From 52dad163cffa50791eb1911562317134c137b2a4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 17 Sep 2024 18:09:48 +0200
Subject: Lots of writing and figure modification.

---
 topology.tex | 238 +++++++++++++++++++++++++++--------------------------------
 1 file changed, 107 insertions(+), 131 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 65cede5..08c3a0b 100644
--- a/topology.tex
+++ b/topology.tex
@@ -398,38 +398,128 @@ principle behind the calculation of the effective action, or it could be the
 result of a negative Euler characteristic.
 
 To address this problem, we compute also the average of the square of the Euler
-characteristic, $\overline{\chi(\Omega)^2}$. This has the benefit of always
-being positive, so that the saddle-point approach to the calculation at large
-$N$ does not produce complex values even when $\overline{\chi(\Omega)}$ is
-negative. We find three saddle points that could contribute to the value of
+characteristic, $\overline{\chi(\Omega)^2}$, with details in
+Appendix~\ref{sec:rms}. This has the benefit of always being positive, so that
+the saddle-point approach to the calculation at large $N$ does not produce
+complex values even when $\overline{\chi(\Omega)}$ is negative. We find three
+saddle points that could contribute to the value of
 $\overline{\chi(\Omega)^2}$: two at $\pm m^*$ where
-$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$, and one at $m=0$ where
+$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$,
+and one at $m=0$ where
 \begin{equation}
   \frac1N\log\overline{\chi(\Omega)^2}=2\operatorname{Re}\mathcal S_\chi(0)
 \end{equation}
+which is consistent with
+$\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$. Such a
+correspondence, which indicates that the `annealed' calculation here is also
+representative of typical realizations of the constraints, is not always true.
+With average squared Euler characteristic we find instabilities of the solution
+described here to replica symmetry breaking (\textsc{rsb}). We do not explore
+these \textsc{rsb} solutions here, except in the context of $M=1$ in
+Section~\ref{sec:ssg}. However, in the following Figures \ref{fig:phases} and
+\ref{fig:crossover} we depict the unstable region with shading.
 
-We therefore have four possible topological regimes:
+\begin{figure}
+  \includegraphics[width=0.196\textwidth]{figs/connected.pdf}
+  \includegraphics[width=0.196\textwidth]{figs/middle.pdf}
+  \includegraphics[width=0.196\textwidth]{figs/complex.pdf}
+  \includegraphics[width=0.196\textwidth]{figs/shattered.pdf}
+  \includegraphics[width=0.196\textwidth]{figs/gone.pdf}
+
+  \hspace{1.5em}
+  \textbf{Regime I}
+  \hfill
+  \textbf{Regime II}
+  \hfill
+  \textbf{Regime III}
+  \hfill
+  \textbf{Regime IV}
+  \hfill
+  \textbf{Regime V}
+  \hspace{1.5em}
+
+  \caption{
+    \textbf{Cartoons of the solution manifold in five topological regimes.}
+    The solution manifold is shown as a shaded region with a black boundary,
+    and the height axis $\mathbf x_0$ is a black arrow. In Regime I, the
+    statistics of the Euler characteristic is consistent with a manifold with a
+    single simply-connected component. In Regime II, holes occupy the equator
+    but its most polar regions are
+    topologically simple. In Regime III, holes dominate and the edge of the
+    manifold is not necessarily simple. In Regime IV, disconnected components
+    dominate. In Regime V, the manifold is empty.
+  } \label{fig:cartoons}
+\end{figure}
+
+We therefore have five possible topological regimes:
 \begin{itemize}
-  \item \textbf{Simple connected:
-    \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}=0$,
-    $\frac1N\log\overline{\chi(\Omega)}=0$}.} This regime occurs when $m_*^2>0$
+  \item \textbf{Regime I:
+    \boldmath{$\overline{\chi(\Omega)}=2$}.}
+
+    $\frac1N\log\overline{\chi(\Omega)^2}=0$,
+    $\frac1N\log\overline{\chi(\Omega)}=0$
+    This regime occurs when $m_*^2>0$
     and $\operatorname{Re}\mathcal S_\chi(0)<0$. Here,
     $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$, strongly indicating a
     topology homeomorphic to the $S^{N-M-1}$ sphere.
-  \item \textbf{Complex connected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}<0$}.}
+
+  \item \textbf{Regime II: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, isolated contribution at \boldmath{$m=\pm m^*$}.}
+
+    Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}<0$.
     This regime occurs when $m_\text{min}^2>0$ and $\operatorname{Re}\mathcal
     S_\chi(0)>0$. Here the average Euler characteristic is large and negative.
     While the topology of the manifold is not necessarily connected in this
-    regime, holes are more numerous than components.
-  \item \textbf{Disconnected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}>0$}.}
-    This regime occurs when $m_\text{min}^2<0$ and $\mathcal S(0)>0$. Here the
+    regime, holes are more numerous than components. In addition, here
+    $m_*^2>m_\text{min}^2$, meaning that the solutions at $m=\pm m_*$ are still
+    present. This indicates that the manifold has topologically simple
+    boundaries with some separation from the sea of holes.
+
+  \item \textbf{Regime III: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, no contribution at \boldmath{$m=\pm m^*$}.}
+
+    The same as Regime II, but with $m_*^2<m_\text{min}^2$. The solutions at
+    $m=\pm m_*$ no longer exist, and nontrivial contributions to the Euler
+    characteristic are made all the way to the solution manifold's boundary.
+
+  \item \textbf{Regime IV: \boldmath{$\overline{\chi(\Omega)}$} very large and positive.}
+
+    Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}>0$.
+    This regime occurs when $m_\text{min}^2<0$ and $\mathcal S_\chi(0)>0$. Here the
     average Euler characteristic is large and positive. Large connected
-    components of the manifold may or may not exist, but disconnected
+    components of the manifold may or may not exist, but small disconnected
     components outnumber holes.
-  \item \textbf{\textsc{Unsat}: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}<0$}.}
-    There is typically not a manifold at all, indicating that the equations cannot be satisfied.
+
+  \item \textbf{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.}
+
+    Here $\frac1N\log\overline{\chi(\Omega)}<0$, indicating that the average
+    Euler characteristic shrinks exponentially with $N$. Under most conditions
+    we conclude this is the \textsc{unsat} regime where no manifold exists, but
+    there may be circumstances where part of this regime is characterized by
+    nonempty solution manifolds that are overwhelmingly likely to have Euler
+    characteristic zero.
 \end{itemize}
 
+\begin{figure}
+  \includegraphics{figs/phases_1.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_2.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_3.pdf}
+
+  \caption{
+    \textbf{Topological phase diagram.}
+    Topological phases of the model for three different homogeneous covariance
+    functions. The onset transition $V_\text{on}$, shattering transition
+    $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$
+    are indicated when they exist. In the limit of $\alpha\to0$, the behavior
+    of level sets of the spherical spin glasses are recovered: the final 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 for mixed models with
+    inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of
+    $V_\text{sh}$.
+  } \label{fig:phases}
+\end{figure}
+
 However, when the magnitude of $V_0$ is sufficiently large, with
 \begin{equation}
   V_0^2>V_\text{on}^2\equiv\frac{1-\alpha+\sqrt{1-\alpha}}\alpha f(1)
@@ -470,28 +560,6 @@ linear $f(q)$ one can see that
 $V_\text{\textsc{sat}}=V_{\text{\textsc{sat}}\ast}$, and indeed this is the
 case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$.
 
-\begin{figure}
-  \includegraphics{figs/phases_1.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/phases_2.pdf}
-  \hspace{-3em}
-  \includegraphics{figs/phases_3.pdf}
-
-  \caption{
-    \textbf{Topological phase diagram.}
-    Topological phases of the model for three different homogeneous covariance
-    functions. The onset transition $V_\text{on}$, shattering transition
-    $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$
-    are indicated when they exist. In the limit of $\alpha\to0$, the behavior
-    of level sets of the spherical spin glasses are recovered: the final 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 for mixed models with
-    inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of
-    $V_\text{sh}$.
-  } \label{fig:phases}
-\end{figure}
-
 \begin{figure}
   \includegraphics{figs/phases_12_1.pdf}
   \hspace{-2.85em}
@@ -506,7 +574,7 @@ case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$.
     Topological phases for models with a covariance function
     $f(q)=(1-\lambda)q+\lambda\frac12q^2$ for several values of $\lambda$,
     interpolating between homogeneous linear ($\lambda=0$) and quadratic ($\lambda=1$) constraints.
-  }
+  } \label{fig:crossover}
 \end{figure}
 
 The phase diagram implied by these transitions is shown in
@@ -589,98 +657,6 @@ 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.
 
-\begin{figure}
-  \includegraphics{figs/regime_1.pdf}
-  \hspace{-3.5em}
-  \includegraphics{figs/regime_2.pdf}
-  \hspace{-3.5em}
-  \includegraphics{figs/regime_3.pdf}
-  \hspace{-3.5em}
-  \includegraphics{figs/regime_4.pdf}
-
-  \caption{
-    \textbf{Behavior of the action in four regimes.} The effective action $\mathcal S_\chi$ as a function of overlap $m$ with
-    the height axis for our model with $f(q)=\frac12q^3$ and $\alpha=\frac12$
-    at three different target values $V_0$. \textbf{Left: the connected
-    regime.} The action is maximized with $\mathcal S_\chi(m^*)=0$, and no
-    stationary points are found with overlap less than $m_\text{min}$ with a
-    random point on the sphere. \textbf{Center: the onset regime.} The action
-    is maximized with $\mathcal S_\chi(m_\text{min})>0$, and is positive up
-    to $m_\text{max}$. No stationary points are found with overlap less than $m_\text{min}$. \textbf{Right: the shattered regime.} The action is
-    maximized with $\mathcal S_\chi(0)>0$, and is positive up to
-    $m_\text{max}$.
-  }
-\end{figure}
-
-With these caveats in mind, we can combine the information from the previous
-calculations to reason about what the topology and geometry of $\Omega$ should
-be. We know what the average Euler characteristic is, and we know that it
-corresponds to the average number of stationary points. We also know these two
-averages correspond with each other when restricted to arbitrary latitudes $m$
-associated with an arbitrary height axis $\mathbf x_0$, and we know their value
-at each latitude. From this information, we draw the following inferences about
-the form of the solution manifold in our four possible regimes.
-
-\paragraph{The connected regime: \boldmath{$V_0^2<V_\text{on}^2$}.}
-
-In our calculation above, $\overline{\chi(\Omega)}=2$ could mean a fine-tuned
-average like this, or it could indicate the presence of manifold homeomorphic
-to $S^{N-M-1}$ for even $N-M-1$. In either case it strongly indicates a single
-connected component, whose few minima and maxima are usually found at the
-latitude $m^*$. Randomly chosen points on the sphere have a typical nearest
-overlap $m^*$ with the solution manifold, but can never have a smaller overlap than
-$m_\text{min}$, indicating that the manifold is extensive.
-
-\paragraph{The onset regime: \boldmath{$V_\text{on}^2<V_0^2<V_\text{sh}^2$}.}
-
-In this regime $\log\overline{\chi(\Omega)}=O(N)$ but a minimum overlap
-$m_\text{min}>0$ still exists. The minimum overlap indicates that the solution
-manifold is exclusively made up of extensive components, because the existence
-of small components would lead to stationary points near the equator with
-respect to a randomly chosen axis $\mathbf x_0$. The solution manifold is
-homeomorphic to a topological space with large Euler characteristic like the
-product or disjoint union of many spheres. In the former case we would have one
-topologically nontrivial connected component, while in the latter we would have
-many simple disconnected components; the reality could be a combination of the
-two. In the framework of this calculation, it is not possible to distinguish between
-these scenarios. In any case, the minima and maxima of the height on the
-solution manifold are typically found at latitude $m_\text{min}$ but are found
-in exponential number up to the latitude $m_\text{max}$.
-
-\paragraph{The shattered regime: \boldmath{$V_\text{sh}^2<V_0^2<V_\text{\textsc{sat}}^2$}.}
-
-Here $\log\overline{\chi(\Omega)}=O(N)$ and the primary contribution to the
-number of stationary points and to the Euler characteristic comes from the
-equator. This indicates the presence of a large number of small disconnected
-components to the solution manifold. While most minima and maxima of the height
-are located at the equator $m=0$, they are found in exponential number up to
-the latitude $m_\text{max}$.
-
-\paragraph{The \textsc{unsat} regime: \boldmath{$V_\text{\textsc{sat}}^2<V_0^2$}.}
-
-In this regime $\log\overline{\chi(\Omega)}<0$, indicating that in most
-realizations of the functions $V_k$ the set $\Omega$ is empty.
-
-\begin{figure}
-  \includegraphics[width=0.245\textwidth]{figs/connected.pdf}
-  \includegraphics[width=0.245\textwidth]{figs/coexist.pdf}
-  \includegraphics[width=0.245\textwidth]{figs/shattered.pdf}
-  \includegraphics[width=0.245\textwidth]{figs/gone.pdf}
-
-  \includegraphics{figs/bar.pdf}
-
-  \caption{
-    \textbf{Cartoon of the solution manifold.}
-    A low-dimensional sketch of the solution manifold in black, with stationary
-    points of the height function as red points.
-    The arrow shows the vector $\mathbf x_0$ defining the height
-    function. For $V_0<V_\text{on}$, the manifold has a single connected
-    component. Above the onset with $V_\text{on}<V_0<V_\text{sh}$, the manifold
-    has large connected components with nontrivial topology. Above the shattering transition, or
-    $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, small disconnected pieces appear. Finally, above the satisfiability transition
-    $V_\text{\textsc{sat}}$ the manifold vanishes.
-  } \label{fig:cartoons}
-\end{figure}
 
 \paragraph{}
 Fig.~\ref{fig:cartoons} shows a low-dimensional facsimile of what these
-- 
cgit v1.2.3-70-g09d2

