From f3144b0bc033de918785b40e083f48c8451d4a7d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 27 Sep 2022 16:54:54 +0200 Subject: More work on the letter version. --- frsb_kac-rice.bib | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) (limited to 'frsb_kac-rice.bib') diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index 0374c8a..0412ab7 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -376,16 +376,18 @@ doi = {10.1103/physrevlett.124.078002} } -@article{ElAlaoui_2020_Algorithmic, +@unpublished{ElAlaoui_2020_Algorithmic, author = {El Alaoui, Ahmed and Montanari, Andrea}, title = {Algorithmic Thresholds in Mean Field Spin Glasses}, year = {2020}, month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, -- cgit v1.2.3-70-g09d2 From 6dbe5a229823612fa77fd54f15f505a82cf3c12e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 27 Sep 2022 17:02:14 +0200 Subject: Changed rest of arxiv papers to be compatible with revtex. --- frsb_kac-rice.bib | 18 ++++++++++++------ 1 file changed, 12 insertions(+), 6 deletions(-) (limited to 'frsb_kac-rice.bib') diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index 0412ab7..d8927bc 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -404,17 +404,19 @@ doi = {10.1214/21-aop1519} } -@article{ElAlaoui_2022_Sampling, +@unpublished{ElAlaoui_2022_Sampling, author = {El Alaoui, Ahmed and Montanari, Andrea and Sellke, Mark}, title = {Sampling from the {Sherrington}-{Kirkpatrick} {Gibbs}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 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, @@ -557,16 +559,18 @@ stochastic localization}, doi = {10.1103/physrevlett.120.225501} } -@article{Huang_2021_Tight, +@unpublished{Huang_2021_Tight, author = {Huang, Brice and Sellke, Mark}, title = {Tight {Lipschitz} Hardness for Optimizing Mean Field Spin Glasses}, year = {2021}, month =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, @@ -596,16 +600,18 @@ stochastic localization}, doi = {10.1103/physrevresearch.3.023064} } -@article{Kent-Dobias_2022_Analytic, +@unpublished{Kent-Dobias_2022_Analytic, author = {Kent-Dobias, Jaron and Kurchan, Jorge}, title = {Analytic continuation over complex landscapes}, year = {2022}, month = {4}, url = {http://arxiv.org/abs/2204.06072v1}, + archiveprefix = {arXiv}, date = {2022-04-12T20:24:54Z}, eprint = {2204.06072v1}, eprintclass = {cond-mat.stat-mech}, - eprinttype = {arxiv} + eprinttype = {arxiv}, + primaryclass = {cond-mat.stat-mech} } @article{Li_2021_Determining, -- cgit v1.2.3-70-g09d2 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(-) (limited to 'frsb_kac-rice.bib') 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 From 6c0cd5488bc630a0fcf6a14629fd5f91f2706483 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 7 Nov 2022 14:22:31 +0100 Subject: Finishing up the letter. --- frsb_kac-rice.bib | 19 +++++++------- frsb_kac-rice_letter.tex | 65 ++++++++++++++++++++++++++++++++---------------- 2 files changed, 54 insertions(+), 30 deletions(-) (limited to 'frsb_kac-rice.bib') diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index 3e8acb7..f8d4265 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -600,18 +600,19 @@ $T_g$: Experimental Evidence for the Gardner Transition in Structural Glasses?}, doi = {10.1103/physrevresearch.3.023064} } -@unpublished{Kent-Dobias_2022_Analytic, +@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}, - archiveprefix = {arXiv}, - date = {2022-04-12T20:24:54Z}, - eprint = {2204.06072v1}, - eprintclass = {cond-mat.stat-mech}, - eprinttype = {arxiv}, - primaryclass = {cond-mat.stat-mech} + 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, diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index 63b7811..4fb22e7 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -27,7 +27,7 @@ \begin{abstract} Complexity is a measure of the number of stationary points in complex - landscapes. We {\color{red} solve the long-standing problem of detremining the...} derive a general solution for the complexity of mean-field + 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 @@ -134,14 +134,16 @@ stationary points using a $\delta$-function weighted by a Jacobian \begin{equation} \begin{aligned} \mathcal N(E, \mu) - &=\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\ + &=\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 complexity is then +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}\frac1N\overline{\log\mathcal N(E, \mu}). + \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. @@ -195,30 +197,51 @@ 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 as a function of three matrices indexed by the replicas: one -which is a clear analogue of the usual overlap matrix of the equilibrium case, -and two which can be related to the response of stationary points to -perturbations of the potential. The general expression for the complexity as a -function of these matrices is also found in \cite{Folena_2020_Rethinking}. - -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. +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 two hierarchical matrices corresponding to responses are diagonal, leaving -only the overlap matrix with nontrivial off-diagonal entries. This +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. This line of `supersymmetric' solutions terminates at the -ground state, and describes the most numerous types of stable minima. +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 at the ground state energy, where the complexity vanishes, +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 -- cgit v1.2.3-70-g09d2