×

Continuous spin models on annealed generalized random graphs. (English) Zbl 1376.82026

Summary: We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution \(P\), obtained by annealing over the random graph distribution.
First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.
We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when \(P\) becomes too heavy-tailed, in which case they move continuously with the tail-exponent of \(P\). For large classes of models they are the same as for the Ising model treated in [S. Dommers et al., Commun. Math. Phys. 348, No. 1, 221–263 (2016; Zbl 1359.82020)]. On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

MSC:

82B26 Phase transitions (general) in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
82B27 Critical phenomena in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Citations:

Zbl 1359.82020
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Biskup, M.; Kotecký, Roman, Phase coexistence of gradient Gibbs states, Probab. Theory Related Fields, 139, 1, 1-39 (2007) · Zbl 1120.82003
[2] Bogachev, V. I., Measure Theory. Vol. II (2007), Springer-Verlag · Zbl 1120.28001
[3] Bollobás, B.; Janson, S.; Riordan, O., The phase transition in inhomogeneous random graphs, Random Struct. Algorithms, 31, 1, 3-122 (2007) · Zbl 1123.05083
[4] Coja-Oghlan, A.; Jaafari, N., On the Potts antiferromagnet on random graphs, Electron. J. Combin., 23, 4, #P4.3 (2016) · Zbl 1351.05203
[5] Contucci, P.; Dommers, S.; Giardinà, C.; Starr, S., Antiferromagnetic Potts model on the Erdős-Rényi random graph, Comm. Math. Phys., 323, 2, 517-554 (2013) · Zbl 1285.82007
[6] Dembo, A.; Montanari, A., Ising models on locally tree-like graphs, Ann. Appl. Probab., 20, 2, 565-592 (2010) · Zbl 1191.82025
[7] Dembo, A.; Montanari, A.; Sly, A.; Sun, N., The replica symmetric solution for Potts models on \(d\)-regular graphs, Comm. Math. Phys., 327, 2, 551-575 (2014) · Zbl 1288.82009
[8] Dembo, A.; Montanari, A.; Sun, N., Factor models on locally tree-like graphs, Ann. Probab., 41, 6, 4162-4213 (2013) · Zbl 1280.05119
[9] Dembo, A.; Zeitoni, O., Large Deviations Techniques and Applications (2009), Springer-Verlag
[10] Dommers, S., Metastability of the Ising model on random regular graphs at zero temperature, Probab. Theory Related Fields, 167, 1, 305-324 (2017) · Zbl 1358.60102
[11] Dommers, S.; Giardinà, C.; Giberti, C.; van der Hofstad, R.; Prioriello, M. L., Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs, Comm. Math. Phys., 348, 1, 221-263 (2016) · Zbl 1359.82020
[12] Dommers, S.; Giardinà, C.; van der Hofstad, R., Ising models on power-law random graphs, J. Stat. Phys., 141, 4, 638-660 (2010) · Zbl 1214.82116
[13] Dommers, S.; Giardinà, C.; van der Hofstad, R., Ising critical exponents on random trees and graphs, Comm. Math. Phys., 328, 1, 355-395 (2014) · Zbl 1292.82004
[14] Dommers, S.; den Hollander, F.; Jovanovski, O.; Nardi, F. R., Metastability for Glauber dynamics on random graphs, Ann. Appl. Probab. (2016), (in press) Preprint arXiv:1602.08900 · Zbl 1377.60020
[15] Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F.F., Critical phenomena in complex networks, Rev. Modern Phys., 80, 4, 1275-1335 (2008)
[16] van Enter, A. C.D.; Fernández, R.; den Hollander, F.; Redig, F., A large-deviation view on dynamical Gibbs-non-Gibbs transitions, Moskow Math. J., 10, 4, 687-711 (2010) · Zbl 1221.82046
[17] van Enter, A. C.D.; Fernández, R.; Sokal, A. D., Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory, J. Stat. Phys., 72, 5, 879-1167 (1993) · Zbl 1101.82314
[18] van Enter, A. C.D.; Külske, C.; Maes, C., Comment on “Critical behavior of the randomly spin diluted 2D Ising model: A grand ensemble approach”, Phys. Rev. Lett., 84, 26, 6134 (2000)
[19] Fraiman, D.; Balenzuela, P.; Foss, J.; Chialvo, D. R., Ising-like dynamics in large-scale functional brain networks, Phys. Rev. E, 79, Article 061922 pp. (2009)
[20] Giardinà, C.; Giberti, C.; van der Hofstad, R.; Prioriello, M. L., Quenched central limit theorems for the Ising model on random graphs, J. Stat. Phys., 160, 6, 1623-1657 (2015) · Zbl 1327.82013
[21] Giardinà, C.; Giberti, C.; van der Hofstad, R.; Prioriello, M. L., Annealed central limit theorems for the Ising model on random graphs, ALEA Lat. Am. J. Probab. Math. Stat., 13, 1, 121-161 (2016) · Zbl 1331.05193
[22] Grün, B.; Hornik, K., Amos-type bounds for modified Bessel function ratios, J. Math. Anal. Appl., 408, 1, 91-101 (2013) · Zbl 1309.33011
[23] Guerra, F.; Toninelli, F. L., The high-temperature region of the Viana-Bray diluted spin glass model, J. Stat. Phys., 115, 1-2, 531-555 (2004) · Zbl 1157.82377
[24] den Hollander, F.; Redig, F.; van Zuijlen, W., Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions, Stochastic Process. Appl., 125, 1, 371-400 (2015) · Zbl 1320.60160
[25] Jahnel, B.; Külske, C., Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts model with a Kac-type interaction, Bernoulli (2016), (in press) Preprint arXiv:1502.04238 · Zbl 1373.82047
[26] Jahnel, B.; Külske, C.; Botirov, G. I., Phase transition and critical values of a nearest-neighbor system with uncountable local state space on Cayley trees, Math. Phys. Anal. Geom., 17, 9158 (2014) · Zbl 1310.82015
[27] Kallenberg, O., Foundations of Modern Probability (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0996.60001
[28] Kühn, R., Critical behavior of the randomly spin diluted 2D Ising model: A grand ensemble approach, Phys. Rev. Lett., 73, 16, 2268-2271 (1994)
[29] Külske, C., Analogues of non-Gibbsianness in joint measures of disordered mean field models, J. Stat. Phys., 112, 5, 1079-1108 (2003) · Zbl 1032.82037
[30] Külske, C.; Le Ny, A., Spin-flip dynamics of the Curie-Weiss model: Loss of Gibbsianness with possibly broken symmetry, Comm. Math. Phys., 271, 2, 431-454 (2007) · Zbl 1138.82012
[31] Külske, C.; Le Ny, A.; Redig, F., Relative entropy and variational properties of generalized Gibbsian measures, Ann. Probab., 32, 2, 1691-1726 (2004) · Zbl 1052.60042
[32] Külske, C.; Redig, F., Loss without recovery of Gibbsianness during diffusion of continuous spins, Probab. Theory Related Fields, 135, 3, 428-456 (2006) · Zbl 1095.60027
[33] Montanari, A.; Saberi, A., The spread of innovations in social networks, Proc. Natl. Acad. Sci., 107, 47, 20196-20201 (2010)
[34] Morita, T., Statistical mechanics of quenched solid solutions with application to magnetically dilute alloys, J. Math. Phys., 5, 10, 1401-1405 (1964)
[35] Mossel, E.; Sly, A., Exact thresholds for Ising-Gibbs samplers on general graphs, Ann. Probab., 41, 1, 294-328 (2013) · Zbl 1270.60113
[36] Napolitano, G. M.; Turova, T. S., The Ising model on the random planar causal triangulation: Bounds on the critical line and magnetization properties, J. Stat. Phys., 162, 3, 739-760 (2016) · Zbl 1334.82043
[37] Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 2, 167-256 (2003) · Zbl 1029.68010
[38] Roelly, S.; Ruszel, W. M., Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction, Markov Process. Related Fields, 20, 4, 653-674 (2014) · Zbl 1317.60101
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.