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Consider random $d$-regular graphs, i.e., random graphs such that there areexactly $d$ edges from each vertex for some $d\ge 3$. We study both theconfiguration model version of this graph, which has occasional multi-edges andself-loops, as well as the simple version of it, which is a $d$-regular graphchosen uniformly at random from the collection of all $d$-regular graphs.  In this paper, we discuss mixing times of Glauber dynamics for the Isingmodel with an external magnetic field on a random $d$-regular graph, both inthe quenched as well as the annealed settings. Let $\beta$ be the inversetemperature, $\beta_c$ be the critical temperature and $B$ be the externalmagnetic 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 ismetastable (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 thecut-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 Isingmodel on the $d$-ary tree (namely, above which the model has a unique Gibbsmeasure). 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$(n_k)_{k\geq 1}$ when $0 \leq B < B_c(\beta)$, whereas it is less than orequal to $Cn\log n$ when $B>\hat{B}_c(\beta)$. The quenched results also holdfor the model conditioned on simplicity, for the annealed results this isunclear.

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