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 $dge 3$. 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 $beta$ be the inverse temperature, $beta_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $beta > beta_c$ there exists $hat{B}_c(beta)in (0,infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $beta> beta_c$ and $0 leq B < hat{B}_c(beta)$, whereas it exhibits the cut-off phenomenon at $c_star n log n$ with a window of order $n$ when $beta < beta_c$ or $beta > beta_c$ and $B>hat{B}_c(beta)$. Interestingly, $hat{B}_c(beta)$ 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 $B_c(beta)$ with $B_c(beta) leq hat{B}_c(beta)$ such that for $beta> beta_c$, the mixing time is at least exponential along some subsequence $(n_k)_{kgeq 1}$ when $0 leq B < B_c(beta)$, whereas it is less than or equal to $Cnlog n$ when $B>hat{B}_c(beta)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.