TY - JOUR

T1 - Glauber dynamics for Ising models on random regular graphs

T2 - cut-off and metastability

AU - Can, Van Hao

AU - van der Hofstad, Remco

AU - Kumagai, Takashi

N1 - Funding Information:
We thank Anton Bovier for stimulating discussions and fruitful comments. The work of V. H. Can is supported by fellowship no. 17F17319 of the Japan Society for the Promotion of Science, and by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03–2019.310. The work of RvdH is supported by the Netherlands Organisation for Scientific Research (NWO) through the Gravitation Networks grant 024.002.003. The work of TK is supported by the JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation.
Publisher Copyright:
© 2021, Alea (Rio de Janeiro). All rights reserved.

PY - 2021

Y1 - 2021

N2 - Consider random d-regular graphs, i.e., random graphs such that there are exactly d edges from each vertex for some d ≥ 3. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a d-regular graph chosen uniformly at random from the collection of all d-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random d-regular graph, both in the quenched as well as the annealed settings. Let ß be the inverse temperature, ßc be the critical temperature and B be the external magnetic field. Concerning the annealed measure, we show that for ß > ßc there exists Bc(ß) ϵ (0, ∞) such that the model is metastable (i.e., the mixing time is exponential in the graph size n) when ß > ßc and 0 ≤ B < Bc(ß), whereas it exhibits the cut-off phenomenon at c*n logn with a window of order n when ß < ßc or ß > ßc and B > Bc(ß). Interestingly, Bc(ß) coincides with the critical external field of the Ising model on the d-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists Bc(ß) with Bc(ß) ≤ Bc(ß) such that for ß > ßc, the mixing time is at least exponential along some subsequence (nk)k≥1 when 0 ≤ B < Bc(ß), whereas it is less than or equal to Cnlogn when B > Bc(ß). The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.

AB - Consider random d-regular graphs, i.e., random graphs such that there are exactly d edges from each vertex for some d ≥ 3. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a d-regular graph chosen uniformly at random from the collection of all d-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random d-regular graph, both in the quenched as well as the annealed settings. Let ß be the inverse temperature, ßc be the critical temperature and B be the external magnetic field. Concerning the annealed measure, we show that for ß > ßc there exists Bc(ß) ϵ (0, ∞) such that the model is metastable (i.e., the mixing time is exponential in the graph size n) when ß > ßc and 0 ≤ B < Bc(ß), whereas it exhibits the cut-off phenomenon at c*n logn with a window of order n when ß < ßc or ß > ßc and B > Bc(ß). Interestingly, Bc(ß) coincides with the critical external field of the Ising model on the d-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists Bc(ß) with Bc(ß) ≤ Bc(ß) such that for ß > ßc, the mixing time is at least exponential along some subsequence (nk)k≥1 when 0 ≤ B < Bc(ß), whereas it is less than or equal to Cnlogn when B > Bc(ß). The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.

KW - Cut-off

KW - Glauber dynamics

KW - Ising model

KW - Metastability

KW - Random regular graphs

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U2 - 10.30757/ALEA.v18-52

DO - 10.30757/ALEA.v18-52

M3 - Article

AN - SCOPUS:85111551310

SN - 1980-0436

VL - 18

SP - 1441

EP - 1482

JO - Alea

JF - Alea

IS - 1

ER -