From 375a5d19d82ebf40ed6d3dcf11aae94ad70f1a03 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 24 Jul 2024 15:49:08 +0200
Subject: Added one citation and amended spacing slightly.

---
 marginal.tex | 21 ++++++++++++---------
 1 file changed, 12 insertions(+), 9 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index f167ec7..8414310 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -75,13 +75,18 @@ compared to stiff minima or saddle points. This ubiquity of behavior suggests
 that the distribution of marginal minima can be used to bound out-of-equilibrium dynamical
 behavior.
 
-Despite their importance in a wide variety of in and out of equilibrium settings \cite{Muller_2015_Marginal, Anderson_1984_Lectures, Sommers_1984_Distribution, Parisi_1995-01_On, Horner_2007_Time, Pankov_2006_Low-temperature, Erba_2024_Quenches, Efros_1985_Coulomb, Shklovskii_2024_Half}, it is not straightforward to condition on the marginality of minima using the
-traditional methods for analyzing the distribution of minima in rugged
-landscapes. Using the method of a Legendre transformation of the Parisi
-parameter corresponding to a set of real replicas, one can force the result to
-correspond with marginal minima by tuning the value of that parameter \cite{Monasson_1995_Structural}. However, this
-results in only a characterization of the threshold energy and cannot characterize marginal minima at
-other energies where they are a minority.
+Despite their importance in a wide variety of in and out of equilibrium
+settings \cite{Muller_2015_Marginal, Anderson_1984_Lectures,
+Sommers_1984_Distribution, Parisi_1995-01_On, Horner_2007_Time,
+Pankov_2006_Low-temperature, Erba_2024_Quenches, Efros_1985_Coulomb,
+Shklovskii_2024_Half, Folena_2022_Marginal}, it is not straightforward to condition on the
+marginality of minima using the traditional methods for analyzing the
+distribution of minima in rugged landscapes. Using the method of a Legendre
+transformation of the Parisi parameter corresponding to a set of real replicas,
+one can force the result to correspond with marginal minima by tuning the value
+of that parameter \cite{Monasson_1995_Structural}. However, this results in
+only a characterization of the threshold energy and cannot characterize
+marginal minima at other energies where they are a minority.
 
 The alternative approach, used to great success in the spherical spin glasses, is to
 start by making a detailed understanding of the Hessian matrix at stationary
@@ -517,7 +522,6 @@ $E$, Hessian trace $\mu$, and smallest eigenvalue $\lambda^*$ as
     \delta\big(N\lambda^*-\mathbf s^T\operatorname{Hess}H(\mathbf x,\pmb\omega)\mathbf s\big)
   \end{aligned}
 \end{equation}
-\end{widetext}
 where the additional $\delta$-functions
 \begin{equation}
   \delta(\mathbf s^T\partial\mathbf g(\mathbf x))
@@ -538,7 +542,6 @@ In practice, this can be computed by introducing replicas to treat the
 logarithm ($\log x=\lim_{n\to0}\frac\partial{\partial n}x^n$) and introducing another set of replicas
 to treat each of the normalizations in the numerator
 ($x^{-1}=\lim_{m\to-1}x^m$). This leads to the expression
-\begin{widetext}
 \begin{equation} \label{eq:min.complexity.expanded}
   \begin{aligned}
     \Sigma_{\lambda^*}(E,\mu)
-- 
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