Information Percolation and cutoff for Potts Glauber Dynamics

Abstract

We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus ${\Lambda }_n=(\mathbb{Z}/n\mathbb{Z}{)}^d$ at sufficiently high temperature. More precisely, for every fixed $d\ge 2$ and $q\ge 3$, there exists ${\beta }_0={\beta }_0(d,q)>0$ such that, for $0<\beta <{\beta }_0$, the cutoff occurs in an optimal $O(1)$ window centered at \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |Λ_n|, \] where $\mathfrak{r}\in (0,1)$ is the exact exponential decay rate of the one-site color bias. Thus, the mixing time is determined up to an additive \(O(1)\). It is characterized by the time at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. On a transitive graph, information percolation naturally gives an upper bound on the mixing time of one half of the grand coupling time which is sharp for the Ising model. However, for many non-monotone systems including the Potts model, this misses additional averaging over the update histories giving an upper bound that is off by a constant multiplicate factor. We develop a new framework that treats the information percolation histories as random polymers and show how to average over them to give the effective bias from the initial condition. This accounts for all the additional mixing, recovering a sharp upper bound and cutoff. We believe this approach will be applicable in a much wider range of settings.