Merge branch 'master' into aps

9 files changed, 569 insertions, 38 deletions

diff --git a/.gitignore b/.gitignore
index faacc6a..9a36fc7 100644
--- a/.gitignore
+++ b/.gitignore
@@ -11,3 +11,4 @@
 *.run.xml
 *.synctex(busy)
 *.toc
+*Notes.bib
diff --git a/appeal.tex b/appeal.tex
new file mode 100644
index 0000000..7ae78f4
--- /dev/null
+++ b/appeal.tex
@@ -0,0 +1,70 @@
+\documentclass[a4paper]{letter}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[
+  colorlinks=true,
+  urlcolor=purple,
+  linkcolor=black,
+  citecolor=black,
+  filecolor=black
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{xcolor}
+\usepackage[style=phys]{biblatex}
+
+\addbibresource{bezout.bib}
+
+\signature{
+  \vspace{-6\medskipamount}
+  \smallskip
+  Jaron Kent-Dobias \& Jorge Kurchan
+}
+
+\address{
+  Laboratoire de Physique\\
+  Ecole Normale Sup\'erieure\\
+  24 rue Lhomond\\ 
+  75005 Paris
+}
+
+\begin{document}
+\begin{letter}{
+  Editorial Office\\
+  Physical Review Letters\\
+  1 Research Road\\
+  Ridge, NY 11961
+}
+
+\opening{To the editors of Physical Review,}
+
+We wish to appeal the decision on our manuscript \emph{How to count in
+hierarchical landscapes: A ‘full’ solution to mean-field complexity}, which was
+rejected without being sent to referees.
+
+The problem of characterizing the geometry of complex energy and cost
+landscapes is long-standing. Until this work, the correct calculation of the
+complexity has only been made for a small minority of systems, those with
+so-called replica symmetry. We show explicitly how such calculations can be
+made for the vast majority of cases.
+
+Landscape complexity even for the simple models we consider is relevant to a
+broad spectrum of physics disciplines. These models appear explicitly in modern
+research of machine learning, like tensor denoising, and understanding how
+complexity, dynamics, and equilibrium interplay in them provides powerful analogies
+and insights into emergent phenomena in more complicated contexts, from
+realistic machine learning models to the behavior of structural glasses.
+Already in this work, we identify the surprising result that the purported
+algorithmic threshold for optimization on mean-field cost functions lies
+\emph{far above} the geometric threshold traditionally understood as the dynamic limit.
+
+We urge you to allow this paper to go to referees and allow it to
+be judged by other scientists at the forefront of these fields.
+
+\closing{Sincerely,}
+
+\vspace{1em}
+
+\end{letter}
+
+\end{document}
diff --git a/figs/24_detail_letter_legend.pdf b/figs/24_detail_letter_legend.pdf
new file mode 100644
index 0000000..ca32ca0
--- /dev/null
+++ b/figs/24_detail_letter_legend.pdf
diff --git a/figs/24_phases_letter.pdf b/figs/24_phases_letter.pdf
new file mode 100644
index 0000000..366f49e
--- /dev/null
+++ b/figs/24_phases_letter.pdf
diff --git a/figs/316_complexity_contour_1_letter.pdf b/figs/316_complexity_contour_1_letter.pdf
new file mode 100644
index 0000000..b5aa80d
--- /dev/null
+++ b/figs/316_complexity_contour_1_letter.pdf
diff --git a/figs/316_detail_letter.pdf b/figs/316_detail_letter.pdf
new file mode 100644
index 0000000..4c33457
--- /dev/null
+++ b/figs/316_detail_letter.pdf
diff --git a/figs/316_detail_letter_legend.pdf b/figs/316_detail_letter_legend.pdf
new file mode 100644
index 0000000..756a2c7
--- /dev/null
+++ b/figs/316_detail_letter_legend.pdf
diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index 0374c8a..f8d4265 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},
@@ -376,16 +376,18 @@
  doi = {10.1103/physrevlett.124.078002}
 }
 
-@article{ElAlaoui_2020_Algorithmic,
+@unpublished{ElAlaoui_2020_Algorithmic,
  author = {El Alaoui, Ahmed and Montanari, Andrea},
  title = {Algorithmic Thresholds in Mean Field Spin Glasses},
  year = {2020},
  month = {9},
  url = {http://arxiv.org/abs/2009.11481v1},
+ archiveprefix = {arXiv},
  date = {2020-09-24T04:22:42Z},
  eprint = {2009.11481v1},
  eprintclass = {cond-mat.stat-mech},
- eprinttype = {arxiv}
+ eprinttype = {arxiv},
+ primaryclass = {cond-mat.stat-mech}
 }
 
 @article{ElAlaoui_2021_Optimization,
@@ -402,17 +404,19 @@
  doi = {10.1214/21-aop1519}
 }
 
-@article{ElAlaoui_2022_Sampling,
+@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},
  url = {http://arxiv.org/abs/2203.05093v1},
+ archiveprefix = {arXiv},
  date = {2022-03-10T00:15:22Z},
  eprint = {2203.05093v1},
  eprintclass = {math.PR},
- eprinttype = {arxiv}
+ eprinttype = {arxiv},
+ primaryclass = {cond-mat.stat-mech}
 }
 
 @article{Folena_2020_Rethinking,
@@ -432,14 +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}
 }
 
@@ -459,7 +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},
@@ -473,7 +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},
@@ -486,8 +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},
@@ -508,14 +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},
@@ -543,7 +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},
@@ -555,16 +559,18 @@ stochastic localization},
  doi = {10.1103/physrevlett.120.225501}
 }
 
-@article{Huang_2021_Tight,
+@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},
+ archiveprefix = {arXiv},
  date = {2021-10-15T04:08:35Z},
  eprint = {2110.07847v1},
  eprintclass = {math.PR},
- eprinttype = {arxiv}
+ eprinttype = {arxiv},
+ primaryclass = {cond-mat.stat-mech}
 }
 
 @article{Kac_1943_On,
@@ -597,18 +603,37 @@ stochastic localization},
 @article{Kent-Dobias_2022_Analytic,
  author = {Kent-Dobias, Jaron and Kurchan, Jorge},
  title = {Analytic continuation over complex landscapes},
+ journal = {Journal of Physics A: Mathematical and Theoretical},
+ publisher = {IOP Publishing},
  year = {2022},
- month = {4},
- url = {http://arxiv.org/abs/2204.06072v1},
- date = {2022-04-12T20:24:54Z},
- eprint = {2204.06072v1},
+ month = {10},
+ number = {43},
+ volume = {55},
+ pages = {434006},
+ url = {https://doi.org/10.1088%2F1751-8121%2Fac9cc7},
+ doi = {10.1088/1751-8121/ac9cc7},
+ collection = {Random Landscapes and Dynamics in Evolution, Ecology and Beyond}
+}
+
+@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}
+ 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},
@@ -763,7 +788,7 @@ 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},
@@ -819,7 +844,7 @@ 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},
@@ -847,7 +872,7 @@ 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
new file mode 100644
index 0000000..4fb22e7
--- /dev/null
+++ b/frsb_kac-rice_letter.tex
@@ -0,0 +1,435 @@
+
+\documentclass[reprint,aps,prl,longbibliography,floatfix]{revtex4-2}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{amsmath,amssymb,latexsym,graphicx}
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[dvipsnames]{xcolor}
+\usepackage[
+  colorlinks=true,
+  urlcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  filecolor=MidnightBlue,
+  linkcolor=MidnightBlue
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{anyfontsize}
+
+\begin{document}
+
+\title{
+  Unveiling the complexity of hierarchical energy landscapes
+}
+
+\author{Jaron Kent-Dobias}
+\author{Jorge Kurchan}
+\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
+
+\begin{abstract}
+  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
+
+The functions used to describe the energies, costs, and fitnesses of disordered
+systems in physics, computer science, and biology are typically \emph{complex},
+meaning that they have a number of minima that grows exponentially with the
+size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex,
+Altieri_2021_Properties}. Though they are often called `rough landscapes' to
+evoke the intuitive image of many minima in something like a mountain range,
+the metaphor to topographical landscapes is strained by the reality that these
+complex landscapes also exist in very high dimensions: think of the dimensions
+of phase space for $N$ particles, or the number of parameters in a neural
+network.
+
+The \emph{complexity} of a function is the average of the logarithm of the
+number of its minima, maxima, and saddle points (collectively stationary
+points), under conditions fixing the value of the energy or the index of the
+stationary point 
+\cite{Bray_1980_Metastable}. 
+Since in complex landscapes this
+number grows exponentially with system size, their complexity is an extensive
+quantity. Understanding the complexity offers an understanding about the
+geometry and topology of the landscape, which can provide insight into
+dynamical behavior.
+
+When complex systems are fully connected, i.e., each degree of freedom
+interacts directly with every other, they are often described by a hierarchical
+structure of the type first proposed by Parisi, the \emph{replica symmetry
+breaking} (RSB) \cite{Parisi_1979_Infinite}. This family of structures is rich, spanning uniform
+\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested
+structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB),
+and arbitrary combinations thereof. Though these rich structures are understood
+in the equilibrium properties of fully connected models, the complexity has
+only been computed in RS cases.
+
+In this paper and its longer companion, we share the first results for the
+complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. 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.}.
+
+We study the mixed $p$-spin spherical models, with Hamiltonian
+\begin{equation} \label{eq:hamiltonian}
+  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
+\end{equation}
+is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
+$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with
+zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
+the energy is typically extensive. The overbar will always denote an average
+over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
+constants that define the particular model. For instance, the so-called `pure'
+models have $a_p=1$ for some $p$ and all others zero.
+
+The complexity of the $p$-spin models has been extensively studied by
+physicists and mathematicians. Among physicists, the bulk of work has been on
+ the so-called  `TAP' complexity, 
+which counts minima in the mean-field Thouless--Anderson--Palmer () free energy \cite{Rieger_1992_The,
+Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
+Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
+Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure
+and mixed models without RSB \cite{Auffinger_2012_Random,
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been
+treated without RSB \cite{Folena_2020_Rethinking}. And the methods of
+complexity have been used to study many geometric properties of the pure
+models, from the relative position of stationary points to one another to shape
+and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
+
+The variance of the couplings implies that the covariance of the energy with
+itself depends on only the dot product (or overlap) between two configurations.
+In particular, one finds
+\begin{equation} \label{eq:covariance}
+  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right),
+\end{equation}
+where $f$ is defined by the series
+\begin{equation}
+  f(q)=\frac12\sum_pa_pq^p.
+\end{equation}
+One needn't start with a Hamiltonian like
+\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
+\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in
+the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.
+
+The family of spherical models thus defined is quite rich, and by varying the
+covariance $f$ nearly any hierarchical structure can be found in
+equilibrium. Because of a correspondence between the ground state complexity
+and the entropy at zero temperature, any hierarchical structure in the
+equilibrium should be reflected in the complexity.
+
+The complexity is calculated using the Kac--Rice formula, which counts the
+stationary points using a $\delta$-function weighted by a Jacobian
+\cite{Kac_1943_On, Rice_1939_The}. The count is given by
+\begin{equation}
+  \begin{aligned}
+    \mathcal N(E, \mu)
+    &=\int_{\mathbb R^N}d\boldsymbol\xi\,e^{-\frac12\|\boldsymbol\xi\|^2/\sigma^2}\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)-\boldsymbol\xi\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\
+    &\hspace{2pc}\times\delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big)
+  \end{aligned}
+\end{equation}
+with two additional $\delta$-functions inserted to fix the energy density $E$
+and the stability $\mu$. The additional `noise' field $\boldsymbol\xi$
+helps regularize the $\delta$-functions for the energy and stability at finite
+$N$, and will be convenient for defining the order parameter matrices later. The complexity is then
+\begin{equation} \label{eq:complexity}
+  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{\sigma\to0}\frac1N\overline{\log\mathcal N(E, \mu}).
+\end{equation}
+Most of the difficulty of this calculation resides in the logarithm in this
+formula.
+
+The stability $\mu$, sometimes called the radial reaction, determines the depth
+of minima or the index of saddles. At large $N$ the Hessian can be shown to
+consist of the sum of a GOE matrix with variance $f''(1)/N$ shifted by a
+constant diagonal matrix of value $\mu$. Therefore, the spectrum of the Hessian
+is a Wigner semicircle of radius $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu$. When
+$\mu>\mu_\mathrm m$, stationary points are minima whose sloppiest eigenvalue is
+$\mu-\mu_\mathrm m$. When $\mu=\mu_\mathrm m$, the stationary points are marginal minima with
+flat directions. When $\mu<\mu_\mathrm m$, the stationary points are saddles with
+indexed fixed to within order one (fixed macroscopic index).
+
+It's worth reviewing the complexity for the best-studied case of the pure model
+for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. Here, because the covariance
+is a homogeneous polynomial, $E$ and $\mu$ cannot be fixed separately, and one
+implies the other: $\mu=pE$. Therefore at each energy there is only one kind of
+stationary point. When the energy reaches $E_\mathrm{th}=-\mu_\mathrm m/p$, the
+population of stationary points suddenly shifts from all saddles to all minima,
+and there is an abrupt percolation transition in the topology of
+constant-energy slices of the landscape. This behavior of the complexity can be
+used to explain a rich variety of phenomena in the equilibrium and dynamics of
+the pure models: the `threshold' \cite{Cugliandolo_1993_Analytical} energy $E_\mathrm{th}$ corresponds to the
+average energy at the dynamic transition temperature, and the asymptotic energy
+reached by slow aging dynamics.
+
+Things become much less clear in even the simplest mixed models. For instance,
+one mixed model known to have a replica symmetric complexity was shown to
+nonetheless not have a clear relationship between features of the complexity
+and the asymptotic dynamics \cite{Folena_2020_Rethinking}. There is no longer a
+sharp topological transition.
+
+In the pure models, $E_\mathrm{th}$ also corresponds to the \emph{algorithmic
+threshold} $E_\mathrm{alg}$, defined by the lowest energy reached by local
+algorithms like approximate message passing \cite{ElAlaoui_2020_Algorithmic,
+ElAlaoui_2021_Optimization}. In the spherical models, this has been proven to
+be
+\begin{equation}
+  E_{\mathrm{alg}}=-\int_0^1dq\,\sqrt{f''(q)}
+\end{equation}
+For full RSB systems, $E_\mathrm{alg}=E_0$ and the algorithm can reach the
+ground state energy. For the pure $p$-spin models,
+$E_\mathrm{alg}=E_\mathrm{th}$, where $E_\mathrm{th}$ is the energy at which
+marginal minima are the most common stationary points. Something about the
+topology of the energy function might be relevant to where this algorithmic
+threshold lies.
+
+To compute the complexity in the generic case, we use the replica method to
+treat the logarithm inside the average of \eqref{eq:complexity}, and the
+$\delta$-functions are written in a Fourier basis. The average of the factor
+including the determinant and the factors involving $\delta$-functions can be
+averaged over the disorder separately \cite{Bray_2007_Statistics}. The result
+can be written
+\begin{equation}
+  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{n\to0}\frac1N\frac{\partial}{\partial n}
+  \int_{\mathrm M_n(\mathbb R)} dQ\,dR\,dD\,e^{N\mathcal S(Q,R,D\mid E,\mu)},
+\end{equation}
+where the effective action $\mathcal S$ is a function of three matrices indexed
+by the $n$ replicas:
+\begin{equation}
+  \begin{aligned}
+    &Q_{ab}=\frac{\mathbf s_a\cdot\mathbf s_b}N
+    \hspace{4em}
+    R_{ab}=\frac{\boldsymbol\xi_a\cdot\mathbf s_b}{N\sigma^2}
+    \\
+    &D_{ab}=\frac1{N\sigma^4}\left(\sigma^2\delta_{ab}-\boldsymbol\xi_a\cdot\boldsymbol\xi_b\right).
+  \end{aligned}
+\end{equation}
+The matrix $Q$ is a clear analogue of the usual overlap matrix of the
+equilibrium case. The matrices $R$ and $D$ have interpretations as response
+functions: $R$ is related to the typical displacement of stationary points by
+perturbations to the potential, and $D$ is related to the change in the
+complexity caused by the same perturbations. The general expression for the
+complexity as a function of these matrices is also found in
+\cite{Folena_2020_Rethinking}.
+
+The complexity is found by the saddle point method, extremizing $\mathcal S$
+with respect to $Q$, $R$, and $D$ and replacing the integral with its integrand
+evaluated at the extremum. We make the \emph{ansatz} that all three matrices have
+a hierarchical structure, and moreover that they share the same hierarchical
+structure. This means that the size of the blocks of equal value of each is the
+same, though the values inside these blocks will vary from matrix to matrix.
+This form can be shown to exactly reproduce the ground state energy predicted
+by the equilibrium solution, a key consistency check.
+
+Along one line in the energy--stability plane the solution takes a simple form:
+the matrices $R$ and $D$ corresponding to responses are diagonal, leaving
+only the overlap matrix $Q$ with nontrivial off-diagonal entries. This
+simplification makes the solution along this line analytically tractable even
+for FRSB. The simplification is related to the presence of an approximate
+supersymmetry in the Kac--Rice formula, studied in the past in the context of
+the TAP free energy \cite{Annibale_2003_Supersymmetric, Annibale_2003_The,
+Annibale_2004_Coexistence}. This line of `supersymmetric' solutions terminates
+at the ground state, and describes the most numerous types of stable minima.
+
+Using this solution, one finds a correspondence between properties of the
+overlap matrix $Q$ at the ground state energy, where the complexity vanishes,
+and the overlap matrix in the equilibrium problem in the limit of zero
+temperature. The saddle point parameters of the two problems are related
+exactly. In the case where the vicinity of the equilibrium ground state is
+described by a $k$RSB solution, the complexity at the ground state is
+$(k-1)$RSB. This can be intuitively understood by considering the difference
+between measuring overlaps between equilibrium \emph{states} and stationary
+\emph{points}. For states, the finest level of the hierarchical description
+gives the typical overlap between two points drawn from the same state, which
+has some distribution about the ground state at nonzero temperature. For
+points, this finest level does not exist.
+
+In general, solving the saddle-point equations for the parameters of the three
+replica matrices is challenging. Unlike the equilibrium case, the solution is
+not extremal, and so minimization methods cannot be used. However, the line of
+simple `supersymmetric' solutions offers a convenient foothold: starting from
+one of these solutions, the parameters $E$ and $\mu$ can be slowly varied to
+find the complexity everywhere. This is how the data in what follows was produced.
+
+\begin{figure}
+  \centering
+  \hspace{-1em}
+  \includegraphics[width=\columnwidth]{figs/316_complexity_contour_1_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}
+
+  \caption{
+    Complexity of the $3+16$ model in the energy $E$ and stability $\mu$
+    plane. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Below the yellow marginal line
+    the complexity counts saddles of increasing index as $\mu$ decreases. Above
+    the yellow marginal line the complexity counts minima of increasing
+    stability as $\mu$ increases.
+  } \label{fig:2rsb.contour}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}
+
+  \caption{
+    Detail of the `phases' of the $3+16$ model complexity as a function of
+    energy and stability. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Above the yellow marginal stability line the
+    complexity counts saddles of fixed index, while below that line it counts
+    minima of fixed stability. The shaded red region shows places where the
+    complexity is described by the 1RSB solution, while the shaded gray region
+    shows places where the complexity is described by the RS solution. In white
+    regions the complexity is zero. Several interesting energies are marked
+    with vertical black lines: the traditional `threshold' $E_\mathrm{th}$
+    where minima become most numerous, the algorithmic threshold
+    $E_\mathrm{alg}$ that bounds the performance of smooth algorithms, and the
+    average energies at the $2$RSB and $1$RSB equilibrium transitions $\langle
+    E\rangle_2$ and $\langle E\rangle_1$, respectively. Though the figure is
+    suggestive, $E_\mathrm{alg}$ lies at slightly lower energy than the termination of the RS
+    -- 1RSB transition line.
+  } \label{fig:2rsb.phases}
+\end{figure}
+
+For the first example, we study a model whose complexity has the simplest
+replica symmetry breaking scheme, 1RSB. By choosing a covariance $f$ as the sum
+of polynomials with well-separated powers, one develops 2RSB in equilibrium.
+This should correspond to 1RSB in the complexity. We take
+\begin{equation}
+  f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
+\end{equation}
+established to have a 2RSB ground state \cite{Crisanti_2011_Statistical}.
+With this covariance, the model sees a replica symmetric to 1RSB transition at
+$\beta_1=1.70615\ldots$ and a 1RSB to 2RSB transition at
+$\beta_2=6.02198\ldots$. The typical equilibrium energies at these phase
+transitions are listed in Table~\ref{tab:energies}.
+
+\begin{table}
+  \begin{tabular}{l|cc}
+    & $3+16$ & $2+4$ \\\hline\hline
+    $\langle E\rangle_\infty$ &---& $-0.531\,25\hphantom{1\,111\dots}$ \\
+    $\hphantom{\langle}E_\mathrm{max}$ &     $-0.886\,029\,051\dots$ & $-1.039\,701\,412\dots$\\
+    $\langle E\rangle_1$ & $-0.906\,391\,055\dots$ & ---\\
+    $\langle E\rangle_2$ & $-1.195\,531\,881\dots$ & ---\\
+    $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
+    $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\
+    $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
+  \end{tabular}
+  \caption{
+    Landmark energies of the equilibrium and complexity problems for the two
+    models studied. $\langle E\rangle_1$, $\langle E\rangle_2$ and $\langle
+    E\rangle_\infty$ are the average energies in equilibrium at the RS--1RSB,
+    1RSB--2RSB, and RS--FRSB transitions, respectively. $E_\mathrm{max}$ is the
+    highest energy at which any stationary points are described by a RSB
+    complexity. $E_\mathrm{dom}$ is the energy at which dominant stationary
+    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. $E_\mathrm m$ is the lowest energy at which
+    saddles or marginal minima are found. $E_0$ is the ground state energy.
+  } \label{tab:energies}
+\end{table}
+
+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, shown in Fig.~\ref{fig:2rsb.contour}. It predicts 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
+the ground state. On the other hand, for high-index saddles the complexity
+becomes described by 1RSB at quite high energies. This suggests that when
+sampling a landscape at high energies, high index saddles may show a sign of
+replica symmetry breaking when minima or inherent states do not.
+
+Fig.~\ref{fig:2rsb.phases} shows a different detail of the complexity in the
+vicinity of the ground state, now as functions of the energy difference and
+stability difference from the ground state. Several of the landmark energies
+described above are plotted, alongside the boundaries between the `phases.'
+Though $E_\mathrm{alg}$ looks quite close to the energy at which dominant
+saddles transition from 1RSB to RS, they differ by roughly $10^{-3}$, as
+evidenced by the numbers cited above. Likewise, though $\langle E\rangle_1$
+looks very close to $E_\mathrm{max}$, where the 1RSB transition line
+terminates, they too differ. The fact that $E_\mathrm{alg}$ is very slightly
+below the place where most saddle transition to 1RSB is suggestive; we
+speculate that an analysis of the typical minima connected to these saddles by
+downward trajectories will coincide with the algorithmic limit. An analysis of
+the typical nearby minima or the typical downward trajectories from these
+saddles at 1RSB is warranted \cite{Ros_2019_Complex, Ros_2021_Dynamical}. Also
+notable is that $E_\mathrm{alg}$ is at a significantly higher energy than
+$E_\mathrm{th}$; according to the theory, optimal smooth algorithms in this
+model stall in a place where minima are exponentially subdominant.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/24_phases_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/24_detail_letter_legend.pdf}
+  \caption{
+    `Phases' of the complexity for the $2+4$ model in the energy $E$ and
+    stability $\mu$ plane. Solid lines show the prediction of 1RSB complexity,
+    while dashed lines show the prediction of RS complexity. The region shaded
+    gray shows where the RS solution is correct, while the region shaded red
+    shows that where the FRSB solution is correct. The white region shows where
+    the complexity is zero.
+  } \label{fig:frsb.phases}
+\end{figure}
+
+If the covariance $f$ is chosen to be concave, then one develops FRSB in equilibrium. To this purpose, we choose
+\begin{equation}
+  f(q)=\frac12\left(q^2+\frac1{16}q^4\right),
+\end{equation}
+also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. Because the ground state is FRSB, for this model $E_0=E_\mathrm{alg}=E_\mathrm{th}=E_\mathrm m$.
+In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty$, also in Table~\ref{tab:energies}.
+
+Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model,
+computed using finite-$k$ RSB approximations. 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.
+
+\paragraph{Funding information}
+JK-D and JK are supported by the Simons Foundation Grant No.~454943.
+
+\bibliography{frsb_kac-rice}
+
+\end{document}