From e998093b001d91c2f1ff0173de67cecd035a6015 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 22 May 2023 00:12:49 +0200
Subject: More figure work.

---
 2-point.tex | 115 ++++++++++++++++++++++++++++++++++++++++++++----------------
 1 file changed, 84 insertions(+), 31 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 3ae4109..3ab4b07 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -90,17 +90,17 @@
 
 \maketitle
 \begin{abstract}
-  The family of mixed spherical models was recently found to
-  violate long-held assumptions about mean-field glassy dynamics. In
-  particular, the threshold energy, where most stationary points are marginal
-  and which in the simpler pure models attracts long-time dynamics, seems to
-  loose significance. Here, we compute the typical distribution of stationary
-  points relative to each other in mixed models with a replica symmetric
-  complexity. We examine the stability of nearby points, accounting for the
-  presence of an isolated eigenvalue in their spectrum due to their proximity.
-  Despite finding rich structure not present in the pure models, we find
-  nothing that distinguishes the points that do attract the dynamics. Instead,
-  we find new geometric significance of the old threshold energy.
+  The mixed spherical models were recently found to violate long-held
+  assumptions about mean-field glassy dynamics. In particular, the threshold
+  energy, where most stationary points are marginal and which in the simpler
+  pure models attracts long-time dynamics, seems to lose significance. Here,
+  we compute the typical distribution of stationary points relative to each
+  other in mixed models with a replica symmetric complexity. We examine the
+  stability of nearby points, accounting for the presence of an isolated
+  eigenvalue in their spectrum due to their proximity. Despite finding rich
+  structure not present in the pure models, we find nothing that distinguishes
+  the points that do attract the dynamics. Instead, we find new geometric
+  significance of the old threshold energy.
 \end{abstract}
 
 \tableofcontents
@@ -114,6 +114,8 @@
 
 \section{Model}
 
+\cite{Crisanti_1992_The, Crisanti_1993_The}
+
 The mixed spherical models are defined by the Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
   H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
@@ -154,6 +156,49 @@ where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivat
 
 \section{Results}
 
+\begin{figure}
+  \includegraphics{figs/gapped_min_energy.pdf}
+  \raisebox{1.2em}{\includegraphics{figs/gapped_min_energy_legend.pdf}}
+  \hfill
+  \includegraphics{figs/gapped_min_stability.pdf}
+  \raisebox{1.2em}{\includegraphics{figs/gapped_min_stability_legend.pdf}}
+
+  \caption{
+    The neighborhood of a reference minimum with $E_0=-1.71865<E_\mathrm{th}$
+    and $\mu_0=6.1>\mu_\mathrm m$. \textbf{Left:} The most common type of
+    stationary point lying at fixed overlap $q$ and energy $E_1$ from the
+    reference minimum. The black line gives the smallest or largest energies
+    where neighbors can be found at a given overlap. \textbf{Right:} The most
+    common type of stationary point lying at fixed overlap $q$ and stability
+    $\mu_1$ from the reference minimum. Note that this describes a different
+    set of stationary points than shown in the left plot. On both plots, the
+    shading of the righthand part depicts the state of an isolated eigenvalue
+    in the spectrum of the Hessian of the neighboring points. Those more
+    lightly shaded are minima with an isolated eigenvalue that does not change
+    their stability, i.e., $\lambda_\mathrm{min}>0$. Those more darkly shaded
+    are saddles with an isolated eigenvalue, either with many unstable
+    directions ($\mu_1<\mu_\mathrm m$) or with only one, corresponding to minima
+    destabilized by the isolated eigenvalue ($\mu_1>\mu_\mathrm m$). The
+    dot-dashed lines on both plots depict the trajectory of the solid line on
+    the other plot. In this case, the points lying nearest to the reference
+    minimum are saddles with $\mu<\mu_\mathrm m$, but with energies smaller than
+    the threshold energy.
+  } \label{fig:min.neighborhood}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/franz_parisi.pdf}
+
+  \caption{
+    Comparison of the lowest-energy stationary points at overlap $q$ with a
+    reference minimum of $E_0=-1.71865<E_\mathrm{th}$ and
+    $\mu_0=6.1>\mu_\mathrm m$ (yellow, top), and the zero-temperature Franz--Parisi potential
+    with respect to the same reference minimum (blue, bottom). The two curves
+    coincide precisely at their minimum $q=0$ and at the local maximum $q\simeq0.5909$.
+  } \label{fig:franz-parisi}
+\end{figure}
+
 \section{Complexity}
 
 We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
@@ -164,13 +209,18 @@ We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
   \big|\det\operatorname{Hess}H(\mathbf s,\omega)\big|
 \end{equation}
 which counts stationary points of the function $H$. More interesting is the
-measure conditioned on the energy density $E$ and stability $\mu$,
+measure conditioned on the energy density $E$ and stability $\mu$ of the points,
 \begin{equation}
   d\nu_H(\mathbf s,\omega\mid E,\mu)
   =d\nu_H(\mathbf s,\omega)\,
   \delta\big(NE-H(\mathbf s)\big)\,
   \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\omega)\big)
 \end{equation}
+While $\mu$ is strictly the trace of the Hessian, we call it the stability
+because in this family of models all stationary points have a bulk spectrum of
+the same shape, shifted by different constants. The stability $\mu$ sets this
+shift, and therefore determines if the spectrum has bulk support on zero.
+
 We want the typical number of stationary points with energy density
 $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference
 stationary point of energy density $E_0$ and stability $\mu_0$.
@@ -450,8 +500,6 @@ then the result is
 \end{equation}
 where each block is a constant $n\times n$ matrix.
 
-\section{Replica symmetric case}
-
 We focus now on models whose equilibrium has at most one level of replica
 symmetry breaking, which corresponds to a replica symmetric complexity.
 For these models, the saddle point parameters for the reference stationary
@@ -509,7 +557,7 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case,
 \end{align}
 
 \begin{equation}
-  \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(4f''(1)-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q)
+  \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q)
   +O\big((1-q)^2\big)
 \end{equation}
 \begin{equation}
@@ -575,21 +623,25 @@ limit of $\beta\to\infty$. There are two possibilities. First, in the replica
 symmetric case $x=1$, and in the limit of large $\beta$ $q_0$ will scale like
 $q_0=1-(y_0\beta)^{-1}$. Inserting this, the limit is
 \begin{equation}
-  V_\infty=-\hat\beta_0 f(q)-r^{11}_\mathrm df'(q)q-\frac12\left(y_0(1-q^2)+\frac{f'(1)^2-f'(q)^2}{y_0f'(1)}\right)
+  V_\infty^{\textsc{rs}}=-\hat\beta_0 f(q)-r^{11}_\mathrm df'(q)q-\frac12\left(y_0(1-q^2)+\frac{f'(1)^2-f'(q)^2}{y_0f'(1)}\right)
 \end{equation}
 The saddle point in $y_0$ can now be taken, taking care to choose the solution for $y_0>0$. This gives
 \begin{equation}
-  V_0^{\textsc{rs}}=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\sqrt{(1-q^2)\left(1-\frac{f'(q)^2}{f'(1)^2}\right)}
+  V_\infty^{\textsc{rs}}=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\sqrt{(1-q^2)\left(1-\frac{f'(q)^2}{f'(1)^2}\right)}
 \end{equation}
 The second case is when the inner statistical mechanics problem has replica symmetry breaking. Here, $q_0$ approaches a nontrivial limit, but $x=z\beta^{-1}$ approaches zero and $q_1=1-(y_1\beta)^{-1}$ approaches one.
 \begin{equation}
-  V_0^{\oldstylenums{1}\textsc{rsb}}=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\frac12\left(
-    z(f(1)-f(q_0))+\frac{f'(1)}{y_1}-\frac{y_1(q^2-q_0)}{1+y_1z(1-q_0)}-(1+y_0z(1-q_0))\frac{f'(q)^2}{y_1f'(1)}+\frac1z\log\left(1+zy_1(1-q_0)\right)
-  \right)
+  \begin{aligned}
+    V_\infty^{\oldstylenums{1}\textsc{rsb}}(q\mid E_0,\mu_0)
+    &=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\frac12\bigg(
+      z(f(1)-f(q_0))+\frac{f'(1)}{y_1}-\frac{y_1(q^2-q_0)}{1+y_1z(1-q_0)} \\
+    &\hspace{8pc}-(1+y_1z(1-q_0))\frac{f'(q)^2}{y_1f'(1)}+\frac1z\log\left(1+zy_1(1-q_0)\right)
+    \bigg)
+  \end{aligned}
 \end{equation}
-Though the saddle point in $y_1$ can technically be evaluated in this
-expression, it delivers no insight. The final potential is found by taking the
-saddle over $z$, $y_1$, and $q_0$.
+Though the saddle point in $y_1$ can be evaluated in this expression, it
+delivers no insight. The final potential is found by taking the saddle over
+$z$, $y_1$, and $q_0$.
 
 \section{Isolated eigenvalue}
 
@@ -603,17 +655,18 @@ bulk. We study the possibility of \emph{one} stray eigenvalue.
 We use a technique recently developed to find the smallest eigenvalue of a
 random matrix \cite{Ikeda_2023_Bose-Einstein-like}. One defines a quadratic
 statistical mechanics model with configurations defined on the sphere, whose
-interaction tensor is given by the matrix in question. By construction, the
+interaction tensor is given by the matrix of interest. By construction, the
 ground state is located in the direction of the eigenvector associated with the
 smallest eigenvalue, and the ground state energy is proportional to that
 eigenvalue.
 
-Our matrix is the Hessian evaluated at a stationary point of the mixed $p$-spin
+Our matrix of interest is the Hessian evaluated at a stationary point of the mixed spherical
 model, conditioned on the relative position, energies, and stabilities
 discussed above. We must restrict the artificial spherical model to lie in the
 tangent plane of the `real' spherical configuration space at the point of
-interest. The free energy of this model given a point $\mathbf s$ and a
-specific realization of the disordered Hamiltonian is
+interest, to avoid our eigenvector pointing in a direction that violates the
+spherical constraint. The free energy of this model given a point $\mathbf s$
+and a specific realization of the disordered Hamiltonian is
 \begin{equation}
   \begin{aligned}
     \beta F_H(\beta\mid\mathbf s)
@@ -627,7 +680,7 @@ specific realization of the disordered Hamiltonian is
 \end{equation}
 where the first $\delta$-function keeps the configurations in the tangent
 plane, and the second enforces the spherical constraint. We have anticipated
-treated the logarithm with replicas already. We are of course interested in
+treating the logarithm with replicas. We are of course interested in
 points $\mathbf s$ that have certain properties: they are stationary points of
 $H$ with given energy density and stability, and fixed overlap from a reference
 configuration $\pmb\sigma$. We therefore average the free energy above over
@@ -639,7 +692,7 @@ such points, giving
     &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1)
   \end{aligned}
 \end{equation}
-again anticipating the use of replicas. Finally, the reference configuration $\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
+again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
 \begin{equation}
   \begin{aligned}
     F_H(\beta\mid\epsilon_1,\mu_1,\epsilon_2,\mu_2,q)
@@ -1062,14 +1115,14 @@ for
   &
   C_{33}
   =
-    -\frac{1-q^2}{(1-q_0)^2}-\frac{r^{11}_d-r^{11}_0}{1-q_0}\left[
+    -\frac{r^{11}_d-r^{11}_0}{1-q_0}\left[
       \frac{r^{11}_d-r^{11}_0}{1-q_0}f'(1)
       -2\left(
         \frac{qr^{01}-r^{11}_0}{1-q_0}+\frac{1-q^2}{1-q_0}\frac{r^{11}_d-r^{11}_0}{1-q_0}
         \right)(f'(1)-f'(q_0))
         -2\frac{qr^{00}-r^{10}}{1-q_0}f'(q)
     \right]\\
-  &\qquad-\frac{(r^{10}-qr^{00}_d)^2}{(1-q_0)^2}f'(1)
+  &\qquad-\frac{1-q^2}{(1-q_0)^2}-\frac{(r^{10}-qr^{00}_d)^2}{(1-q_0)^2}f'(1)
     \\
   &
   C_{34}
-- 
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