From b5b2b7eef46084fe02594f15861a17a20a1a1289 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 16 Jun 2023 21:45:02 +0200
Subject: Changed line lengths.

---
 when_annealed.tex | 121 ++++++++++++++++++++++++++++--------------------------
 1 file changed, 63 insertions(+), 58 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index ae7e3b8..537381a 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -53,15 +53,15 @@
   settings might be affected.
 \end{abstract}
 
-Random high-dimensional energies, cost functions, and interaction networks are important in many fields. The energy landscape of glasses, the likelihood
-landscape of machine learning and inference, and the
-interactions between organisms in an ecosystem are just a few examples. A
-traditional tool for making sense of their behavior is to analyze the
-statistics of points where their dynamics are stationary. For energy or cost
-landscapes, these correspond to the minima, maxima, and saddles, while for
-ecosystems and other non-gradient dynamical systems these correspond to
-equilibria of the dynamics. When many stationary points are present, the system
-is considered complex.
+Random high-dimensional energies, cost functions, and interaction networks are
+important in many fields. The energy landscape of glasses, the likelihood
+landscape of machine learning and inference, and the interactions between
+organisms in an ecosystem are just a few examples. A traditional tool for
+making sense of their behavior is to analyze the statistics of points where
+their dynamics are stationary. For energy or cost landscapes, these correspond
+to the minima, maxima, and saddles, while for ecosystems and other non-gradient
+dynamical systems these correspond to equilibria of the dynamics. When many
+stationary points are present, the system is considered complex.
 
 Despite the importance of stationary point statistics for understanding complex
 behavior, they are often calculated using an uncontrolled approximation.
@@ -69,13 +69,15 @@ Because their number is so large, it cannot be reliably averaged. The annealed
 approximation takes this average anyway, risking a systematic bias by rare and
 atypical samples. The annealed approximation is known to be exact for certain
 models and in certain circumstances, but it is used outside those circumstances
-without much reflection \cite{Wainrib_2013_Topological, Gershenzon_2023_On-Site}. In a few cases researches have made instead the
+without much reflection \cite{Wainrib_2013_Topological,
+Gershenzon_2023_On-Site}. In a few cases researches have made instead the
 better-controlled quenched average, which averages the logarithm of the number
 of stationary points, and find deviations from the annealed approximation with
-important implications for the system's behavior \cite{Muller_2006_Marginal, Ros_2019_Complexity,
-Kent-Dobias_2023_How, Ros_2023_Quenched}. Generically, the annealed
-approximation to the complexity is wrong when a nonvanishing fraction of pairs
-of stationary points have nontrivial correlations in their mutual position.
+important implications for the system's behavior \cite{Muller_2006_Marginal,
+Ros_2019_Complexity, Kent-Dobias_2023_How, Ros_2023_Quenched}. Generically,
+the annealed approximation to the complexity is wrong when a nonvanishing
+fraction of pairs of stationary points have nontrivial correlations in their
+mutual position.
 
 A heuristic line of reasoning for the appropriateness of the annealed
 approximation is sometimes made when the approximation is correct for an
@@ -91,10 +93,10 @@ stationary points?
 In this paper, we show that the behavior of the ground state, or \emph{any}
 equilibrium behavior, does not govern whether stationary points will have a
 correct annealed average. In a prototypical family of models of random
-functions, we calculate a condition for determining when annealed averages should fail
-and stationary points should have nontrivial correlations in their mutual
-position. We produce examples of models whose equilibrium is guaranteed to
-never see such correlations between thermodynamic states, but where a
+functions, we calculate a condition for determining when annealed averages
+should fail and stationary points should have nontrivial correlations in their
+mutual position. We produce examples of models whose equilibrium is guaranteed
+to never see such correlations between thermodynamic states, but where a
 population of saddle points is nevertheless correlated.
 
 We study the mixed spherical models, which are models of Gaussian-correlated
@@ -106,16 +108,17 @@ product (or overlap) between the two configurations:
 \begin{equation} \label{eq:covariance}
   \overline{H(\pmb\sigma_1)H(\pmb\sigma_2)}=\frac1Nf\bigg(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\bigg)
 \end{equation}
-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
+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. 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} 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, Crisanti_2011_Statistical}.
+$\lambda\in(0,1)$, called $3+s$ models. 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,
+Crisanti_2011_Statistical}.
 
 There are several well-established results on the equilibrium of this model.
 First, if the function $f$ is convex then it is not possible for the
@@ -146,9 +149,9 @@ complexity has nontrivial \textsc{rsb}. As evidenced in that figure,
 \textsc{rsb} among saddles is possible well outside the bounds from
 equilibrium.
 
-There are two important features which differentiate stationary points $\pmb\sigma^*$ in the
-spherical models: their \emph{energy density} $E=\frac1NH(\pmb\sigma^*)$ and
-their \emph{stability}
+There are two important features which differentiate stationary points
+$\pmb\sigma^*$ in the spherical models: their \emph{energy density}
+$E=\frac1NH(\pmb\sigma^*)$ and their \emph{stability}
 $\mu=\frac1N\operatorname{\mathrm{Tr}}\operatorname{\mathrm{Hess}}H(\pmb\sigma^*)$.
 The energy density should be familiar, as the `height' in the landscape. The
 stability is so-called because it governs the spectrum of the stationary point.
@@ -176,15 +179,15 @@ $\mu=\mu_\mathrm m$, the spectrum has a pseudogap, and we have marginal minima.
 \end{figure}
 
 The number $\mathcal N(E,\mu)$ of stationary points with energy density $E$ and
-stability $\mu$ is exponential in $N$ for these models. Their complexity $\Sigma(E,\mu)$ is
-defined by the average of the logarithm of their number, or
-$\Sigma(E,\mu)=\frac1N\overline{\log\mathcal N(E,\mu)}$. More often the annealed
-complexity is calculated, where the average is taken before the logarithm:
-$\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$. The annealed
-complexity has been computed for these models \cite{BenArous_2019_Geometry,
-Folena_2020_Rethinking}, and the quenched complexity has been computed for a
-couple examples which have correlations among ground-state minima
-\cite{Kent-Dobias_2023_How}.
+stability $\mu$ is exponential in $N$ for these models. Their complexity
+$\Sigma(E,\mu)$ is defined by the average of the logarithm of their number, or
+$\Sigma(E,\mu)=\frac1N\overline{\log\mathcal N(E,\mu)}$. More often the
+annealed complexity is calculated, where the average is taken before the
+logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$.
+The annealed complexity has been computed for these models
+\cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched
+complexity has been computed for a couple examples which have correlations
+among ground-state minima \cite{Kent-Dobias_2023_How}.
 
 In these models, trivial correlations between stationary points correspond with
 zero overlap: almost all stationary points are orthogonal to each other. This
@@ -197,15 +200,16 @@ ansatz, which corresponds to two kinds of pairs of stationary point: a fraction
 $x$ of pairs have the trivial zero overlap, and the remaining fraction $1-x$
 have a nontrivial overlap $q_1$. In the annealed or replica-symmetric case,
 $x=1$ and all but a vanishing fraction of stationary points are uncorrelated
-with each other. Since other kinds of \textsc{rsb} order encompass {\oldstylenums1}\textsc{rsb}, we are guaranteed that
-$\Sigma\leq\Sigma_{\oldstylenums1\textsc{rsb}}\leq\Sigma_\mathrm a$. We
-will discuss later in what settings the {\oldstylenums1}\textsc{rsb} complexity
-is correct.
+with each other. Since other kinds of \textsc{rsb} order encompass
+{\oldstylenums1}\textsc{rsb}, we are guaranteed that
+$\Sigma\leq\Sigma_{\oldstylenums1\textsc{rsb}}\leq\Sigma_\mathrm a$. We will
+discuss later in what settings the {\oldstylenums1}\textsc{rsb} complexity is
+correct.
 
 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,
+point method for large $N$ \cite{Kent-Dobias_2023_How}. The complexity is given
+by extremizing an effective action,
 \begin{equation}
   \Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu)
   =\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)}
@@ -268,13 +272,13 @@ solution, and some parameters, e.g., $q_1$, are unconstrained and can take any
 value in the old solution.
 
 There is one place where we can consistently search for a bifurcating solution
-to the saddle point equations: along the zero complexity line
-$\Sigma_\mathrm a(E,\mu)=0$. Going along this line in the replica symmetric solution, the
+to the saddle point equations: along the zero complexity line $\Sigma_\mathrm
+a(E,\mu)=0$. Going along this line in the replica symmetric solution, the
 {\oldstylenums1}\textsc{rsb} complexity transitions at a critical point where
 $x=q_1=1$ \cite{Kent-Dobias_2023_How}. Since all the parameters in the
 bifurcating solution are known at this point, we can search for it by looking
-for a flat direction in the way described above. In the annealed
-solution for points describing saddles ($\mu<\mu_\mathrm m$), this line is
+for a flat direction in the way described above. 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)
 \end{equation}
@@ -319,11 +323,11 @@ by
   \qquad
   d=\frac{w_f}{f'f''}
 \end{equation}
-Changing variables to $y$ from $\mu$ is a
-convenient choice because the branch of \eqref{eq:extremal.line} is chosen
-by the sign of $y$ (the lower-energy branch we are interested in corresponds
-with $y>0$) and the relationship between $y$ and $E$ on the extremal line is
-$g=2hy^2+eE^2$, where the constants $e$, $g$, and $h$ are given by
+Changing variables to $y$ from $\mu$ is a convenient choice because the branch
+of \eqref{eq:extremal.line} is chosen by the sign of $y$ (the lower-energy
+branch we are interested in corresponds with $y>0$) and the relationship
+between $y$ and $E$ on the extremal line is $g=2hy^2+eE^2$, where the constants
+$e$, $g$, and $h$ are given by
 \begin{equation}
   e=f''-f'
   \qquad
@@ -371,7 +375,8 @@ energy point, so that these two points give the precise range of energies at
 which \textsc{rsb} saddles are found. An example that conforms with this
 picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}.
 
-The expression inside the inner square root of \eqref{eq:energies} is proportional to
+The expression inside the inner square root of \eqref{eq:energies} is
+proportional to
 \begin{equation} \label{eq:condition}
   G_f
   =
@@ -465,7 +470,8 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$.
   } \label{fig:order}
 \end{figure}
 
-There are implications for the emergence of \textsc{rsb} in equilibrium. Consider a specific $H$ with
+There are implications for the emergence of \textsc{rsb} in equilibrium.
+Consider a specific $H$ with
 \begin{equation}
   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}
@@ -501,11 +507,10 @@ $s>2$, this transition line \emph{always} intersects the extremal line
 among some population of stationary points. However, it is likely that for much
 of the parameter space the so-called one-full \textsc{rsb}
 ({\oldstylenums1\textsc{frsb}}) is the correct solution, as it likely is for
-large $s$ in the $3+s$ model at hand. Further work to find the
-conditions for transitions of the complexity to these forms of order is
-necessary. For values of $s$ where there is no
-\textsc{rsb} of any kind in the ground state, we expect that the
-{\oldstylenums1\textsc{rsb}} complexity is correct.
+large $s$ in the $3+s$ model at hand. Further work to find the conditions for
+transitions of the complexity to these forms of order is necessary. For values
+of $s$ where there is no \textsc{rsb} of any kind in the ground state, we
+expect that the {\oldstylenums1\textsc{rsb}} complexity is correct.
 
 What are the implications for dynamics? We find that nontrivial correlations
 tend to exist among saddle points with the maximum or minimum index possible at
-- 
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