From f0d4764926b291c57146079b3dabe77c02b86bac Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 00:30:43 +0200
Subject: Lots of writing, figures added, and changed to SciPost submission.

---
 topology.tex | 377 ++++++++++++++++++++++++++++++++++++++---------------------
 1 file changed, 246 insertions(+), 131 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 3e8ce9b..f420411 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1,27 +1,19 @@
-\documentclass[a4paper,fleqn]{article}
+\documentclass[submission, Phys]{SciPost}
 
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
-\usepackage{amsmath,amssymb,latexsym,graphicx}
-\usepackage{newtxtext,newtxmath}
-\usepackage{bbold}
+\usepackage{amsmath,latexsym,graphicx}
+\usepackage[bitstream-charter]{mathdesign}
 \usepackage[dvipsnames]{xcolor}
-\usepackage[
-  colorlinks=true,
-  urlcolor=BlueViolet,
-  citecolor=BlueViolet,
-  filecolor=BlueViolet,
-  linkcolor=BlueViolet
-]{hyperref}
-\usepackage[
-  style=phys,
-  eprint=true,
-  maxnames = 100
-]{biblatex}
 \usepackage{anyfontsize,authblk}
-\usepackage{fullpage}
 
-\addbibresource{topology.bib}
+\hypersetup{
+  colorlinks=true,
+  urlcolor={blue!50!black},
+  citecolor={blue!50!black},
+  filecolor={blue!50!black},
+  linkcolor={blue!50!black}
+}
 
 \title{
   On the topology of solutions to random continuous constraint satisfaction problems
@@ -39,15 +31,17 @@
   independent Gaussian coefficients on the $(N-1)$-sphere. 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 scaling of $M$ with $N$. When $\alpha=M/N$ is held constant,
-  the average characteristic is 2 whenever solutions exist. When $M$ is
-  constant, the average characteristic is also 2 up until a transition value
-  $M_\textrm{th}$, above which it is exponentially large in $N$. To better
-  interpret these results, we compute the average number of stationary points
-  of a test function on the solution manifold. In both regimes, this reveals
-  another transition between a regime with few and one with exponentially many
-  stationary points. We conjecture that this transition corresponds to a
-  geometric rather than a topological transition.
+  depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the
+  average characteristic is 2 and there is a single connected component, while
+  for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ a large connected
+  component coexists with many disconnected components. When
+  $\alpha>\alpha_\text{shatter}$ the large connected component vanishes, and the
+  entire manifold vanishes for $\alpha>\alpha_\text{\textsc{sat}}$. In the
+  limit $\alpha\to0$ there is a correspondence between this problem and the
+  topology of constant-energy level sets in the spherical spin glasses. We
+  conjecture that the energy $E_\text{shatter}$ associated with the vanishing of
+  the large connected component corresponds to the asymptotic limit of gradient
+  descent from a random initial condition.
 \end{abstract}
 
 \tableofcontents
@@ -71,7 +65,12 @@ to solve them \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties,
 Beneventano_2023_On}. Here, we show how \emph{topological} information about
 the set of solutions can be calculated in a simple model of satisfying random
 nonlinear equalities. This allows us to reason about the connectivity of this
-solution set.
+solution set. The topological properties revealed by this calculation yield
+surprising results for the well-studied spherical spin glasses, where a
+topological transition thought to occur at a threshold energy $E_\text{th}$
+where marginal minima are dominant is shown to occur at a different energy
+$E_\text{shatter}$. We conjecture that this difference resolves an outstanding
+problem in gradient descent dynamics in these systems.
 
 We consider the problem of finding configurations $\mathbf x\in\mathbb R^N$
 lying on the $(N-1)$-sphere $\|\mathbf x\|^2=N$ that simultaneously satisfy $M$
@@ -80,20 +79,21 @@ constant $V_0\in\mathbb R$. The nonlinear constraints are taken to be centered
 Gaussian random functions with covariance
 \begin{equation} \label{eq:covariance}
   \overline{V_i(\mathbf x)V_j(\mathbf x')}
-  =\delta_{ij}F\left(\frac{\mathbf x\cdot\mathbf x'}N\right)
+  =\delta_{ij}f\left(\frac{\mathbf x\cdot\mathbf x'}N\right)
 \end{equation}
-for some choice of function $F$. When the covariance function $F$ is polynomial, the
-$V_k$ are also polynomial, with a term of degree $p$ in $F$ corresponding to
+for some choice of function $f$. When the covariance function $f$ is polynomial, the
+$V_k$ are also polynomial, with a term of degree $p$ in $f$ corresponding to
 all possible terms of degree $p$ in $V_k$. In particular, taking
 \begin{equation}
   V_k(\mathbf x)
-  =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{F^{(p)}(0)}{N^p}}
+  =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{f^{(p)}(0)}{N^p}}
   \sum_{i_1\cdots i_p}^NJ^{(k,p)}_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
 \end{equation}
 with the elements of the tensors $J^{(k,p)}$ as independently distributed
 unit normal random variables satisfies \eqref{eq:covariance}. The size of the
-series coefficients of $F$ therefore control the variances in the coefficients
-of random polynomial constraints.
+series coefficients of $f$ therefore control the variances in the coefficients
+of random polynomial constraints. When $M=1$, this problem corresponds to the
+level set of a spherical spin glass with energy density $E=\sqrt{N}V_0$.
 
 
 This problem or small variations thereof have attracted attention recently for
@@ -112,21 +112,94 @@ directly.
 This set
 can be written as
 \begin{equation}
-  \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=0
+  \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=V_0
   \;\forall\;k=1,\ldots,M\big\}
 \end{equation}
-Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points. The conditions for a singular point are that
+Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points.\footnote{The conditions for a singular point are that
 $0=\frac\partial{\partial\mathbf x}V_k(\mathbf x)$ for all $k$. This is
 equivalent to asking that the constraints $V_k$ all have a stationary point at
 the same place. When the $V_k$ are independent and random, this is vanishingly
 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.
+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, respectively the alternating sum and
+direct sum of the Betti numbers of $\Omega$. 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.
+
+\section{Results}
 
-When $M$ is too large, no solutions exist and $\Omega$ becomes the empty set.
-Following previous work, a replica symmetric equilibrium calculation using the
-cost function \eqref{eq:cost} predicts that solutions vanish when the ratio
-$\alpha=M/N$ is larger than $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$. Based on the results of this paper, and the fact that this $\alpha_\text{\textsc{sat}}$ is consistent 
+\subsection{Topology of solutions to many equations and the satisfiability transition}
+
+\begin{figure}
+  \includegraphics{figs/phases_1.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_2.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_3.pdf}
+
+  \caption{
+    Topological phases of the model for three different homogeneous covariance
+    functions. The onset transition $V_\text{onset}$, shattering transition
+    $V_\text{shatter}$, 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
+    $V_\text{sh}$.
+  }
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=0.24\textwidth]{figs/connected.pdf}
+  \hfill
+  \includegraphics[width=0.24\textwidth]{figs/coexist.pdf}
+  \hfill
+  \includegraphics[width=0.24\textwidth]{figs/shattered.pdf}
+  \hfill
+  \includegraphics[width=0.24\textwidth]{figs/gone.pdf}
+
+  \includegraphics{figs/bar.pdf}
+
+  \caption{
+    Cartoon of the topology of the solution manifold implied by our
+    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
+    function. The region of solutions is marked in black, and the critical points
+    of the height function restricted to this region are marked with a point.
+    For $\alpha<1$, there are few simply connected regions with most of the
+    minima and maxima contributing to the Euler characteristic concentrated at
+    the height $m^*$. For $\alpha\geq1$, there are many simply
+    connected regions and most of their minima and maxima are concentrated at
+    the equator.
+  } \label{fig:cartoons}
+\end{figure}
+
+\subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
+
+\begin{figure}
+  \includegraphics{figs/dynamics_2.pdf}
+  \hspace{-0.5em}
+  \includegraphics{figs/dynamics_3.pdf}
+
+  \caption{
+    Comparison of the shattering energy $E_\text{sh}$ with the asymptotic
+    performance of gradient descent from a random initial condition in $p+s$
+    models with $p=2$ and $p=3$ and varying $s$. The values of $\lambda$ depend on $p$ and $s$ and are taken from \cite{Folena_2023_On}. The points show the asymptotic performance
+    extrapolated using two different methods and have unknown uncertainty, from \cite{Folena_2023_On}. Also
+    shown is the annealed threshold energy $E_\text{th}$, where marginal minima
+    are the most common type of stationary point. The section of $E_\text{sh}$
+    that is dashed on the left plot indicates the continuation of the annealed
+    result, whereas the solid portion gives the value calculated with a
+    {\oldstylenums 1}\textsc{frsb} ansatz.
+  }
+\end{figure}
+
+\section{The average Euler characteristic}
 
 The Euler characteristic $\chi$ of a manifold is a topological invariant \cite{Hatcher_2002_Algebraic}. It is
 perhaps most familiar in the context of connected compact orientable surfaces, where it
@@ -153,7 +226,7 @@ points. Since the sign of the determinant of the Hessian matrix of $H$ at a
 stationary point is equal to its index, if we count stationary points including
 the sign of the determinant, we arrive at the Euler characteristic, or
 \begin{equation} \label{eq:kac-rice}
-  \chi=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x)
+  \chi(\Omega)=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x)
 \end{equation}
 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
@@ -167,8 +240,6 @@ 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.
 
-\section{The average Euler characteristic}
-
 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
@@ -176,7 +247,7 @@ each of the $V_k$, resulting in the Lagrangian
 \begin{equation}
   L(\mathbf x,\pmb\omega)
   =H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big)
-  +\sum_{k=1}^M\omega_kV_k(\mathbf x)
+  +\sum_{k=1}^M\omega_k\big(V_k(\mathbf x)-V_0\big)
 \end{equation}
 The integral over $\Omega$ in \eqref{eq:kac-rice} then becomes
 \begin{equation} \label{eq:kac-rice.lagrange}
@@ -202,11 +273,11 @@ odd-dimensional manifolds. This is the signature of it in this problem.
 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}}
+  &=\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\,
+  &=\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}
 
@@ -240,7 +311,7 @@ Since this is an exponential integrand linear in the functions $V_k$, we can ave
         +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big)
         -iV_0\sum_{k=1}^M\sigma_k(1)
       \right] \\
-      -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
+      -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
     \Bigg\}
   \end{aligned}
 \end{equation}
@@ -254,8 +325,8 @@ Performing that integral yields
         H(\pmb\phi(1))
         +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big)
       \right] \\
-    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1}
-      -\frac M2\log\operatorname{sdet}F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
+    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1}
+      -\frac M2\log\operatorname{sdet}f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
     \Bigg\}
   \end{aligned}
 \end{equation}
@@ -278,8 +349,8 @@ These new variables can replace $\pmb\phi$ in the integral using a generalized H
         \mathbb M(1)
         +\frac{i}2\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
       \right] \\
-    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F(\mathbb Q)^{-1}(1,2)
-      -\frac M2\log\operatorname{sdet}F(\mathbb Q)
+    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
+      -\frac M2\log\operatorname{sdet}f(\mathbb Q)
       +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
     \Bigg\}
   \end{aligned}
@@ -339,8 +410,7 @@ We can treat the integral over $\sigma_0$ immediately. It gives
   =2\pi\,\delta(C-1)\,\delta(G+R)\,\bar HH
 \end{equation}
 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.
-
+as setting everything depending on $\bar H$ and $H$ to zero. With these solutions inserted, the remaining remaining terms in the exponential give
 \begin{equation}
   \begin{aligned}
     \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
@@ -360,24 +430,59 @@ as setting everything depending on $\bar H$ and $H$ to zero.
   +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H
 \end{equation}
 \begin{equation}
-  \int d1\,d2\,F(\mathbb Q)^{-1}(1,2)
+  \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}
+\begin{equation}
+  D=-\frac{m+R}{1-m^2} \qquad \hat m=0
+\end{equation}
+\begin{equation}
+  \mathcal S(R,D,m,\hat m\mid\alpha,V_0)
+  =\hat m-\frac\alpha2\left[
+    \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
+    +V_0^2\left(f(1)+\frac{f'(1)R^2}{D}\right)^{-1}
+  \right]
+  +\frac12\log\left(
+    1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+  \right)
+\end{equation}
+When $\alpha<\alpha_\text{onset}$ this potential has maxima at $\pm m^*$ with
+$m^*=-R^*$ where its value is zero.
+\begin{equation}
+  \alpha_\text{onset}=1-\left(\frac{V_0^2}{V_0^2+f(1)}\right)^2
+\end{equation}
+\begin{equation}
+  \alpha_\text{shatter}=4V_0^2f(1)f'(1)\frac{f'(1)-f(1)}{\big((V_0^2+f(1))f'(1)-f(1)^2\big)^2}
+\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 $E=v_0$.
+\section{Complexity of the height function}
 
-$v_0^{\chi>2}=\sqrt{2f(1)}$
+\section{Implications for the spherical spin glasses}
 
-$v_0^{m=0}=2\sqrt{f(1)-\frac{f(1)^2}{f'(1)}}$
+As indicated earlier, for $M=1$ the solution manifold corresponds to the energy
+level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
+results from the previous sections follow, and can be translated to the spin
+glasses by taking the limit $\alpha\to0$ while scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields
+\begin{equation}
+  E_\text{onset}=\pm\sqrt{2f(1)}
+\end{equation}
+\begin{equation}
+  E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
+\end{equation}
+for the energies at which level sets of the spherical spin glasses have
+disconnected pieces appear, and that at which a large connected component
+vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
+$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
+models. This is expected, since threshold energy, defined as the place where
+marginal minima are dominant in the landscape, is widely understood as the
+place where level sets are broken into pieces.
+
+However, for general mixed models the threshold energy is
+\begin{equation}
+  E_\mathrm{th}=\pm\frac{f(1)[f''(1)-f'(1)]+f'(1)^2}{f'(1)\sqrt{f''(1)}}
+\end{equation}
+which generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
 
 \subsection{What does the average Euler characteristic tell us?}
 
@@ -426,27 +531,6 @@ 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[width=0.32\columnwidth]{figs/connected.pdf}
-  \hfill
-  \includegraphics[width=0.32\columnwidth]{figs/shattered.pdf}
-  \hfill
-  \includegraphics[width=0.32\columnwidth]{figs/gone.pdf}
-
-  \includegraphics{figs/bar.pdf}
-
-  \caption{
-    Cartoon of the topology of the solution manifold implied by our
-    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
-    function. The region of solutions is marked in black, and the critical points
-    of the height function restricted to this region are marked with a point.
-    For $\alpha<1$, there are few simply connected regions with most of the
-    minima and maxima contributing to the Euler characteristic concentrated at
-    the height $m^*$. For $\alpha\geq1$, there are many simply
-    connected regions and most of their minima and maxima are concentrated at
-    the equator.
-  } \label{fig:cartoons}
-\end{figure}
 
 
 \cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
@@ -471,58 +555,89 @@ have characteristics of either 0 or 4.
 
 \section{Interpretation of our results}
 
-\paragraph{Quenched average of the Euler characteristic.}
-
-\begin{equation}
-  D=\beta R
-  \qquad
-  \hat\beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}}
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1R-\omega_1^2D)f'(C)+\omega_1^2(R^2-G^2)f''(C)+\log\det\frac{CD+R^2}{G^2}
+  \right]
+\]
+$D=\beta R$, $R=r_dI$, $G=-r_dI$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1-\omega_1^2\beta)r_df'(1)+\log\det(\beta r_d^{-1}C+I)
+  \right]
+\]
+$\beta=\hat\omega_1/\omega_1$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+\omega_1\hat\omega_1r_df'(1)+\log\det(\hat\omega_1\omega_1^{-1}r_d^{-1}C+I)
+  \right]
+\]
+$z=r_d\omega_1$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+\hat \omega_1 zf'(1)+\log\det(\hat\omega_1C/z+I)
+  \right]
+\]
+\[
+  \log\chi
+  =-\hat\omega_1 E
+  +\frac12[2\omega_1\hat\omega_1r_d-\omega_1^2d_d]f'(1)
+  +\frac12\int_0^1dq\,\left[
+    \hat\omega_1^2f''(q)\chi(q)
+    +\frac1{\chi(q)+r_d^2/d_d}
+  \right]
+  -\frac12\log r_d^2
+\]
+\[
+  0=\frac12\hat\omega_1^2f''(q)-\frac12\frac1{[\chi(q)+r_d^2/d_d]^2}
+\]
+\[
+  \chi_0(q)=\frac1{\hat\omega_1}f''(q)^{-1/2}-\frac{r_d^2}{d_d}
+\]
+
+\[
+  \chi(q)=\begin{cases}
+    \chi_0(q) & q < q_0 \\
+    1-(1-m)q_1-mq & q_0 < q < q_1 \\
+    1-q & q > q_1
+  \end{cases}
+\]
+\[
+  0=\hat\omega_1r_d-\omega_1d_d
   \qquad
-  \hat m=0
-\end{equation}
-
-\begin{align}
-  &\mathcal S(m,C,R)
-  =\frac12\log\det\big[I+\hat\beta R^{-1}(C-m^2)\big] \notag \\
-  &\quad-\frac\alpha2\log\det\big[I+\hat\beta\big(R\odot f'(C)\big)^{-1}f(C)\big]
-\end{align}
-
-The quenched average of the Euler characteristic in the replica symmetric ansatz becomes for $1<\alpha<\alpha_\text{\textsc{sat}}$
-\begin{align}
-  \frac1N\overline{\log\chi}
-  =\frac12\bigg[
-    \log\left(-\frac 1{\tilde r_d}\right)
-    -\alpha\log\left(
-      1-\Delta f\frac{1+\tilde r_d}{f'(1)\tilde r_d}
-    \right) \notag \\
-    -\alpha f(0)\left(\Delta f-\frac{f'(1)\tilde r_d}{1+\tilde r_d}\right)^{-1}
-  \bigg]
-\end{align}
-where $\Delta f=f(1)-f(0)$ and $\tilde r_d$ is given by
-\begin{align}
-  \tilde r_d
-  =-\frac{f'(1)f(1)-\Delta f^2}{2(f'(1)-\Delta f)^2}
-  \bigg(
-    \alpha-2+\frac{2f'(1)f(0)}{f'(1)f(1)-\Delta f^2} \notag\\
-    +\sqrt{
-      \alpha^2
-      -4\alpha\frac{f'(1)f(0)\Delta f\big(f'(1)-\Delta f\big)}{\big(f'(1)f(1)-\Delta f^2\big)^2}
-    }
-  \bigg)
-\end{align}
-When $\alpha\to\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ from below, $\tilde r_d\to -1$, which produces $N^{-1}\overline{\log\chi}\to0$.
+  \omega_1=\hat\omega_1\frac{r_d}{d_d}
+\]
+\[
+  \log\chi
+  =-\hat\omega_1 E
+  +\frac12\hat\omega_1^2r_d^2/d_df'(1)
+  +\frac12\int_0^1dq\,\left[
+    \hat\omega_1^2f''(q)\chi(q)
+    +\frac1{\chi(q)+r_d^2/d_d}
+  \right]
+  -\frac12\log r_d^2
+\]
+\[
+  0=-\frac{\hat\omega_1^2f'(1)}{d_d}+\int_0^1dq\,\frac1{(r_d^2/d_d+\chi(q))^2}
+\]
+\[
+  d_d=-\frac{1+r_d}{\int dq\,\chi(q)}r_d
+\]
+
+\paragraph{Acknowledgements}
+The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
 
-\section*{Acknowledgements}
-\addcontentsline{toc}{section}{Acknowledgements}
+\paragraph{Funding information}
 JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
-The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
 
 \appendix
 
 \section{Calculation of the prefactor of the average Euler characteristic}
 \label{sec:prefactor}
 
-\printbibliography
-\addcontentsline{toc}{section}{References}
+\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
+\label{sec:1frsb}
+
+\bibliography{topology}
 
 \end{document}
-- 
cgit v1.2.3-70-g09d2


From eaab0737ca427954587eb7aca26bea072d9abe65 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 00:56:54 +0200
Subject: Some tweaks.

---
 topology.tex | 46 ++++++++++++++++++++++------------------------
 1 file changed, 22 insertions(+), 24 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index f420411..4d2d660 100644
--- a/topology.tex
+++ b/topology.tex
@@ -134,6 +134,28 @@ putting strong constraints on the resulting topology and geometry.
 
 \subsection{Topology of solutions to many equations and the satisfiability transition}
 
+\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{
+    Cartoon of the topology of the solution manifold implied by our
+    calculation. 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 a large connected component around the equator, and many disconnected
+    pieces in a certain range of latitudes. Above the shattering transition, or
+    $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, the large connected
+    component vanishes and small disconnected pieces occupy the entire equatorial
+    region. Finally, above the satisfiability transition
+    $V_\text{\textsc{sat}}$ the manifold vanishes.
+  } \label{fig:cartoons}
+\end{figure}
+
 \begin{figure}
   \includegraphics{figs/phases_1.pdf}
   \hspace{-3em}
@@ -155,30 +177,6 @@ putting strong constraints on the resulting topology and geometry.
   }
 \end{figure}
 
-\begin{figure}
-  \includegraphics[width=0.24\textwidth]{figs/connected.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/coexist.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/shattered.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/gone.pdf}
-
-  \includegraphics{figs/bar.pdf}
-
-  \caption{
-    Cartoon of the topology of the solution manifold implied by our
-    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
-    function. The region of solutions is marked in black, and the critical points
-    of the height function restricted to this region are marked with a point.
-    For $\alpha<1$, there are few simply connected regions with most of the
-    minima and maxima contributing to the Euler characteristic concentrated at
-    the height $m^*$. For $\alpha\geq1$, there are many simply
-    connected regions and most of their minima and maxima are concentrated at
-    the equator.
-  } \label{fig:cartoons}
-\end{figure}
-
 \subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
 
 \begin{figure}
-- 
cgit v1.2.3-70-g09d2


From b3e6c5a2533e90cef7c27ce4d3d52bbfda54e7aa Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 09:32:16 +0200
Subject: Font tweaks.

---
 topology.tex | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 4d2d660..4c14207 100644
--- a/topology.tex
+++ b/topology.tex
@@ -7,6 +7,12 @@
 \usepackage[dvipsnames]{xcolor}
 \usepackage{anyfontsize,authblk}
 
+\urlstyle{sf}
+
+% Fix \cal and \mathcal characters look (so it's not the same as \mathscr)
+\DeclareSymbolFont{usualmathcal}{OMS}{cmsy}{m}{n}
+\DeclareSymbolFontAlphabet{\mathcal}{usualmathcal}
+
 \hypersetup{
   colorlinks=true,
   urlcolor={blue!50!black},
@@ -104,7 +110,7 @@ tissues \cite{Fyodorov_2019_A, Fyodorov_2020_Counting,
 Montanari_2024_On, Kent-Dobias_2024_Conditioning, Kent-Dobias_2024_Algorithm-independent}. In each of these cases, the authors studied properties of
 the cost function
 \begin{equation} \label{eq:cost}
-  \mathcal C(\mathbf x)=\frac12\sum_{k=1}^MV_k(\mathbf x)^2
+  \mathscr C(\mathbf x)=\frac12\sum_{k=1}^MV_k(\mathbf x)^2
 \end{equation}
 which achieves zero only for configurations that satisfy all the constraints.
 Here we dispense with the cost function and study the set of solutions
-- 
cgit v1.2.3-70-g09d2

