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

Consider random $d$-regular graphs, i.e., graphs such that there are exactly $d$ edges from each vertex for some $d\ge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well simple it, is a graph chosen uniformly at collection all graphs. In paper, we discuss mixing times Glauber dynamics Ising with an external magnetic field on in quenched annealed settings. Let $\beta$ be inverse temperature, $\beta_c$ critical temperature $B$ field. Concerning measure, show $\beta > \beta_c$ exists $\hat{B}_c(\beta)\in (0,\infty)$ metastable (i.e., time exponential size $n$) when $\beta> $0 \leq B $B>\hat{B}_c(\beta)$. Interestingly, $\hat{B}_c(\beta)$ coincides $d$-ary tree (namely, above unique Gibbs measure). $B_c(\beta)$ $B_c(\beta) \hat{B}_c(\beta)$ \beta_c$, least along subsequence $(n_k)_{k\geq 1}$ \hat{B}_c(\beta)$. The results also hold conditioned simplicity, unclear.