More writing.

2 files changed, 87 insertions, 52 deletions

diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index d8927bc..3e8acb7 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -1,6 +1,6 @@
 @article{Albert_2021_Searching,
  author = {Albert, Samuel and Biroli, Giulio and Ladieu, François and Tourbot, Roland and Urbani, Pierfrancesco},
- title = {Searching for the {Gardner} Transition in Glassy Glycerol},
+ title = {Searching for the Gardner Transition in Glassy Glycerol},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2021},
@@ -14,7 +14,7 @@
 
 @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},
+ 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},
@@ -28,7 +28,7 @@
 
 @article{Annibale_2003_Supersymmetric,
  author = {Annibale, Alessia and Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio},
- title = {Supersymmetric complexity in the {Sherrington}-{Kirkpatrick} model},
+ title = {Supersymmetric complexity in the Sherrington-Kirkpatrick model},
  journal = {Physical Review E},
  publisher = {American Physical Society (APS)},
  year = {2003},
@@ -42,7 +42,7 @@
 
 @article{Annibale_2003_The,
  author = {Annibale, Alessia and Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio and Trevigne, Elisa},
- title = {The role of the {Becchi}--{Rouet}--{Stora}--{Tyutin} supersymmetry in the calculation of the complexity for the {Sherrington}--{Kirkpatrick} model},
+ title = {The role of the Becchi--Rouet--Stora--Tyutin supersymmetry in the calculation of the complexity for the Sherrington--Kirkpatrick model},
  journal = {Journal of Physics A: Mathematical and General},
  publisher = {IOP Publishing},
  year = {2003},
@@ -112,7 +112,7 @@
 
 @article{Berthier_2019_Gardner,
  author = {Berthier, Ludovic and Biroli, Giulio and Charbonneau, Patrick and Corwin, Eric I. and Franz, Silvio and Zamponi, Francesco},
- title = {{Gardner} physics in amorphous solids and beyond},
+ title = {Gardner physics in amorphous solids and beyond},
  journal = {The Journal of Chemical Physics},
  publisher = {AIP Publishing},
  year = {2019},
@@ -140,7 +140,7 @@
 
 @article{Biroli_2018_Liu-Nagel,
  author = {Biroli, Giulio and Urbani, Pierfrancesco},
- title = {{Liu}-{Nagel} phase diagrams in infinite dimension},
+ title = {Liu-Nagel phase diagrams in infinite dimension},
  journal = {SciPost Physics},
  publisher = {Stichting SciPost},
  year = {2018},
@@ -168,7 +168,7 @@
 
 @article{Bray_2007_Statistics,
  author = {Bray, Alan J. and Dean, David S.},
- title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces},
+ title = {Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2007},
@@ -210,7 +210,7 @@
 
 @article{Cavagna_1998_Stationary,
  author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio},
- title = {Stationary points of the {Thouless}-{Anderson}-{Palmer} free energy},
+ title = {Stationary points of the Thouless-Anderson-Palmer free energy},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {1998},
@@ -252,7 +252,7 @@
 
 @article{Charbonneau_2015_Numerical,
  author = {Charbonneau, Patrick and Jin, Yuliang and Parisi, Giorgio and Rainone, Corrado and Seoane, Beatriz and Zamponi, Francesco},
- title = {Numerical detection of the {Gardner} transition in a mean-field glass former},
+ title = {Numerical detection of the Gardner transition in a mean-field glass former},
  journal = {Physical Review E},
  publisher = {American Physical Society (APS)},
  year = {2015},
@@ -294,7 +294,7 @@
 
 @article{Crisanti_1995_Thouless-Anderson-Palmer,
  author = {Crisanti, A. and Sommers, H.-J.},
- title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical {$p$}-Spin Spin Glass Model},
+ title = {Thouless-Anderson-Palmer Approach to the Spherical $p$-Spin Spin Glass Model},
  journal = {Journal de Physique I},
  publisher = {EDP Sciences},
  year = {1995},
@@ -308,7 +308,7 @@
 
 @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},
+ title = {Spherical $2+p$ Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2004},
@@ -322,7 +322,7 @@
 
 @article{Crisanti_2006_Spherical,
  author = {Crisanti, A. and Leuzzi, L.},
- title = {Spherical {$2+p$} spin-glass model: An analytically solvable model with a glass-to-glass transition},
+ title = {Spherical $2+p$ spin-glass model: An analytically solvable model with a glass-to-glass transition},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2006},
@@ -406,7 +406,7 @@
 
 @unpublished{ElAlaoui_2022_Sampling,
  author = {El Alaoui, Ahmed and Montanari, Andrea and Sellke, Mark},
- title = {Sampling from the {Sherrington}-{Kirkpatrick} {Gibbs} measure via algorithmic
+ title = {Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic
 stochastic localization},
  year = {2022},
  month = {3},
@@ -436,14 +436,14 @@ stochastic localization},
 
 @phdthesis{Folena_2020_The,
  author = {Folena, Giampaolo},
- title = {The mixed {$p$}-spin model: selecting, following and losing states},
+ title = {The mixed $p$-spin model: selecting, following and losing states},
  year = {2020},
  month = {3},
  number = {2020UPASS060},
  url = {https://tel.archives-ouvertes.fr/tel-02883385},
  hal_id = {tel-02883385},
  hal_version = {v1},
- school = {Université Paris-Saclay \& Università degli studi La Sapienza (Rome)},
+ school = {Université Paris-Saclay & Università degli studi La Sapienza (Rome)},
  type = {Theses}
 }
 
@@ -463,7 +463,7 @@ stochastic localization},
 
 @article{Gamarnik_2021_The,
  author = {Gamarnik, David and Jagannath, Aukosh},
- title = {The overlap gap property and approximate message passing algorithms for {$p$}-spin models},
+ title = {The overlap gap property and approximate message passing algorithms for $p$-spin models},
  journal = {The Annals of Probability},
  publisher = {Institute of Mathematical Statistics},
  year = {2021},
@@ -477,7 +477,7 @@ stochastic localization},
 
 @article{Gardner_1985_Spin,
  author = {Gardner, E.},
- title = {Spin glasses with {$p$}-spin interactions},
+ title = {Spin glasses with $p$-spin interactions},
  journal = {Nuclear Physics B},
  publisher = {Elsevier BV},
  year = {1985},
@@ -490,8 +490,8 @@ stochastic localization},
 
 @article{Geirhos_2018_Johari-Goldstein,
  author = {Geirhos, K. and Lunkenheimer, P. and Loidl, A.},
- title = {{Johari}-{Goldstein} Relaxation Far Below 
-{$T_g$}: Experimental Evidence for the {Gardner} Transition in Structural Glasses?},
+ title = {Johari-Goldstein Relaxation Far Below 
+$T_g$: Experimental Evidence for the Gardner Transition in Structural Glasses?},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2018},
@@ -512,14 +512,14 @@ stochastic localization},
  pages = {204--209},
  url = {https://doi.org/10.1142%2F9789812701558_0023},
  doi = {10.1142/9789812701558_0023},
- booksubtitle = {Proceedings of the 31st Workshop of the International School of Solid State Physics, {Erice}, {Sicily}, {Italy}, 20 – 26 {July} 2004},
+ booksubtitle = {Proceedings of the 31st Workshop of the International School of Solid State Physics, Erice, Sicily, Italy, 20 – 26 July 2004},
  booktitle = {Complexity, Metastability and Nonextensivity},
  editor = {Beck, C and Benedek, G and Rapisarda, A and Tsallis, C}
 }
 
 @article{Gross_1985_Mean-field,
  author = {Gross, D. J. and Kanter, I. and Sompolinsky, H.},
- title = {Mean-field theory of the {Potts} glass},
+ title = {Mean-field theory of the Potts glass},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {1985},
@@ -547,7 +547,7 @@ stochastic localization},
 
 @article{Hicks_2018_Gardner,
  author = {Hicks, C. L. and Wheatley, M. J. and Godfrey, M. J. and Moore, M. A.},
- title = {{Gardner} Transition in Physical Dimensions},
+ title = {Gardner Transition in Physical Dimensions},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2018},
@@ -561,7 +561,7 @@ stochastic localization},
 
 @unpublished{Huang_2021_Tight,
  author = {Huang, Brice and Sellke, Mark},
- title = {Tight {Lipschitz} Hardness for Optimizing Mean Field Spin Glasses},
+ title = {Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses},
  year = {2021},
  month = {10},
  url = {http://arxiv.org/abs/2110.07847v1},
@@ -614,9 +614,25 @@ stochastic localization},
  primaryclass = {cond-mat.stat-mech}
 }
 
+@unpublished{Kent-Dobias_2022_How,
+ author = {Kent-Dobias, Jaron and Kurchan, Jorge},
+ title = {How to count in hierarchical landscapes: a `full' solution to mean-field
+complexity},
+ year = {2022},
+ month = {7},
+ url = {http://arxiv.org/abs/2207.06161v2},
+ archiveprefix = {arXiv},
+ date = {2022-07-13T12:45:58Z},
+ eprint = {2207.06161v2},
+ eprintclass = {cond-mat.stat-mech},
+ eprinttype = {arxiv},
+ primaryclass = {cond-mat.stat-mech},
+ urldate = {2022-10-05T20:12:41.619402Z}
+}
+
 @article{Li_2021_Determining,
  author = {Li, Huaping and Jin, Yuliang and Jiang, Ying and Chen, Jeff Z. Y.},
- title = {Determining the nonequilibrium criticality of a {Gardner} transition via a hybrid study of molecular simulations and machine learning},
+ title = {Determining the nonequilibrium criticality of a Gardner transition via a hybrid study of molecular simulations and machine learning},
  journal = {Proceedings of the National Academy of Sciences},
  publisher = {Proceedings of the National Academy of Sciences},
  year = {2021},
@@ -771,7 +787,7 @@ stochastic localization},
 
 @article{Rieger_1992_The,
  author = {Rieger, H.},
- title = {The number of solutions of the {Thouless}-{Anderson}-{Palmer} equations for {$p$}-spin-interaction spin glasses},
+ title = {The number of solutions of the Thouless-Anderson-Palmer equations for $p$-spin-interaction spin glasses},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {1992},
@@ -827,7 +843,7 @@ stochastic localization},
 
 @article{Seguin_2016_Experimental,
  author = {Seguin, A. and Dauchot, O.},
- title = {Experimental Evidence of the {Gardner} Phase in a Granular Glass},
+ title = {Experimental Evidence of the Gardner Phase in a Granular Glass},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2016},
@@ -855,7 +871,7 @@ stochastic localization},
 
 @article{Xiao_2022_Probing,
  author = {Xiao, Hongyi and Liu, Andrea J. and Durian, Douglas J.},
- title = {Probing {Gardner} Physics in an Active Quasithermal Pressure-Controlled Granular System of Noncircular Particles},
+ title = {Probing Gardner Physics in an Active Quasithermal Pressure-Controlled Granular System of Noncircular Particles},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2022},
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index f216171..f539fbc 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,11 +26,14 @@
 \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
 
 \begin{abstract}
-  We derive the general solution for counting the stationary points of
-  mean-field complex landscapes. It incorporates Parisi's solution
-  for the ground state, as it should. Using this solution, we count the
-  stationary points of two models: one with multi-step replica symmetry
-  breaking, and one with full replica symmetry breaking.
+  Complexity is a measure of the number of stationary points in complex
+  landscapes. We derive a general solution for the complexity of mean-field
+  complex landscapes. It incorporates Parisi's solution for the ground state,
+  as it should. Using this solution, we count the stationary points of two
+  models: one with multi-step replica symmetry breaking, and one with full
+  replica symmetry breaking. These examples demonstrate the consistency of the
+  solution and reveal that the signature of replica symmetry breaking at high
+  energy densities is found in high-index saddles, not minima.
 \end{abstract}
 
 \maketitle
@@ -64,14 +67,14 @@ 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, we detail 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.}.
+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}.
 
 We study the mixed $p$-spin spherical models, with Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
@@ -275,6 +278,7 @@ transitions are listed in Table~\ref{tab:energies}.
     $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
     $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
     $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\
     $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
   \end{tabular}
   \caption{
@@ -287,7 +291,8 @@ transitions are listed in Table~\ref{tab:energies}.
     points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic
     threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the
     traditional threshold energy, defined by the energy at which marginal
-    minima become most common. $E_0$ is the ground state energy.
+    minima become most common. $E_\mathrm m$ is the lowest energy at which
+    saddles or marginal minima are found. $E_0$ is the ground state energy.
   } \label{tab:energies}
 \end{table}
 
@@ -295,13 +300,15 @@ In this model, the RS complexity gives an inconsistent answer for the
 complexity of the ground state, predicting that the complexity of minima
 vanishes at a higher energy than the complexity of saddles, with both at a
 lower energy than the equilibrium ground state. The 1RSB complexity resolves
-these problems, predicting the same ground state as equilibrium and with a ground state stability $\mu_0=6.480\,764\ldots>\mu_m$. It predicts that the
-complexity of marginal minima (and therefore all saddles) vanishes at
-$E_m=-1.287\,605\,527\ldots$, which is very slightly greater than $E_0$. Saddles
-become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,575\,114\ldots$.
-The 1RSB complexity transitions to a RS description for dominant stationary
-points at an energy $E_1=-1.273\,886\,852\ldots$. The highest energy for which
-the 1RSB description exists is $E_\mathrm{max}=-0.886\,029\,051\ldots$
+these problems, predicting the same ground state as equilibrium and with a
+ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that
+the complexity of marginal minima (and therefore all saddles) vanishes at
+$E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become
+dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity
+transitions to a RS description for dominant stationary points at an energy
+$E_\mathrm{dom}$. The highest energy for which the 1RSB description exists is
+$E_\mathrm{max}$. The numeric values for all these energies are listed in
+Table~\ref{tab:energies}.
 
 For minima, the complexity does
 not inherit a 1RSB description until the energy is with in a close vicinity of
@@ -349,16 +356,28 @@ also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_
 \end{equation}
 In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty=-0.53125\ldots$.
 
-Fig.~\ref{fig:frsb.phases}
-shows these trajectories, along with the phase boundaries of the complexity in
-this plane. Notably, the phase boundary predicted by a perturbative expansion
-correctly predicts where all of the finite $k$RSB approximations terminate.
+Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model.
+Notably, the phase boundary predicted by a perturbative expansion
+correctly predicts where the finite $k$RSB approximations terminate.
 Like the 1RSB model in the previous subsection, this phase boundary is oriented
 such that very few, low energy, minima are described by a FRSB solution, while
 relatively high energy saddles of high index are also. Again, this suggests
 that studying the mutual distribution of high-index saddle points might give
 insight into lower-energy symmetry breaking in more general contexts.
 
+We have used our solution for mean-field complexity to explore how hierarchical
+RSB in equilibrium corresponds to analogous hierarchical structure in the
+energy landscape. In the examples we studied, a relative minority of energy
+minima are distributed in a nontrivial way, corresponding to the lowest energy
+densities. On the other hand, very high-index saddles begin exhibit RSB at much
+higher energy densities, on the order of the energy densities associated with
+RSB transitions in equilibrium. More wore is necessary to explore this
+connection, as well as whether a purely \emph{geometric} explanation can be
+made for the algorithmic threshold. Applying this method to the most realistic
+RSB scenario for structural glasses, the so-called 1FRSB which has features of
+both 1RSB and FRSB, might yield insights about signatures that should be
+present in the landscape.
+
 \paragraph{Acknowledgements}
 The authors would like to thank Valentina Ros for helpful discussions.