2019/12/17 by Can, Van Hao, van der Hofstad, Remco, Kumagai, Takashi · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1912.07798
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 β be the inverse temperature, βc be the critical temperature and B be the external magnetic field. Concerning the annealed measure, we show that for β> βc there exists Bc(β)∈ (0,∞) such that the model is metastable (i.e., the mixing time is exponential in the graph size n) when β> βc and 0 ≤ B < Bc(β), whereas it exhibits the cut-off phenomenon at c_⋆ n log n with a window of order n when β< βc or β> βc and B>Bc(β). Interestingly, Bc(β) 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(β) with Bc(β) ≤ Bc(β) such that for β> βc, the mixing time is at least exponential along some subsequence (nk)k≥ 1 when 0 ≤ B < Bc(β), whereas it is less than or equal to Cnlog n when B>Bc(β). The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.