The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions

We study Swendsen–Wang dynamics for the critical q-state Potts model on the square lattice. For q = 2, 3, 4, where the phase transition is continuous, the mixing time t_mix is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large q, where the phase transition is discontinuous, the authors recently showed that t_mix is highly sensitive to boundary conditions: t_mix ≥ exp(cn) on an n × n box with periodic boundary, yet under free or monochromatic boundary conditions, t_mix ≤ exp(n^o(1)). In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the q colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce t_mix ≥ exp(cn), yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.