Merge branch 'epl' of git:research/replicated_kac-rice/paper_2 into epl

5 files changed, 54 insertions, 39 deletions

diff --git a/figs/complexity_35.pdf b/figs/complexity_35.pdf
index 5f1f6aa..20a2789 100644
--- a/figs/complexity_35.pdf
+++ b/figs/complexity_35.pdf
diff --git a/figs/order_plot_1.pdf b/figs/order_plot_1.pdf
index fcc8195..54657cc 100644
--- a/figs/order_plot_1.pdf
+++ b/figs/order_plot_1.pdf
diff --git a/figs/order_plot_2.pdf b/figs/order_plot_2.pdf
index 173ef94..8a09d73 100644
--- a/figs/order_plot_2.pdf
+++ b/figs/order_plot_2.pdf
diff --git a/when_annealed.bib b/when_annealed.bib
index 5521002..decc756 100644
--- a/when_annealed.bib
+++ b/when_annealed.bib
@@ -89,7 +89,7 @@
 
 @article{Crisanti_1992_The,
  author = {Crisanti, A. and Sommers, H.-J.},
- title = {The spherical $p$-spin interaction spin glass model: the statics},
+ title = {The spherical {$p$}-spin interaction spin glass model: the statics},
  journal = {Zeitschrift für Physik B Condensed Matter},
  publisher = {Springer Science and Business Media LLC},
  year = {1992},
@@ -102,7 +102,7 @@
 
 @article{Crisanti_2004_Spherical,
  author = {Crisanti, A. and Leuzzi, L.},
- title = {Spherical $2+p$ Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition},
+ title = {Spherical {$2+p$} Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2004},
@@ -115,7 +115,7 @@
 
 @article{Crisanti_2006_Spherical,
  author = {Crisanti, A. and Leuzzi, L.},
- title = {Spherical $2+p$ spin-glass model: An analytically solvable model with a glass-to-glass transition},
+ title = {Spherical {$2+p$} spin-glass model: An analytically solvable model with a glass-to-glass transition},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2006},
@@ -180,7 +180,7 @@
 
 @article{Folena_2020_Rethinking,
  author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico},
- title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed $p$-Spin Model},
+ title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed {$p$}-Spin Model},
  journal = {Physical Review X},
  publisher = {American Physical Society},
  year = {2020},
@@ -194,7 +194,7 @@
 
 @article{Folena_2021_Gradient,
  author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico},
- title = {Gradient descent dynamics in the mixed $p$-spin spherical model: finite-size simulations and comparison with mean-field integration},
+ title = {Gradient descent dynamics in the mixed {$p$}-spin spherical model: finite-size simulations and comparison with mean-field integration},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  publisher = {IOP Publishing},
  year = {2021},
diff --git a/when_annealed.tex b/when_annealed.tex
index 0304218..516526b 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -3,7 +3,6 @@
 \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
 \usepackage[T1]{fontenc} % vector fonts
 \usepackage{amsmath,amssymb,latexsym,graphicx}
-\usepackage{newtxtext,newtxmath} % Times
 \usepackage[dvipsnames]{xcolor}
 \usepackage[
   colorlinks=true,
@@ -30,7 +29,7 @@
   counts are reliably found by taking the average of the logarithm (the
   quenched average), which is more difficult and not often done in practice.
   When most stationary points are uncorrelated with each other, quenched and
-  anneals averages are equal. Equilibrium heuristics can guarantee when most of
+  annealed averages are equal. Equilibrium heuristics can guarantee when most of
   the lowest minima will be uncorrelated. We show that these equilibrium
   heuristics cannot be used to draw conclusions about other minima and saddles
   by producing examples among Gaussian-correlated functions on the hypersphere
@@ -106,15 +105,28 @@ Specifying the covariance function $f$ uniquely specifies the model. The series
 coefficients of $f$ need to be nonnnegative in order for $f$ to be a
 well-defined covariance. The case where $f$ is a homogeneous polynomial has
 been extensively studied, and corresponds to the pure spherical models of glass
-physics or the spiked tensor models of statistical inference \cite{Castellani_2005_Spin-glass}. Here we will
-study cases where $f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for
-$\lambda\in(0,1)$, called $3+s$ models. These are examples of \emph{mixed}
+physics or the spiked tensor models of statistical inference \cite{Castellani_2005_Spin-glass}. Here our examples will be models with $f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for
+$\lambda\in(0,1)$, called $3+s$ models.\footnote{
+  Though the examples and discussion will focus on the $3+s$ models, most
+  formulas (including the principal result in \eqref{eq:condition}) are valid for
+  arbitrary covariance functions $f$ under the condition that $f'(0)=0$, i.e.,
+  that there is no linear field in the problem. This condition is necessary to
+  ensure that what we call `trivial' correlations are actually \emph{zero}
+  correlations: in the absence of a field, trivially correlated points on the
+  sphere are orthogonal. This simplifies our formulas by setting the overlap
+  $q_0$ between trivially correlated configurations to zero, which would
+  otherwise be another order parameter, but reduces the scope of this study. The
+  trivial overlap $q_0$ is also important in situations where a deterministic
+  field (or spike) is present, as in \cite{Ros_2019_Complex}, but deterministic
+  fields are likewise not considered here.
+}These are examples of \emph{mixed}
 spherical models, which have been studied in the physics and statistics
 literature and host a zoo of complex orders and phase transitions
 \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
 Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous,
 Crisanti_2011_Statistical, BenArous_2019_Geometry, Subag_2020_Following, ElAlaoui_2020_Algorithmic}.
 
+
 There are several well-established results on the equilibrium of this model.
 First, if the function $\chi(q)=f''(q)^{-1/2}$ is convex then it is not possible for the
 equilibrium solution to have nontrivial correlations between states at any
@@ -187,7 +199,7 @@ logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$.
 The annealed complexity has been computed for these models
 \cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched
 complexity has been computed for a couple examples which have nontrivial ground
-states \cite{Kent-Dobias_2023_How}. The annealed complexity bounds the
+states \cite{Crisanti_2006_Spherical ,Kent-Dobias_2023_How}. The annealed complexity bounds the
 complexity from above. A positive complexity indicates the presence of an
 exponentially large number of stationary points of the indicated kind, while a
 negative one means it is vanishingly unlikely they will appear. The line of
@@ -215,13 +227,11 @@ When the complexity is calculated using the Kac--Rice formula and a physicists'
 tool set, the problem is reduced to the evaluation of an integral by the saddle
 point method for large $N$ \cite{Kent-Dobias_2023_How}. The complexity is given
 by extremizing an effective action,
-\begin{equation}
-  \begin{aligned}
-    &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \\
-    &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \\
-    &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
-  \end{aligned}
-\end{equation}
+\begin{align}
+  &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \notag \\
+  &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \notag \\
+  &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
+\end{align}
 for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb.action}.
 \begin{widetext}
   \begin{equation} \label{eq:1rsb.action}
@@ -247,7 +257,7 @@ for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb
   \end{aligned}
 \end{equation}
 where $\Delta x=1-x$ and
-\begin{equation}
+\begin{equation} \label{eq:hess.term}
   \mathcal D(\mu)
   =\begin{cases}
     \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
@@ -285,9 +295,9 @@ a(E,\mu)=0$. Going along this line in the replica symmetric solution, the
 $x=q_1=1$ \cite{Kent-Dobias_2023_How}. Since all the parameters in the
 bifurcating solution are known at this point, we can search for it by looking
 for a flat direction. In the annealed solution for
-points describing saddles ($\mu<\mu_\mathrm m$), this line is
+points describing saddles (with $\mu^2\leq\mu_\mathrm m^2$ and therefore the simpler form of \eqref{eq:hess.term}), 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)
+  \mu_0=-\frac{2Ef'f''}{z_f}-\sqrt{\frac{2f''u_f}{z_f^2}\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}
 \end{equation}
 where we have chosen the lower branch as a convention (see
 Fig.~\ref{fig:complexity_35}) and where we define for brevity (here and
@@ -362,7 +372,7 @@ $e$, $g$, and $h$ are given by
     transition, coincides with it in this case. The gray shaded region
     highlights the minima, which are stationary points with $\mu\geq\mu_\mathrm
     m$. $E_\textrm{min}$ is marked on the plot as the lowest energy at which
-    extensive saddles are found.
+    saddles of extensive index are found.
   } \label{fig:complexity_35}
 \end{figure}
 
@@ -405,27 +415,30 @@ proportional to
 \end{equation}
 If $G_f>0$, then the bifurcating solutions exist, and there are some saddles whose
 complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution.
-Therefore, $G_f>0$ is a condition to see {\oldstylenums1}\textsc{rsb} in the
+Therefore, $G_f>0$ is a sufficient condition to see at least {\oldstylenums1}\textsc{rsb} in the
 complexity. If $G_f<0$, then there is nowhere along the extremal line where
-saddles can be described by such a complexity. The range of $3+s$ models where
-$G_f$ is positive is shown in Fig.~\ref{fig:phases}.
+saddles can be described by such a complexity, but this does not definitively
+rule out \textsc{rsb}: the model may be unstable to different \textsc{rsb}
+orders, or its phase boundary may simply not have a critical point on the extremal line. We
+discuss the former possibility later in the paper. The range of $3+s$ models where $G_f$ is positive is
+shown in Fig.~\ref{fig:phases}.
 
 \begin{figure*}
   \centering
   \includegraphics[width=0.29\textwidth]{figs/range_plot_1.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_2.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_3.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_4.pdf} \\
   \vspace{-2em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_log_1.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_log_2.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_log_3.pdf}
-  \hspace{-3.25em}
+  \hspace{-3.4em}
   \includegraphics[width=0.29\textwidth]{figs/range_plot_log_4.pdf}
 
   \caption{
@@ -450,12 +463,14 @@ of saddles $E_\mathrm{min}$. Also, these models have a transition boundary that
 smoothly connects $E_{\oldstylenums1\textsc{rsb}}^+$ and
 $E_{\oldstylenums1\textsc{rsb}}^-$, so $E_{\oldstylenums1\textsc{rsb}}^-$
 corresponds to the lower bound of \textsc{rsb} complexity. For large enough
-$s$, the range passes into minima, which is excepted as these models have
-nontrivial complexity of their ground states. This also seems to correspond
-with the decoupling of the \textsc{rsb} solutions connected to
+$s$, the range passes into minima, which is expected as these models have
+nontrivial complexity of their ground states. Interestingly, this appears to
+happen at precisely the value of $s$ for which nontrivial ground state
+configurations appear, $s=12.403\ldots$. This also seems to correspond with the
+decoupling of the \textsc{rsb} solutions connected to
 $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, with
-the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}. In
-these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower
+the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}.
+In these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower
 bound, but sometimes it is given by the termination of the phase boundary
 extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
 
@@ -466,8 +481,8 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
   \includegraphics[width=0.95\columnwidth]{figs/order_plot_2.pdf}
 
   \caption{
-    Examples of $3+14$ models where the solution
-    $E_{\oldstylenums1\textsc{rsb}}^-$ does and doesn't define the lower limit
+    Examples of $3+14$ models where the critical point
+    $E_{\oldstylenums1\textsc{rsb}}^-$ is and is not the lower limit
     of energies where \textsc{rsb} saddles are found. In both plots the red dot
     shows $E_{\oldstylenums1\textsc{rsb}}^-$, while the solid red lines shows
     the transition boundary with the \textsc{rs} complexity. The dashed black
@@ -493,7 +508,7 @@ Consider a specific $H$ with
   \begin{aligned}
     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_p} \\
-    &\hspace{6pc}+\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}
+    &\hspace{5pc}+\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{aligned}
 \end{equation}
 where the interaction tensors $J$ are drawn from zero-mean normal distributions