Shattering versus metastability in spin glasses. (English) Zbl 07782027
Summary: Our goal in this work is to better understand the relationship between replica symmetry breaking, shattering, and metastability. To this end, we study the static and dynamic behaviour of spherical pure \(p\)-spin glasses above the replica symmetry breaking temperature \(T_s\). In this regime, we find that there are at least two distinct temperatures related to non-trivial behaviour. First we prove that there is a regime of temperatures in which the spherical \(p\)-spin model exhibits a shattering phase. Our results holds in a regime above but near \(T_s\). We then find that metastable states exist up to an even higher temperature \(T_{BBM}\) as predicted by Barrat-Burioni-Mézard which is expected to be higher than the phase boundary for the shattering phase \(T_d <T_{BBM}\). We develop this work by first developing a Thouless-Anderson-Palmer decomposition which builds on the work of Subag. We then present a series of questions and conjectures regarding the sharp phase boundaries for shattering and slow mixing.
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.
© 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.
References:
| [1] | D.Achlioptas and A.Coja‐Oghlan, Algorithmic barriers from phase transitions. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS ’08). IEEE Computer Society, USA, 2008793-802. https://doi.org/10.1109/FOCS.2008.11 · doi:10.1109/FOCS.2008.11 |
| [2] | D.Achlioptas, A.Coja‐Oghlan, and F.Ricci‐Tersenghi, On the solution‐space geometry of random constraint satisfaction problems, Random Struct. Algorithms38 (2011), no. 3, 251-268. https://doi.org/10.1002/rsa.20323 · Zbl 1217.68156 · doi:10.1002/rsa.20323 |
| [3] | M.Aizenman, R.Sims, and S. L.Starr, Extended variational principle for the sherrington‐kirkpatrick spin‐glass model, Physical Review B68 (2003), no. 21, 214 403. |
| [4] | A. E.Alaoui, A.Montanari, and M.Sellke, Optimization of mean‐field spin glasses. The Annals of Probability49 (2021), no. 6, 2922-2960. · Zbl 07467487 |
| [5] | L.‐P.Arguin, A remark on the infinite‐volume gibbs measures of spin glasses, J. Math. Phys.49 (2008), no. 12, 125 204. http://scitation.aip.org/content/aip/journal/jmp/49/12/10.1063/1.2966281 · Zbl 1159.81300 |
| [6] | A.Auffinger and G.Ben Arous, Complexity of random smooth functions on the high‐dimensional sphere, Ann. Probab.41 (2013), no. 6, 4214-4247. https://doi.org/10.1214/13‐AOP862 · Zbl 1288.15045 · doi:10.1214/13‐AOP862 |
| [7] | A.Auffinger, G.Ben Arous, and J.Černý, Random matrices and complexity of spin glasses, Comm. Pure Appl. Math.66 (2013), no. 2, 165-201. https://doi.org/10.1002/cpa.21422 · Zbl 1269.82066 · doi:10.1002/cpa.21422 |
| [8] | A.Auffinger and W.‐K.Chen, On properties of Parisi measures, Probab. Theory Related Fields161 (2015), no. 3‐4, 817-850. https://doi.org/10.1007/s00440‐014‐0563‐y · Zbl 1322.60204 · doi:10.1007/s00440‐014‐0563‐y |
| [9] | A.Auffinger and J.Gold, The number of saddles of the sphericalp‐spin model, arXiv preprint arXiv:2007.09269 (2020). |
| [10] | A.Auffinger and A.Jagannath, Thouless‐Anderson‐Palmer equations for generic p‐spin glasses, Ann. Probab.47 (2019), no. 4, 2230-2256. https://doi.org/10.1214/18‐AOP1307 · Zbl 1426.82067 · doi:10.1214/18‐AOP1307 |
| [11] | M.Baity‐Jesi, A. Achard‐deLustrac, and G.Biroli, Activated dynamics: An intermediate model between the random energy model and the p‐spin model, Phys. Rev. E98 (2018), no. 1, 012 133. |
| [12] | A.Barrat, R.Burioni, and M.Mézard, Dynamics within metastable states in a mean‐field spin glass, J. Phys. A: Math. Gen.29 (1996), no. 5, L81. · Zbl 0943.82586 |
| [13] | R.Bauerschmidt and T.Bodineau, A very simple proof of the LSI for high temperature spin systems, J. Funct. Anal.276 (2019), no. 8, 2582-2588. https://doi.org/10.1016/j.jfa.2019.01.007 · Zbl 1408.81033 · doi:10.1016/j.jfa.2019.01.007 |
| [14] | D.Belius and N.Kistler, The TAP-plefka variational principle for the spherical SK model, Comm. Math. Phys.367 (2019), no. 3, 991-1017. · Zbl 1419.82069 |
| [15] | G.Ben Arous, A.Bovier, and J.Černý, Universality of the REM for dynamics of mean‐field spin glasses, Comm. Math. Phys.282 (2008), no. 3, 663-695. https://doi.org/10.1007/s00220‐008‐0565‐7 · Zbl 1208.82024 · doi:10.1007/s00220‐008‐0565‐7 |
| [16] | G.Ben Arous, A.Bovier, and V.Gayrard, Aging in the random energy model, Phys. Rev. Lett.88 (2002), no. 8, 087 201. |
| [17] | G.Ben Arous, A.Dembo, and A.Guionnet, Aging of spherical spin glasses, Probab. Theory Related Fields120 (2001), no. 1, 1-67. https://doi.org/10.1007/PL00008774 · doi:10.1007/PL00008774 |
| [18] | G.Ben Arous, A.Dembo, and A.Guionnet, Cugliandolo‐Kurchan equations for dynamics of spin‐glasses, Probab. Theory Related Fields136 (2006), no. 4, 619-660. https://doi.org/10.1007/s00440‐005‐0491‐y · Zbl 1109.82021 · doi:10.1007/s00440‐005‐0491‐y |
| [19] | G.Ben Arous, R.Gheissari, and A.Jagannath, Algorithmic thresholds for tensor PCA, Ann. Probab.48 (2020), no. 4, 2052-2087. https://doi.org/10.1214/19‐AOP1415 · Zbl 1444.62080 · doi:10.1214/19‐AOP1415 |
| [20] | G.Ben Arous, R.Gheissari, and A.Jagannath, Bounding flows for spherical spin glass dynamics, Comm. Math. Phys.373 (2020), no. 3, 1011-1048. https://doi.org/10.1007/s00220‐019‐03649‐4 · Zbl 1434.82086 · doi:10.1007/s00220‐019‐03649‐4 |
| [21] | G.Ben Arous and A.Jagannath, Spectral gap estimates in mean field spin glasses, Comm. Math. Phys.361 (2018), no. 1, 1-52. · Zbl 1397.82054 |
| [22] | G.Ben Arous, E.Subag, and O.Zeitouni, Geometry and temperature chaos in mixed spherical spin glasses at low temperature: the perturbative regime, Comm. Pure Appl. Math.73 (2020), no. 8, 1732-1828. · Zbl 1453.82089 |
| [23] | E.Bolthausen, An iterative construction of solutions of the TAP equations for the Sherrington‐Kirkpatrick model, Comm. Math. Phys.325 (2014), no. 1, 333-366. https://doi.org/10.1007/s00220‐013‐1862‐3 · Zbl 1288.82038 · doi:10.1007/s00220‐013‐1862‐3 |
| [24] | E.Bolthausen and A.‐S.Sznitman, On Ruelle’s probability cascades and an abstract cavity method, Comm. Math. Phys.197 (1998), no. 2, 247-276. · Zbl 0927.60071 |
| [25] | J.‐P.Bouchaud, L. F.Cugliandolo, J.Kurchan, and M.Mézard, Out of equilibrium dynamics in spin‐glasses and other glassy systems, Spin Glasses and Random Fields (1998), 161-223. https://doi.org/10.1142/3517 · doi:10.1142/3517 |
| [26] | A.Bovier and A.Faggionato, Spectral characterization of aging: the REM‐like trap model, Ann. Appl. Probab.15 (2005), no. 3, 1997-2037. https://doi.org/10.1214/105051605000000359 · Zbl 1086.60064 · doi:10.1214/105051605000000359 |
| [27] | T.Castellani and A.Cavagna, Spin‐glass theory for pedestrians, J. Stat. Mech. Theory Exp.2005 (2005), no. 05, P05 012. http://stacks.iop.org/1742‐5468/2005/i=05/a=P05012 · Zbl 1456.82490 |
| [28] | J.Černý and T.Wassmer, Aging of the Metropolis dynamics on the random energy model, Probab. Theory Related Fields167 (2017), no. 1‐2, 253-303. https://doi.org/10.1007/s00440‐015‐0681‐1 · Zbl 1359.82018 · doi:10.1007/s00440‐015‐0681‐1 |
| [29] | W.‐K.Chen, The Aizenman‐Sims‐Starr scheme and Parisi formula for mixed p‐spin spherical models, Electron. J. Probab.18 (2013), no. 94, 14. https://doi.org/10.1214/EJP.v18‐2580 · Zbl 1288.60127 · doi:10.1214/EJP.v18‐2580 |
| [30] | W.‐K.Chen, D.Panchenko, and E.Subag, The generalized tap free energy. Comm. Pure Appl. Math.76 (2023), no. 7, 1329-1415. · Zbl 07748347 |
| [31] | A.Coja‐Oghlan and W.Perkins, Bethe states of random factor graphs, Comm. Math. Phys.366 (2019), no. 1, 173-201. · Zbl 1412.82016 |
| [32] | A.Crisanti and H. J.Sommers, The spherical p‐spin interaction spin glass model: the statics, Zeitschrift für Physik B Condensed Matter87 (1992), no. 3, 341-354. https://doi.org/10.1007/BF01309287 · doi:10.1007/BF01309287 |
| [33] | L.Cugliandolo and J.Kurchan, Weak ergodicity breaking in mean‐field spin‐glass models, Philos. Mag. B71 (1995), no. 4, 501-514. |
| [34] | L. F.Cugliandolo and J.Kurchan, Analytical solution of the off‐equilibrium dynamics of a long‐range spin‐glass model, Phys. Rev. Lett.71 (1993), 173-176. https://doi.org/10.1103/PhysRevLett.71.173 · doi:10.1103/PhysRevLett.71.173 |
| [35] | A.Dembo, A.Guionnet, and C.Mazza, Limiting dynamics for spherical models of spin glasses at high temperature, J. Stat. Phys.128 (2007), no. 4, 847-881. https://doi.org/10.1007/s10955‐006‐9239‐z · Zbl 1130.82032 · doi:10.1007/s10955‐006‐9239‐z |
| [36] | A.Dembo, A.Montanari, and N.Sun, Factor models on locally tree‐like graphs, Ann. Probab.41 (2013), no. 6, 4162-4213. · Zbl 1280.05119 |
| [37] | A.Dembo and E.Subag, Dynamics for spherical spin glasses: disorder dependent initial conditions, J. Stat. Phys.181 (2020), no. 2, 465-514. https://doi.org/10.1007/s10955‐020‐02587‐z · Zbl 1456.82846 · doi:10.1007/s10955‐020‐02587‐z |
| [38] | J.Ding, A.Sly, and N.Sun, Proof of the satisfiability conjecture for large k. Annals of Mathematics1962022, no. 1, pp. 1-388. · Zbl 1509.60169 |
| [39] | J.Ding, A.Sly, and N.Sun, Maximum independent sets on random regular graphs, Acta Math.217 (2016), no. 2, 263-340. https://doi.org/10.1007/s11511‐017‐0145‐9 · Zbl 1371.05261 · doi:10.1007/s11511‐017‐0145‐9 |
| [40] | R.Eldan, F.Koehler, and O.Zeitouni, A spectral condition for spectral gap: Fast mixing in high‐temperature ising models. Probab. Theor. Relat. Fields. 182 (2022), no. 3‐4, pp. 1035-1051. · Zbl 1515.82046 |
| [41] | S.Franz and G.Parisi, Recipes for metastable states in spin glasses, Journal de Physique I5 (1995), no. 11, 1401-1415. |
| [42] | D.Gamarnik and M.Sudan, Limits of local algorithms over sparse random graphs. The Annals of Probability, Ann. Probab.452017 July no. 4, 2353-2376. · Zbl 1371.05265 |
| [43] | V.Gayrard, Aging in metropolis dynamics of the rem: A proof, Probab. Theory Relat. Fields174 (2019), no. 1, 501-551. · Zbl 1416.82033 |
| [44] | R.Gheissari and A.Jagannath, On the spectral gap of spherical spin glass dynamics, Ann. Inst. Henri Poincaré Probab. Stat.55 (2019), no. 2, 756-776. https://doi.org/10.1214/18‐aihp897 · Zbl 1435.82025 · doi:10.1214/18‐aihp897 |
| [45] | F.Guerra, Broken replica symmetry bounds in the mean field spin glass model, Comm. Math. Phys.233 (2003), no. 1, 1-12. https://doi.org/10.1007/s00220‐002‐0773‐5 · Zbl 1013.82023 · doi:10.1007/s00220‐002‐0773‐5 |
| [46] | R.Holley and D.Stroock, Logarithmic sobolev inequalities and stochastic ising models, J. Stat. Phys.46 (1987), no. 5‐6, 1159-1194. · Zbl 0682.60109 |
| [47] | A.Jagannath, Approximate ultrametricity for random measures and applications to spin glasses, Comm. Pure Appl. Math70 (2017), no. 4, 611-664. · Zbl 1361.60035 |
| [48] | A.Jagannath, Dynamics of mean field spin glasses on short and long timescales, J. Math. Phys.60 (2019), no. 8, 083 305. · Zbl 1426.82068 |
| [49] | A.Jagannath and I.Tobasco, Bounds on the complexity of replica symmetry breaking for spherical spin glasses, Proc. Amer. Math. Soc.146 (2018), no. 7, 3127-3142. · Zbl 1390.60353 |
| [50] | T. R.Kirkpatrick and D.Thirumalai, p‐spin‐interaction spin‐glass models: Connections with the structural glass problem, Phys. Rev. B36 (1987), 5388-5397. https://link.aps.org/doi/10.1103/PhysRevB.36.5388 |
| [51] | J.Ko, Free energy of multiple systems of spherical spin glasses with constrained overlaps, Electron. J. Probab.25 (2020), 1-34. · Zbl 1437.60075 |
| [52] | F.Krzakała, A.Montanari, F.Ricci‐Tersenghi, G.Semerjian, and L.Zdeborová, Gibbs states and the set of solutions of random constraint satisfaction problems, Proc. Natl. Acad. Sci.104 (2007), no. 25, 10 318-10 323. https://www.pnas.org/content/104/25/10318.full.pdf. https://www.pnas.org/content/104/25/10318 · Zbl 1190.68031 |
| [53] | J.Kurchan, G.Parisi, and M. A.Virasoro, Barriers and metastable states as saddle points in the replica approach, Journal de Physique I3 (1993), no. 8, 1819-1838. |
| [54] | P.Mathieu, Convergence to equilibrium for spin glasses, Comm. Math. Phys.215 (2000), no. 1, 57-68. https://doi.org/10.1007/s002200000292 · Zbl 1018.82019 · doi:10.1007/s002200000292 |
| [55] | P.Mathieu and J.‐C.Mourrat, Aging of asymmetric dynamics on the random energy model, Probab. Theory Related Fields161 (2015), no. 1‐2, 351-427. · Zbl 1316.82026 |
| [56] | M.Mézard and A.Montanari, Information, physics, and computation, Oxford Graduate Texts, Oxford University Press, Oxford, 2009. https://doi.org/10.1093/acprof:oso/9780198570837.001.0001 · Zbl 1163.94001 · doi:10.1093/acprof:oso/9780198570837.001.0001 |
| [57] | M.Mézard, G.Parisi, and M. A.Virasoro, Spin glass theory and beyond, vol. 9, World scientific, Singapore, 1987. · Zbl 0992.82500 |
| [58] | M.Mézard, G.Parisi, and R.Zecchina, Analytic and algorithmic solution of random satisfiability problems, Science297 (2002), no. 5582, 812-815. http://www.sciencemag.org/content/297/5582/812.full.pdf. http://www.sciencemag.org/content/297/5582/812.abstract |
| [59] | A.Montanari, Optimization of the sherrington-kirkpatrick hamiltonian, SIAM J. Comput. (2021), no. 0, FOCS19-1. https://doi.org/10.1137/20M132016X · Zbl 1528.82043 · doi:10.1137/20M132016X |
| [60] | D.Panchenko, The Parisi ultrametricity conjecture, Ann. of Math. (2)177 (2013), no. 1, 383-393. https://doi.org/10.4007/annals.2013.177.1.8 · Zbl 1270.60060 · doi:10.4007/annals.2013.177.1.8 |
| [61] | D.Panchenko, The Sherrington‐Kirkpatrick model, SpringerNew York, NY, 2013. · Zbl 1266.82005 |
| [62] | D.Panchenko, The Parisi formula for mixed p‐spin models, Ann. Probab.42 (2014), no. 3, 946-958. https://doi.org/10.1214/12‐AOP800 · Zbl 1292.82020 · doi:10.1214/12‐AOP800 |
| [63] | M.Rahman and B.Virag, Local algorithms for independent sets are half‐optimal, Ann. Probab.45 (2017), no. 3, 1543-1577. · Zbl 1377.60049 |
| [64] | A.Sly and Y.Zhang, Reconstruction of colourings without freezing, arXiv preprint arXiv:1610.02770 (2016). |
| [65] | E.Subag, The complexity of spherical p‐spin models—a second moment approach, Ann. Probab.45 (2017), no. 5, 3385-3450. · Zbl 1417.60029 |
| [66] | E.Subag, The geometry of the gibbs measure of pure spherical spin glasses, Invent. Math.210 (2017), 135-209. https://doi.org/10.1007/s00222‐017‐0726‐4 · Zbl 1379.60043 · doi:10.1007/s00222‐017‐0726‐4 |
| [67] | E.Subag, Free energy landscapes in spherical spin glasses, arXiv preprint arXiv:1804.10576 (2018). |
| [68] | E.Subag, The free energy of spherical pure p‐spin models: computation from the TAP approach. Probability Theory and Related Fields, (2023), pp. 1-20. |
| [69] | M.Talagrand, On guerra’s broken replica‐symmetry bound, Comptes Rendus Mathematique337 (2003), no. 7, 477-480. http://www.sciencedirect.com/science/article/pii/S1631073X03003984 · Zbl 1138.82326 |
| [70] | M.Talagrand, Free energy of the spherical mean field model, Probab. Theory Related Fields134 (2006), no. 3, 339-382. https://doi.org/10.1007/s00440‐005‐0433‐8 · Zbl 1130.82019 · doi:10.1007/s00440‐005‐0433‐8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
