The effect of boundary conditions on mixing of 2D Potts models at discontinuous phase transitions
R Gheissari, E Lubetzky - 2018 - projecteuclid.org
2018•projecteuclid.org
Abstract 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 …
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 …
Abstract
We study Swendsen–Wang dynamics for the critical -state Potts model on the square lattice. For , where the phase transition is continuous, the mixing time is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large , where the phase transition is discontinuous, the authors recently showed that is highly sensitive to boundary conditions: on an box with periodic boundary, yet under free or monochromatic boundary conditions, .
In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce , yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.
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