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

We study lazy single-site Metropolis dynamics $P_β$ at inverse temperature $β\geq 0$ 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 $κ\le1/2$, the worst-case total-variation mixing time of $