The transitive-graph cut-off conjecture for Glauber dynamics

Let $(G_n)$ be a sequence of transitive graphs, and let $t_{{\rm mix}}(n)$ denote the mixing time of Glauber dynamics on $G_n$. A sequence of chains has cut-off if there are times $t_n$ and windows $w_n=o(t_n)$ such that the worst-case distance to stationarity tends to $1$ at times $t_n-\gamma w_n$ and to $0$ at times $t_n+\gamma w_n$ as $n$ and then $\gamma$ tend to infinity. The transitive-graph cut-off conjecture. If the Glauber dynamics on $G_n$ has $t_{{\rm mix}}(n)=O(n\log n)$, then it exhibits a cut-off. This conjecture proposes that order-$n\log n$ mixing on transitive graphs is generically accompanied by a sharp transition to stationarity; the paper presents it as motivation, without giving evidence of a resolution.