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

RESUMO

We study lazy single-site Metropolis dynamics $P_{\beta}$ at inverse temperature $\beta \geq 0$ 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 $\kappa\le1/2$, the worst-case total-variation mixing time of $P_{\beta}$ is at least of order $d\exp\{c\beta(d-1)\}$ for $\beta\ge1$, with universal $c>0$. Averaging over global spin reversal reduces the mixing time to $\mathcal O(d^2(1+\beta))$ for both $GP_\beta G$ and $(P_\beta+G)/2$, 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.