Many small changes for the reveiwers.

1 files changed, 35 insertions, 21 deletions

diff --git a/when_annealed.tex b/when_annealed.tex
index 512ff12..5e02233 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -113,15 +113,28 @@ 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
@@ -222,13 +235,11 @@ 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}
@@ -294,7 +305,7 @@ 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
 \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
@@ -412,27 +423,30 @@ 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{
@@ -500,7 +514,7 @@ 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