More writing.

1 files changed, 44 insertions, 25 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index f216171..f539fbc 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,11 +26,14 @@
 \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
 
 \begin{abstract}
-  We derive the general solution for counting the stationary points of
-  mean-field complex landscapes. It incorporates Parisi's solution
-  for the ground state, as it should. Using this solution, we count the
-  stationary points of two models: one with multi-step replica symmetry
-  breaking, and one with full replica symmetry breaking.
+  Complexity is a measure of the number of stationary points in complex
+  landscapes. We derive a general solution for the complexity of mean-field
+  complex landscapes. It incorporates Parisi's solution for the ground state,
+  as it should. Using this solution, we count the stationary points of two
+  models: one with multi-step replica symmetry breaking, and one with full
+  replica symmetry breaking. These examples demonstrate the consistency of the
+  solution and reveal that the signature of replica symmetry breaking at high
+  energy densities is found in high-index saddles, not minima.
 \end{abstract}
 
 \maketitle
@@ -64,14 +67,14 @@ in the equilibrium properties of fully connected models, the complexity has
 only been computed in RS cases.
 
 In this paper we share the first results for the complexity with nontrivial
-hierarchy. Using a general form for the solution, we detail the structure of
-landscapes with a 1RSB complexity and a full RSB complexity \footnote{The
-  Thouless--Anderson--Palmer (TAP) complexity is the complexity of a kind of
-  mean-field free energy. Because of some deep thermodynamic relationships
-  between the TAP complexity and the equilibrium free energy, the TAP
-  complexity can be computed with extensions of the equilibrium method. As a
-  result, the TAP complexity has been previously computed for nontrivial
-hierarchical structure.}.
+hierarchy. Using a general form for the solution detailed in a companion
+article, we describe the structure of landscapes with a 1RSB complexity and a
+full RSB complexity \footnote{The Thouless--Anderson--Palmer (TAP) complexity
+  is the complexity of a kind of mean-field free energy. Because of some deep
+  thermodynamic relationships between the TAP complexity and the equilibrium
+  free energy, the TAP complexity can be computed with extensions of the
+equilibrium method. As a result, the TAP complexity has been previously
+computed for nontrivial hierarchical structure.} \cite{Kent-Dobias_2022_How}.
 
 We study the mixed $p$-spin spherical models, with Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
@@ -275,6 +278,7 @@ transitions are listed in Table~\ref{tab:energies}.
     $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
     $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
     $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\
     $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
   \end{tabular}
   \caption{
@@ -287,7 +291,8 @@ transitions are listed in Table~\ref{tab:energies}.
     points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic
     threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the
     traditional threshold energy, defined by the energy at which marginal
-    minima become most common. $E_0$ is the ground state energy.
+    minima become most common. $E_\mathrm m$ is the lowest energy at which
+    saddles or marginal minima are found. $E_0$ is the ground state energy.
   } \label{tab:energies}
 \end{table}
 
@@ -295,13 +300,15 @@ In this model, the RS complexity gives an inconsistent answer for the
 complexity of the ground state, predicting that the complexity of minima
 vanishes at a higher energy than the complexity of saddles, with both at a
 lower energy than the equilibrium ground state. The 1RSB complexity resolves
-these problems, predicting the same ground state as equilibrium and with a ground state stability $\mu_0=6.480\,764\ldots>\mu_m$. It predicts that the
-complexity of marginal minima (and therefore all saddles) vanishes at
-$E_m=-1.287\,605\,527\ldots$, which is very slightly greater than $E_0$. Saddles
-become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,575\,114\ldots$.
-The 1RSB complexity transitions to a RS description for dominant stationary
-points at an energy $E_1=-1.273\,886\,852\ldots$. The highest energy for which
-the 1RSB description exists is $E_\mathrm{max}=-0.886\,029\,051\ldots$
+these problems, predicting the same ground state as equilibrium and with a
+ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that
+the complexity of marginal minima (and therefore all saddles) vanishes at
+$E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become
+dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity
+transitions to a RS description for dominant stationary points at an energy
+$E_\mathrm{dom}$. The highest energy for which the 1RSB description exists is
+$E_\mathrm{max}$. The numeric values for all these energies are listed in
+Table~\ref{tab:energies}.
 
 For minima, the complexity does
 not inherit a 1RSB description until the energy is with in a close vicinity of
@@ -349,16 +356,28 @@ also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_
 \end{equation}
 In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty=-0.53125\ldots$.
 
-Fig.~\ref{fig:frsb.phases}
-shows these trajectories, along with the phase boundaries of the complexity in
-this plane. Notably, the phase boundary predicted by a perturbative expansion
-correctly predicts where all of the finite $k$RSB approximations terminate.
+Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model.
+Notably, the phase boundary predicted by a perturbative expansion
+correctly predicts where the finite $k$RSB approximations terminate.
 Like the 1RSB model in the previous subsection, this phase boundary is oriented
 such that very few, low energy, minima are described by a FRSB solution, while
 relatively high energy saddles of high index are also. Again, this suggests
 that studying the mutual distribution of high-index saddle points might give
 insight into lower-energy symmetry breaking in more general contexts.
 
+We have used our solution for mean-field complexity to explore how hierarchical
+RSB in equilibrium corresponds to analogous hierarchical structure in the
+energy landscape. In the examples we studied, a relative minority of energy
+minima are distributed in a nontrivial way, corresponding to the lowest energy
+densities. On the other hand, very high-index saddles begin exhibit RSB at much
+higher energy densities, on the order of the energy densities associated with
+RSB transitions in equilibrium. More wore is necessary to explore this
+connection, as well as whether a purely \emph{geometric} explanation can be
+made for the algorithmic threshold. Applying this method to the most realistic
+RSB scenario for structural glasses, the so-called 1FRSB which has features of
+both 1RSB and FRSB, might yield insights about signatures that should be
+present in the landscape.
+
 \paragraph{Acknowledgements}
 The authors would like to thank Valentina Ros for helpful discussions.