of solutions do not
 intersect, nor are there self-intersections, without extraordinary fine-tuning.}
-We study the topology of the manifold $\Omega$ by two related means: its
-average Euler characteristic, and the average number of stationary points of a
-linear height function restricted to the manifold. These measures tell us
-complementary pieces of information. We find that for the varied cases
-we study, these two always coincide at the largest exponential order in $N$,
-putting strong constraints on the resulting topology and geometry.
+We study the topology of the manifold $\Omega$ by computing its
+average Euler characteristic, a topological invariant whose value puts
+constraints on the structure of the manifold. The topological phases determined
+by this means are distinguished by the size and sign of the Euler
+characteristic, and the distribution in space of its constituent parts.
 
 \section{The average Euler characteristic}
 
@@ -488,7 +483,7 @@ While the topology of the manifold is not necessarily connected in this
 regime, holes are more numerous than components. Since $V_0^2<V_\text{on}^2$,
 there are isolated contributions to $\overline{\chi(\Omega)}$ at $m=\pm m^*$.
 This implies a temperate band of relative simplicity: given a random point on
-the sphere, the nearest parts of the solution manifold do not have holes or
+the sphere, the nearest parts of the solution manifold are unlikely to have holes or
 disconnected components.
 
 \paragraph{Regime III: \boldmath{$\overline{\chi(\Omega)}$} large and negative, no contribution at \boldmath{$m=\pm m^*$}.}
@@ -562,9 +557,9 @@ characteristic zero.
     Fig.~\ref{fig:cartoons}. The shaded region in the center panel shows where
     these results are unstable to \textsc{rsb}. In the limit of $\alpha\to0$,
     the behavior of level sets of the spherical spin glasses are recovered: the
-    righthand plot shows how in the pure cubic model the ground state energy
-    $E_\text{gs}$ and threshold energy $E_\text{th}$ correspond with the limits
-    of the satisfiability and shattering transitions, respectively. Note that
+    righthand plot shows how the ground state energy
+    $E_\text{gs}$ and threshold energy $E_\text{th}$ of the 3-spin spherical model correspond with the limits
+    of the satisfiability and shattering transitions in the pure cubic model. Note that
     for mixed models with inhomogeneous covariance functions, $E_\text{th}$ is
     not the lower limit of $V_\text{sh}$.
   } \label{fig:phases}
@@ -647,7 +642,7 @@ glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{-1/2}$ fixe
   E_\text{sh}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
   \label{eq:ssg.energies}
 \end{align}
-for the onset and shattering energies. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
+for the onset and shattering energies. For the pure $p$-spin spherical spin glasses, which have homogeneous covariances $f(q)=\frac12q^p$,
 $E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
 models \cite{Castellani_2005_Spin-glass}. This is intuitive, since threshold energy, defined as the place where
 marginal minima are dominant in the landscape, is widely understood as the
@@ -1152,6 +1147,8 @@ leading order in $N$, as specified in the main text.
 \section{The average squared Euler characteristic}
 \label{sec:rms}
 
+\subsection{Derivation}
+
 Here we calculate $\overline{\chi(\Omega)^2}$, the average of the squared Euler
 characteristic. This is accomplished by taking two copies of the integral \eqref{eq:kac-rice.super}, with
 \begin{equation}
@@ -1253,6 +1250,8 @@ as referenced in the main text. This corresponds with
 $\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$, justifying the
 `annealed' approach we have taken here.
 
+\subsection{Instability to replica symmetry breaking}
+
 However, these solutions are not always the correct saddle point for evaluating
 the average squared Euler characteristic. When another solution is dominant,
 the dissonance between the average square and squared average indicates the
@@ -1274,17 +1273,21 @@ in the subspace spanned by $C_{12}$, $R_{12}$, $R_{21}$, and $D_{12}$. This may
 not be surprising, since these are the parameters that represent nontrivial
 correlations between the two copies of the system. We can therefore find the \textsc{rsb} instability by looking for nontrivial zeros of
 \begin{equation}
-  \det\frac{\partial^2\mathcal S_{\chi^2}}{\partial[C_{12},R_{12},R_{21},D_{12}]^2}
+  \det\partial\partial\mathcal S_{\chi^2}\equiv\det\frac{\partial^2\mathcal S_{\chi^2}}{\partial[C_{12},R_{12},R_{21},D_{12}]^2}
 \end{equation}
 evaluated at the $m=0$ solution described above. The resulting expression is
 usually quite heinous, but there is a regime where a dramatic simplification is
 possible. The instability always occurs along the direction
 $R_{21}=R_{12}^\dagger$, but when $R_*$ is real, $R_{11}=R_{22}$ and the instability occurs along
 the direction $R_{21}=R_{12}$. This allows us to examine a simpler action, and
-we find the determinant is proportional to two nontrivial factors. If we let
-$r_*=\lim_{m\to0}R_*/m$, then these are
+we find the determinant is proportional to two nontrivial factors, with
+\begin{equation} \label{eq:stab.det}
+  \det\partial\partial\mathcal S_{\chi^2}=-\frac{2B_1B_2}{[r_*f'(1)]^3[(1+r_*)f(1)-r_*f'(1)]^7}
+\end{equation}
+If we let
+$r_*=\lim_{m\to0}R_*/m$, then the factors $B_1$ and $B_2$ are
 \begin{align}
-  [(1+r_*)f(1)]^3-3r_*[(1+r_*)f(1)]^2f'(1)
+  B_1=[(1+r_*)f(1)]^3-3r_*[(1+r_*)f(1)]^2f'(1)
   +\alpha V_0^2\big[2(1+r_*)f'(0)^2+r_*f'(1)f''(0)\big]
   \quad
   \notag \\
@@ -1292,7 +1295,7 @@ $r_*=\lim_{m\to0}R_*/m$, then these are
   -(1+r_*)f(1)\Big(\alpha\big[f'(0)^2+V_0^2f''(0)\big]-3[r_*f'(1)]^2\Big)
 \end{align}
 \begin{align}
-  &\big[(1+r_*)f(1)-r_*f'(1)\big]^3[f'(1)^2-\alpha f'(0)^2]
+  &B_2=\big[(1+r_*)f(1)-r_*f'(1)\big]^3[f'(1)^2-\alpha f'(0)^2]
   \Big(f'(1)\big[(1+r_*)f(1)-r_*f'(1)\big]-\alpha f'(0)^2\Big)
   \notag \\
   &\qquad-[\alpha V_0^2r_*f'(1)]^2f''(0)\big[(1+r_*)f'(0)^2+r_*f'(1)f''(0)\big]
@@ -1469,9 +1472,23 @@ simultaneously extremizing, ensuring continuity, and maximizing $E$. This draws
 out the shattering energy across the entire range of $s$ plotted in
 Fig.~\ref{fig:ssg}. The transition to the \textsc{rs} solution occurs when the
 value $q_0$ that maximizes $E$ hits zero. We find that the transition between
-\textsc{rs} and \textsc{frsb} is consistent with the \textsc{rsb} instability
-predicted in Appendix~\ref{sec:rms} by analyzing the solution to the average
-square of the Euler characteristic.
+\textsc{rs} and \textsc{frsb} is precisely predicted by the \textsc{rsb} instability
+calculated in Appendix~\ref{sec:rms} by analyzing the solution to the average
+square of the Euler characteristic, as shown in Fig.~\ref{fig:rsb}.
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/rsb_comp.pdf}
+
+  \caption{
+    \textbf{Self-consistency between \textsc{rsb} instabilities.}
+    Comparison between the predicted value $q_0$ for the \textsc{frsb} solution
+    at the shattering energy in $2+s$ models and the value of the determinant
+    \eqref{eq:stab.det} used in the previous appendix to predict the point of
+    \textsc{rsb} instability. The value of $s$ at which $q_0$ becomes nonzero is
+    precisely the point where the determinant has a nontrivial zero.
+  } \label{fig:rsb}
+\end{figure}
 
 \bibliography{topology}
 
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