From torpid to rapid mixing: group averaging for a weakly interacting Ising star

We study lazy single-site Metropolis dynamics $P_{\beta }$ at inverse temperature $\beta \ge 0$ on $\{-1,+1{\}}^d$ for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most $\kappa \le 1\mathrm{/}2$, the worst-case total-variation mixing time of $P_{\beta }$ is at least of order $d\exp \{c\beta (d-1)\}$ for $\beta \ge 1$, with universal $c>0$. Averaging over global spin reversal reduces the mixing time to $\mathcal{O}(d^2(1+\beta ))$ for both $G{P}_{\beta }G$ and $(P_{\beta }+G)\mathrm{/}2$, where $G$ is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits $\{-x,x\}$. Partition-function interpolation gives the lower bound for $P_{\beta }$. The upper bounds follow from Wu’s Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.