The Effect of Boundary Conditions on Mixing of 2d Potts Models at Discontinuous Phase Transitions

We study the critical stochastic q-state Potts model on the square lattice. Unlike the expected behavior when the phase transition is continuous (q = 2, 3, 4)—a universal power-law for the mixing time independently of the boundary conditions— the mixing time at a discontinuous phase transition, tmix, is highly sensitive to those. It was recently shown by the authors that tmix ≥ exp(cn) on an n×n box with periodic boundary, yet under free or monochromatic boundary conditions tmix ≤ exp(n 1 2 ). In this work we classify this effect under boundary conditions interpolating between these two (torus vs. free/monochromatic) for Swendsen–Wang dynamics at large q. Specifically, we show that alternating boundary conditions, such as red-free-red-free, also induce tmix ≥ exp(cn), whereas red-periodic-red-periodic, as well as Dobrushin boundary conditions, such as red-red-free-free, induce sub-exponential mixing.