From 2e83540493363b186a9f049046640a8107c228b1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 16 Jun 2023 21:33:04 +0200
Subject: Some fixes and tweaks.

---
 when_annealed.tex | 48 ++++++++++++++++++++++++++++++++----------------
 1 file changed, 32 insertions(+), 16 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index 6e63ded..ae7e3b8 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -135,11 +135,11 @@ temperature.\footnote{
 This is a very strong condition on the form of equilibrium order. Note that
 non-convex $f$ does not imply that you will see nontrivial correlations between
 states at some temperature. In the $3+s$ models we consider here, models with
-$s>8$ have non-convex $f$ and those with $s\geq8$ have convex $f$ independent
+$s>8$ have non-convex $f$ and those with $s\leq8$ have convex $f$ independent
 of $\lambda$. Second, the characterization of the ground state has been made
 \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
 Crisanti_2011_Statistical, Auffinger_2022_The}. In the $3+s$ models we
-consider, for $s>12.430\ldots$ nontrivial ground state configurations appear in
+consider, for $s>12.430...$ nontrivial ground state configurations appear in
 certain ranges of $\lambda$. These bounds on equilibrium order are shown in
 Fig.~\ref{fig:phases}, along with our result in this paper for where the
 complexity has nontrivial \textsc{rsb}. As evidenced in that figure,
@@ -166,12 +166,12 @@ $\mu=\mu_\mathrm m$, the spectrum has a pseudogap, and we have marginal minima.
   \caption{
     A phase diagram of the boundaries we discuss in this paper for the $3+s$
     model with $f=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$. The blue region
-    shows where there exist some stationary points whose complexity is
-    {\oldstylenums1}\textsc{rsb}, and is given by $G_f>0$ where $G_f$ is found
-    in \eqref{eq:condition}. The yellow region shows where $f$ is not convex
-    and therefore \textsc{rsb} solutions are possible in equilibrium. The green
-    region shows where \textsc{rsb} solutions are correct at the ground state,
-    adapted from \cite{Auffinger_2022_The}.
+    shows models which have some stationary points with nontrivial
+    {\oldstylenums1}\textsc{rsb} structure, and is given by $G_f>0$ where $G_f$
+    is found in \eqref{eq:condition}. The yellow region shows where $f$ is not
+    convex and therefore nontrivial \textsc{rsb} solutions are possible in
+    equilibrium. The green region shows where \textsc{rsb} solutions are
+    correct at the ground state, adapted from \cite{Auffinger_2022_The}.
   } \label{fig:phases}
 \end{figure}
 
@@ -278,7 +278,9 @@ solution for points describing saddles ($\mu<\mu_\mathrm m$), this line is
 \begin{equation} \label{eq:extremal.line}
   \mu_0=-\frac1{z_f}\left(2Ef'f''+\sqrt{2f''u_f\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}\right)
 \end{equation}
-where we define for brevity (here and elsewhere) the constants
+where we have chosen the lower branch as a convention (see
+Fig.~\ref{fig:complexity_35}) and where we define for brevity (here and
+elsewhere) the constants
 \begin{align}
   u_f&=f(f'+f'')-f'^2
   &&
@@ -294,14 +296,16 @@ these constants are positive. We also define $E_\textrm{min}$, the minimum
 energy at which saddle points with an extensive number of downward directions
 are found, as the energy for which $\mu_0(E_\mathrm{min})=\mu_\mathrm m$.
 
-Let $M$ be the matrix of double partial derivatives of $\mathcal S$ with
+Let $M$ be the matrix of double partial derivatives of the action with
 respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point
 $x=1$ with the additional constraint that $q_1=1$ and along the extremal
 complexity line \eqref{eq:extremal.line}. We determine when a zero eigenvalue
 appears, indicating the presence of a bifurcating {\oldstylenums1}\textsc{rsb}
 solution, by solving $0=\det M$. We find
 \begin{equation}
-  \det M=-\bigg(\frac{\partial^2\mathcal S}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\bigg)^2\propto(ay^2+bE^2+2cyE-d)^2
+  \det M
+  =-\bigg(\frac{\partial^2\mathcal S_{\oldstylenums1\textsc{rsb}}}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\bigg)^2
+  \propto(ay^2+bE^2+2cyE-d)^2
 \end{equation}
 where $y=-\frac12z_f\mu-f'f''E$ is proportional to the square-root term in
 \eqref{eq:extremal.line} and the constants $a$, $b$, $c$, and $d$ are defined
@@ -464,7 +468,7 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
 There are implications for the emergence of \textsc{rsb} in equilibrium. Consider a specific $H$ with
 \begin{equation}
   H(\pmb\sigma)
-  =\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_2}
+  =\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p}
   +\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s}
 \end{equation}
 where the interaction tensors $J$ are drawn from zero-mean normal distributions
@@ -497,18 +501,24 @@ $s>2$, this transition line \emph{always} intersects the extremal line
 among some population of stationary points. However, it is likely that for much
 of the parameter space the so-called one-full \textsc{rsb}
 ({\oldstylenums1\textsc{frsb}}) is the correct solution, as it likely is for
-large $s$ in the $3+s$ model at hand. Further work to find the conditions for
-transitions of the complexity to these forms of order is necessary.
+large $s$ in the $3+s$ model at hand. Further work to find the
+conditions for transitions of the complexity to these forms of order is
+necessary. For values of $s$ where there is no
+\textsc{rsb} of any kind in the ground state, we expect that the
+{\oldstylenums1\textsc{rsb}} complexity is correct.
 
 What are the implications for dynamics? We find that nontrivial correlations
 tend to exist among saddle points with the maximum or minimum index possible at
 a given energy density, which are quite atypical in the landscape. However,
 these strangely correlated saddle points must descend to uncorrelated minima,
 which raises questions about whether structure on the boundary of a basin of
-attraction is influential to the dynamics that descends into that basin. These saddles might act as early-time separatrices for descent trajectories.  With
+attraction is influential to the dynamics that descends into that basin. These
+saddles might act as early-time separatrices for descent trajectories.  With
 large open problems in even the gradient decent dynamics on these models, it
 remains to be seen whether such structures could be influential
-\cite{Folena_2020_Rethinking, Folena_2023_On}.
+\cite{Folena_2020_Rethinking, Folena_2023_On}. This structure among saddles
+cannot be the only influence, since it seems that the $3+4$ model is `safe'
+from nontrivial \textsc{rsb} among saddles.
 
 We have determined the conditions under which the complexity of the mixed $3+s$
 spherical models has different quenched and annealed averages, as the result of
@@ -517,6 +527,12 @@ can arise among certain populations of saddle points even when the model is
 guaranteed to lack such correlations between equilibrium states, and exist for
 saddle points at a wide range of energies.
 
+
+\paragraph{Funding information}
+
+JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the
+INFN.
+
 \printbibliography
 
 \end{document}
-- 
cgit v1.2.3-70-g09d2

