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

Reza Gheissari, Eyal Lubetzky

Research output: Contribution to journalArticlepeer-review


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 tmix 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 tmix is highly sensitive to boundary conditions: tmix ≥ exp(cn) on an n × n box with periodic boundary, yet under free or monochromatic boundary conditions, tmix ≤ exp(no(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 tmix ≥ exp(cn), yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.

Original languageEnglish (US)
Article number57
JournalElectronic Journal of Probability
StatePublished - 2018


  • Boundary conditions
  • Mixing times
  • Potts model
  • Surface tension
  • Swendsen
  • Wang

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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