Glauber dynamics for Ising models on random regular graphs: cut-off and metastability

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, ß_c be the critical temperature and B be the external magnetic field. Concerning the annealed measure, we show that for ß > ß_c there exists B_c(ß) ϵ (0, ∞) such that the model is metastable (i.e., the mixing time is exponential in the graph size n) when ß > ß_c and 0 ≤ B < B_c(ß), whereas it exhibits the cut-off phenomenon at c*n logn with a window of order n when ß < ß_c or ß > ß_c and B > B_c(ß). Interestingly, B_c(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 B_c(ß) with B_c(ß) ≤ B_c(ß) such that for ß > ß_c, the mixing time is at least exponential along some subsequence (n_k)_k≥1 when 0 ≤ B < B_c(ß), whereas it is less than or equal to Cnlogn when B > B_c(ß). The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.