Mixing times of critical 2D Potts models

Abstract

We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box at its critical $\beta_c(q)$. Heat-bath Glauber dynamics and cluster dynamics such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all expected to undergo "critical slowdowns" in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is $O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and exponential in $n$ for $q>4$ in accordance with the predicted discontinuous phase transition. This was confirmed for $q=2$ (the Ising model) by the second author and Sly, and for sufficiently large $q$ by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for $q=3$, the inverse gap of Glauber dynamics is $n^{O(1)}$; for $q=4$, it is at most $n^{O(\log n)}$; and for every $q>4$ in the phase-coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen--Wang dynamics are exponential in $n$. For free or monochromatic boundary conditions and large $q$, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen--Wang dynamics is $\exp(n^{o(1)})$.