From 1c6dd13cd759855ab7d49bc2b7758993cef2bafd Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 17 Jun 2023 10:36:44 +0200
Subject: A few fixes and many citations.

---
 when_annealed.tex | 20 +++++++++++---------
 1 file changed, 11 insertions(+), 9 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index dd57ec2..469fd6e 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -56,12 +56,14 @@
 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
+organisms in an ecosystem are just a few examples \cite{Stein_1995_Broken, Krzakala_2007_Landscape, Altieri_2021_Properties, Yang_2023_Stochastic}. 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.
+their dynamics are stationary \cite{Cavagna_1998_Stationary,
+Fyodorov_2004_Complexity, Fyodorov_2007_Density, Bray_2007_Statistics}. 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.
@@ -69,12 +71,12 @@ 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,
+without much reflection \cite{Wainrib_2013_Topological, Kent-Dobias_2021_Complex,
 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,
+Ros_2019_Complex, 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.
@@ -386,8 +388,8 @@ proportional to
   -2(f''-f')u_fw_f
   -2\log^2\frac{f''}{f'}f'^2f''v_f
 \end{equation}
-If $G_f>0$, then the bifurcating solutions exist, and there is someplace where
-the annealed solution is corrected by a {\oldstylenums1\textsc{rsb}} solution.
+If $G_f>0$, then the bifurcating solutions exist, and there are some saddles whose
+complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution.
 Therefore, $G_f>0$ is a condition to see {\oldstylenums1}\textsc{rsb} in the
 complexity. If $G_f<0$, then there is nowhere along the extremal line where
 saddles can be described by such a complexity. The range of $3+s$ models where
-- 
cgit v1.2.3-70-g09d2

