TY - JOUR

T1 - Mixing Times of Critical Two-Dimensional Potts Models

AU - Gheissari, Reza

AU - Lubetzky, Eyal

N1 - Funding Information:
Acknowledgments. The authors thank C. Hongler, F. Martinelli, Y. Peres, and S. Shlosman for useful discussions, as well as A. Sly, whose joint paper with E.L. on the critical Ising model was the starting point of this project. We thank an anonymous referee for useful comments. The research of R.G. was supported in part by National Science Foundation Grant DMS-1507019. The research of E.L. was supported in part by National Science Foundation Grant DMS-1513403.

PY - 2018/5

Y1 - 2018/5

N2 - 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 nO(1); for q = 4, it is at most nO(log 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 or monochromatic 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 behavior at all temperatures): the inverse gap of Swendsen-Wang dynamics is exp(no(1)).

AB - 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 nO(1); for q = 4, it is at most nO(log 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 or monochromatic 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 behavior at all temperatures): the inverse gap of Swendsen-Wang dynamics is exp(no(1)).

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U2 - 10.1002/cpa.21718

DO - 10.1002/cpa.21718

M3 - Article

AN - SCOPUS:85033221407

VL - 71

SP - 994

EP - 1046

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 5

ER -