Lots of figure work.

9 files changed, 50 insertions, 12 deletions

diff --git a/figs/24_complexity.pdf b/figs/24_complexity.pdf
index 50db7cb..cd24d57 100644
--- a/figs/24_complexity.pdf
+++ b/figs/24_complexity.pdf
diff --git a/figs/24_opt_d(x).pdf b/figs/24_opt_d(x).pdf
new file mode 100644
index 0000000..7ba11d2
--- /dev/null
+++ b/figs/24_opt_d(x).pdf
diff --git a/figs/24_opt_q(x).pdf b/figs/24_opt_q(x).pdf
new file mode 100644
index 0000000..4a4cc7a
--- /dev/null
+++ b/figs/24_opt_q(x).pdf
diff --git a/figs/24_opt_r(x).pdf b/figs/24_opt_r(x).pdf
new file mode 100644
index 0000000..b771adf
--- /dev/null
+++ b/figs/24_opt_r(x).pdf
diff --git a/figs/24_opt_xMax.pdf b/figs/24_opt_xMax.pdf
new file mode 100644
index 0000000..72bf79f
--- /dev/null
+++ b/figs/24_opt_xMax.pdf
diff --git a/figs/24_phases.pdf b/figs/24_phases.pdf
new file mode 100644
index 0000000..c283543
--- /dev/null
+++ b/figs/24_phases.pdf
diff --git a/figs/316_complexity.pdf b/figs/316_complexity.pdf
index 2928239..0289ab3 100644
--- a/figs/316_complexity.pdf
+++ b/figs/316_complexity.pdf
diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index 2c12f3b..e09dc8a 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -147,6 +147,7 @@
  month = {4},
  number = {1},
  volume = {5},
+ pages = {3725},
  url = {https://doi.org/10.1038%2Fncomms4725},
  doi = {10.1038/ncomms4725}
 }
@@ -286,8 +287,7 @@
  date = {2020-09-24T04:22:42Z},
  eprint = {2009.11481v1},
  eprintclass = {cond-mat.stat-mech},
- eprinttype = {arxiv},
- urldate = {2022-07-08T16:19:15.304072Z}
+ eprinttype = {arxiv}
 }
 
 @article{ElAlaoui_2021_Optimization,
@@ -580,6 +580,20 @@ stochastic localization},
  doi = {10.1103/physrevx.9.011003}
 }
 
+@article{Ros_2019_Complexity,
+ author = {Ros, V. and Biroli, G. and Cammarota, C.},
+ title = {Complexity of energy barriers in mean-field glassy systems},
+ journal = {EPL (Europhysics Letters)},
+ publisher = {IOP Publishing},
+ year = {2019},
+ month = {5},
+ number = {2},
+ volume = {126},
+ pages = {20003},
+ url = {https://doi.org/10.1209%2F0295-5075%2F126%2F20003},
+ doi = {10.1209/0295-5075/126/20003}
+}
+
 @article{Seguin_2016_Experimental,
  author = {Seguin, A. and Dauchot, O.},
  title = {Experimental Evidence of the {Gardner} Phase in a Granular Glass},
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 130b7e9..2e0b971 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -664,7 +664,7 @@ is given by a Parisi matrix with parameters $x_1,\ldots,x_k$ and
 $q_1,\ldots,q_k$, then the parameters $\hat\beta$, $r_d$, $d_d$, $\tilde
 x_1,\ldots,\tilde x_{k-1}$, and $\tilde c_1,\ldots,\tilde c_{k-1}$ for the
 complexity in the ground state are
-\begin{align}
+\begin{align}\label{eq:equilibrium.complexity.map}
   \hat\beta=\lim_{\beta\to\infty}\beta x_k
   &&
   \tilde x_i=\lim_{\beta\to\infty}\frac{x_i}{x_k}
@@ -846,6 +846,22 @@ that this is the line of stability for the replica symmetric saddle.
 \section{General solution: examples}
 \label{sec:examples}
 
+Though we have only written down an easily computable complexity along a
+specific (and often uninteresting) line in energy and stability, this
+computable (supersymmetric) solution gives a numeric foothold for computing the
+complexity in the rest of that space. First,
+\eqref{eq:ground.state.free.energy.cont} is \emph{maximized} with respect
+to its parameters, since the equilibrium solution is equivalent to a
+variational problem. Second, the mapping \eqref{eq:equilibrium.complexity.map}
+is used to find the corresponding Kac--Rice saddle parameters in the ground
+state. With these parameters in hand, small steps are then made in energy $E$
+or stability $\mu$, after which known values are used as the initial condition
+for a saddle-finding problem. In this section, we use this basic numeric idea
+to map out the complexity for two representative examples: a model with a
+$2RSB$ equilibrium ground state and therefore $1RSB$ complexity in its
+vicinity, and a model with a $FRSB$ equilibrium ground state, and therefore
+$FRSB$ complexity as well.
+
 \subsection{1RSB complexity}
 
 It is known that by choosing a covariance $f$ as the sum of polynomials with
@@ -1002,17 +1018,22 @@ the phase boundary where $q_1$ goes to one.
 
 \begin{figure}
   \centering
-  \includegraphics{figs/24_func.pdf}
+  \includegraphics{figs/24_phases.pdf}
+  \caption{
+  } \label{fig:frsb.phases}
+\end{figure}
+
+\begin{figure}
+  \raggedright
+  \hspace{2em}\includegraphics{figs/24_opt_q(x).pdf}
+  \hspace{1em}
+  \includegraphics{figs/24_opt_xMax.pdf}
+  \\\vspace{1em}
+  \includegraphics{figs/24_opt_r(x).pdf}
   \hspace{1em}
-  \includegraphics{figs/24_qmax.pdf}
+  \includegraphics{figs/24_opt_d(x).pdf}
 
   \caption{
-    \textbf{Left:} The spectrum $\chi$ of the replica matrix in the complexity
-    of dominant saddles for the $2+4$ model at several energies.
-    \textbf{Right:} The cutoff $q_{\mathrm{max}}$ for the nonlinear part of the
-    spectrum as a function of energy $E$ for both dominant saddles and marginal
-    minima. The colored vertical lines show the energies that correspond to the
-    curves on the left.
   } \label{fig:24.func}
 \end{figure}
 
@@ -1031,7 +1052,7 @@ the phase boundary where $q_1$ goes to one.
     fixed energy $E$. Solid lines show the result of a FRSB ansatz and dashed
     lines that of a RS ansatz. All paired parameters coincide at the ground
     state energy, as expected.
-  } \label{fig:2rsb.comparison}
+  } \label{fig:frsb.comparison}
 \end{figure}
 
 
@@ -1246,6 +1267,9 @@ extracted in all detail.
 A first and very important application of the method here is to perform the calculation for high dimensional spheres, where it would give us
 a clear understanding of what happens in a low-temperature realistic jamming dynamics \cite{Maimbourg_2016_Solution}. 
 
+\paragraph{Acknowledgements}
+J K-D and J K would like to thank Valentina Ros for helpful discussions.
+
 \paragraph{Funding information}
 J K-D and J K are supported by the Simons Foundation Grant No. 454943.