Cutoff with an $O(1)$ window for Potts Glauber Dynamics on lattice at High Temperature

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. 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 Glauber dynamics of the Potts model on ${\Lambda }_n$ exhibits cutoff with optimal $O(1)$ window around \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |Λ_n|, \] where $\mathfrak{r}\in (0,1)$ is the exponential decay rate of the one-site magnetization. In particular, this determines the mixing time up to an additive $O(1)$. It is characterized by the point 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. In contrast with the Ising model, a direct implementation of information percolation does not yield matching upper and lower bounds for the Potts dynamics when $q\ge 3$. We overcome this by developing an information-percolation framework for signed influences and combining it with Fourier bounds on signed convolution powers and geometric control of history diagrams.