From 780f33ccb345052b938551776c4965fc0615fc2d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 21 Oct 2022 16:40:26 +0200 Subject: More writing. --- frsb_kac-rice.bib | 70 +++++++++++++++++++++++++++++------------------- frsb_kac-rice_letter.tex | 69 ++++++++++++++++++++++++++++++----------------- 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. -- cgit v1.2.3-70-g09d2