Mixing times of Critical 2d Potts Models

We study dynamical aspects of the q-state Potts model on an n × n box at its critical βc(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 ≤ 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; for q = 4, it is at most 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 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 behaviors at all temperatures): the inverse gap of Glauber dynamics is at most exp(n), and for Swendsen–Wang it is exp(n).