From 40f81655eaaa4264b6feba819f6d904ec6ea0b9f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 21 Oct 2022 17:00:32 +0200
Subject: Some writing and lots of references.

---
 frsb_kac-rice_letter.tex | 56 ++++++++++++++++++++++++++++++++----------------
 1 file changed, 37 insertions(+), 19 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index f539fbc..b3c288a 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -41,9 +41,10 @@
 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
+size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex,
+Altieri_2021_Properties}. 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.
@@ -51,43 +52,60 @@ network.
 The \emph{complexity} of a function is the average of the logarithm of the
 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.
+stationary point \cite{Bray_1980_Metastable}. 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
+breaking} (RSB) \cite{Parisi_1979_Infinite}. 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 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}.
+In this paper and its longer companion, we share the first results for the
+complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. 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.}.
 
 We study the mixed $p$-spin spherical models, with Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
   H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
 is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
-$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$.  The coupling coefficients $J$ are taken at random, with
+$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with
 zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
 the energy is typically extensive. The overbar will always denote an average
 over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
 constants that define the particular model. For instance, the so-called `pure'
 models have $a_p=1$ for some $p$ and all others zero.
 
+The complexity of the $p$-spin models has been extensively studied by
+physicists and mathematicians. Among physicists, the bulk of work has been on
+the so-called Thouless--Anderson--Palmer (TAP) complexity for the pure models,
+which counts minima in a kind of mean-field free energy \cite{Rieger_1992_The,
+Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
+Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
+Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure
+and mixed models without RSB \cite{Auffinger_2012_Random,
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been
+treated in specific cases, again without RSB \cite{Folena_2020_Rethinking,
+Ros_2019_Complex}. And the methods of complexity have been used to study many
+geometric properties of the pure models, from the relative position of
+stationary points to one another to shape and prevalence of instantons
+\cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
+
 The variance of the couplings implies that the covariance of the energy with
 itself depends on only the dot product (or overlap) between two configurations.
 In particular, one finds
@@ -101,7 +119,7 @@ where $f$ is defined by the series
 One needn't start with a Hamiltonian like
 \eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
 \eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in
-the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}.
+the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.
 
 The family of spherical models thus defined is quite rich, and by varying the
 covariance $f$ nearly any hierarchical structure can be found in
-- 
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