2 files changed, 123 insertions, 9 deletions
diff --git a/when_annealed.bib b/when_annealed.bib
index d13ddf8..7c311fa 100644
--- a/when_annealed.bib
+++ b/when_annealed.bib
@@ -1,3 +1,17 @@
+@article{Altieri_2021_Properties,
+ author = {Altieri, Ada and Roy, Felix and Cammarota, Chiara and Biroli, Giulio},
+ title = {Properties of Equilibria and Glassy Phases of the Random {Lotka}-{Volterra} Model with Demographic Noise},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2021},
+ month = {6},
+ number = {25},
+ volume = {126},
+ pages = {258301},
+ url = {https://doi.org/10.1103%2Fphysrevlett.126.258301},
+ doi = {10.1103/physrevlett.126.258301}
+}
+
 @article{Auffinger_2022_The,
  author = {Auffinger, Antonio and Zhou, Yuxin},
  title = {The Spherical {$p+s$} Spin Glass At Zero Temperature},
@@ -24,6 +38,34 @@
  doi = {10.1002/cpa.21875}
 }
 
+@article{Bray_2007_Statistics,
+ author = {Bray, Alan J. and Dean, David S.},
+ title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2007},
+ month = {4},
+ number = {15},
+ volume = {98},
+ pages = {150201},
+ url = {https://doi.org/10.1103%2Fphysrevlett.98.150201},
+ doi = {10.1103/physrevlett.98.150201}
+}
+
+@article{Cavagna_1998_Stationary,
+ author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio},
+ title = {Stationary points of the {Thouless}-{Anderson}-{Palmer} free energy},
+ journal = {Physical Review B},
+ publisher = {American Physical Society (APS)},
+ year = {1998},
+ month = {5},
+ number = {18},
+ volume = {57},
+ pages = {11251--11257},
+ url = {https://doi.org/10.1103%2Fphysrevb.57.11251},
+ doi = {10.1103/physrevb.57.11251}
+}
+
 @article{Crisanti_2004_Spherical,
  author = {Crisanti, A. and Leuzzi, L.},
  title = {Spherical $2+p$ Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition},
@@ -121,6 +163,20 @@
  doi = {10.1103/physrevlett.92.240601}
 }
 
+@article{Fyodorov_2007_Density,
+ author = {Fyodorov, Y. V. and Sommers, H.-J. and Williams, I.},
+ title = {Density of stationary points in a high dimensional random energy landscape and the onset of glassy behavior},
+ journal = {JETP Letters},
+ publisher = {Pleiades Publishing Ltd},
+ year = {2007},
+ month = {5},
+ number = {5},
+ volume = {85},
+ pages = {261--266},
+ url = {https://doi.org/10.1134%2Fs0021364007050098},
+ doi = {10.1134/s0021364007050098}
+}
+
 @article{Fyodorov_2012_Critical,
  author = {Fyodorov, Yan V. and Nadal, Celine},
  title = {Critical Behavior of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom Distribution},
@@ -149,6 +205,20 @@
  doi = {10.1103/physrevlett.130.237103}
 }
 
+@article{Kent-Dobias_2021_Complex,
+ author = {Kent-Dobias, Jaron and Kurchan, Jorge},
+ title = {Complex complex landscapes},
+ journal = {Physical Review Research},
+ publisher = {American Physical Society (APS)},
+ year = {2021},
+ month = {4},
+ number = {2},
+ volume = {3},
+ pages = {023064},
+ url = {https://doi.org/10.1103%2Fphysrevresearch.3.023064},
+ doi = {10.1103/physrevresearch.3.023064}
+}
+
 @article{Kent-Dobias_2023_How,
  author = {Kent-Dobias, Jaron and Kurchan, Jorge},
  title = {How to count in hierarchical landscapes: a full solution to mean-field complexity},
@@ -163,6 +233,20 @@
  doi = {10.1103/PhysRevE.107.064111}
 }
 
+@article{Krzakala_2007_Landscape,
+ author = {Krzakala, Florent and Kurchan, Jorge},
+ title = {Landscape analysis of constraint satisfaction problems},
+ journal = {Physical Review E},
+ publisher = {American Physical Society (APS)},
+ year = {2007},
+ month = {8},
+ number = {2},
+ volume = {76},
+ pages = {021122},
+ url = {https://doi.org/10.1103%2Fphysreve.76.021122},
+ doi = {10.1103/physreve.76.021122}
+}
+
 @article{Muller_2006_Marginal,
  author = {Müller, Markus and Leuzzi, Luca and Crisanti, Andrea},
  title = {Marginal states in mean-field glasses},
@@ -230,6 +314,20 @@ interactions: the typical number of equilibria},
  eprinttype = {arxiv}
 }
 
+@article{Stein_1995_Broken,
+ author = {Stein, D. L. and Newman, C. M.},
+ title = {Broken ergodicity and the geometry of rugged landscapes},
+ journal = {Physical Review E},
+ publisher = {American Physical Society (APS)},
+ year = {1995},
+ month = {6},
+ number = {6},
+ volume = {51},
+ pages = {5228--5238},
+ url = {https://doi.org/10.1103%2Fphysreve.51.5228},
+ doi = {10.1103/physreve.51.5228}
+}
+
 @article{Wainrib_2013_Topological,
  author = {Wainrib, Gilles and Touboul, Jonathan},
  title = {Topological and Dynamical Complexity of Random Neural Networks},
@@ -244,3 +342,17 @@ interactions: the typical number of equilibria},
  doi = {10.1103/physrevlett.110.118101}
 }
 
+@article{Yang_2023_Stochastic,
+ author = {Yang, Ning and Tang, Chao and Tu, Yuhai},
+ title = {Stochastic Gradient Descent Introduces an Effective Landscape-Dependent Regularization Favoring Flat Solutions},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2023},
+ month = {6},
+ number = {23},
+ volume = {130},
+ pages = {237101},
+ url = {https://doi.org/10.1103%2Fphysrevlett.130.237101},
+ doi = {10.1103/physrevlett.130.237101}
+}
+
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