From 07dbb34e714ace184f32cb62c5c484daf053bd48 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 27 Sep 2022 13:22:48 +0200
Subject: Start of letter version.

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+
+\documentclass[reprint,aps,prl,longbibliography]{revtex4-2}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{amsmath,amssymb,latexsym,graphicx}
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[dvipsnames]{xcolor}
+\usepackage[
+  colorlinks=true,
+  urlcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  filecolor=MidnightBlue,
+  linkcolor=MidnightBlue
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{anyfontsize}
+
+\begin{document}
+
+\title{
+  Unvieling the complexity of heirarchical energy landscapes
+}
+
+\author{Jaron Kent-Dobias}
+\author{Jorge Kurchan}
+\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.
+\end{abstract}
+
+\maketitle
+
+The functions used to describe the energies, costs, and fitnesses of disordered
+systems in physics, computer science, and biology are typically \emph{complex},
+meaning that they have a number of minima that grows exponentially with the
+size of the system. Though they are often called `rough landscapes' to evoke
+the intuitive image of many minima in something like a mountain range, the
+metaphor to topographical landscapes is strained by the reality that these
+complex landscapes also exist in very high dimensions: think of the dimensions
+of phase space for $N$ particles, or the number of parameters in a neural
+network.
+
+The \emph{complexity} of a function is the logarithm of the average number of
+its minima, maxima, and saddle points (collectively stationary points), under
+conditions like the value of the energy or the index of the stationary point.
+Since in complex landscapes this number grows exponentially with system size,
+their complexity is an extensive quantity. Understanding the complexity offers
+an understanding about the geometry and topology of the landscape, which can
+provide insight into dynamical behavior.
+
+When complex systems are fully connected, i.e., each degree of freedom
+interacts directly with every other, they are often described by a hierarchical
+structure of the type first proposed by Parisi, the \emph{replica symmetry
+breaking} (RSB). This family of structures is rich, spanning uniform
+\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested
+structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB),
+and arbitrary combinations thereof. Though these rich structures are understood
+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.
+
+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{Albert_2021_Searching}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/316_detail.pdf}
+
+  \caption{
+    Detail of the `phases' of the $3+16$ model complexity as a function of
+    energy and stability. Above the yellow marginal stability line the
+    complexity counts saddles of fixed index, while below that line it counts
+    minima of fixed stability. The shaded red region shows places where the
+    complexity is described by the 1RSB solution, while the shaded gray region
+    shows places where the complexity is described by the RS solution. In white
+    regions the complexity is zero. Several interesting energies are marked
+    with vertical black lines: the traditional `threshold' $E_\mathrm{th}$
+    where minima become most numerous, the algorithmic threshold
+    $E_\mathrm{alg}$ that bounds the performance of smooth algorithms, and the
+    average energies at the $2$RSB and $1$RSB equilibrium transitions $\langle
+    E\rangle_2$ and $\langle E\rangle_1$, respectively. Though the figure is
+    suggestive, $E_\mathrm{alg}$ lies at slightly lower energy than the termination of the RS
+    -- 1RSB transition line.
+  } \label{fig:2rsb.phases}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/24_phases.pdf}
+  \caption{
+    `Phases' of the complexity for the $2+4$ model in the energy $E$ and
+    stability $\mu^*$ plane. The region shaded gray shows where the RS solution
+    is correct, while the region shaded red shows that where the FRSB solution
+    is correct. The white region shows where the complexity is zero.
+  } \label{fig:frsb.phases}
+\end{figure}
+
+
+\paragraph{Acknowledgements}
+The authors would like to thank Valentina Ros for helpful discussions.
+
+\paragraph{Funding information}
+JK-D and JK are supported by the Simons Foundation Grant No.~454943.
+
+\bibliography{frsb_kac-rice}
+
+\end{document}
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