Mixing Times of Critical Two‐Dimensional Potts Models

Abstract

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)). © 2017 Wiley Periodicals, Inc.

Tunneling behavior of Ising and Potts models in the low-temperature regime

“…Another branch of research focuses more on the dynamical properties of the Potts model, investigating in particular mixing times for various types of dynamics, the most studied ones being Glauber [12,13,29,39,40,41,43,44,47,48], Swendsen-Wang dynamics [13,14,26,27,38,43,45,46,63]. The focus of this part of literature is to describe at a given temperature how the mixing time grows as a function the graph size n = |V | and the number of colors q.…”

Spatial Mixing and Non-local Markov chains

“…Hence, we get SW = T RT * . This useful decomposition of the SW dynamics was discovered rst in [47,48,46] and has already been used in other comparison arguments involving the SW dynamics (see, e.g., [4,20]). The following | Ω J | × | Ω J | matrices allow us to obtain similar decompositions for I and I D .…”

“…In Z 2 , Ullrich's result [47] implies that the relaxation time of the SW dynamics is O(n) for β < β c (q), O(n 2 log n) for β > β c (q), and at most polynomial in n for β = β c (q) and q = 2. Recently, Gheissari and Lubetzky [20,21], using the results of Duminil-Copin et al [11,10] settling the continuity of phase transition, analyzed the dynamics at the critical point β c (q) for all q. They showed that the mixing time is at most polynomial in n for q = 3, at most quasi-polynomial for q = 4, and exp(Ω(n)) for q > 4.…”

Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

“…These bounds follow as a consequence of the exponential decay of correlations of the model in the high-temperature regime, which holds even near the boundary for arbitrary Ising/Potts boundary conditions; this property is known as strong spatial mixing. In the low-temperature region β > β c the mixing time is exponential in n for free and periodic (toroidal) boundaries [35,6,17]. The more general problem of understanding the mixing time of the Glauber dynamics for other boundary conditions at low temperatures is a long-standing open problem, e.g., see [25,29].The FK-dynamics is quite powerful since the self-duality of the model on Z 2 implies that it is rapidly mixing in the low-temperature regime where the Ising/Potts Glauber dynamics is torpidly mixing.…”

“…For the FK-dynamics on Λ n , [4] showed that the mixing time is Θ(n 2 log n) for all q > 1 whenever p = p c (q); see also [15] for recent results concerning the cutoff phenomenon in the FK-dynamics. (At the critical p = p c (q) the FK-dynamics may exhibit torpid mixing depending on the "order" of the phase transition [17,18].) Since the proof in [4] (as well as in [15]) used a strong spatial mixing property for the random-cluster model, the Θ(n 2 log n) upper bound only holds under boundary conditions that are free (no boundary condition), wired (all boundary vertices are connected to one another) or periodic (the torus).…”