Dynamics of 2D Potts and Random-Cluster Models

Abstract

This thesis considers the dynamical phase portrait of the Potts and random-cluster (FK) models in two-dimensions, as the system size diverges.

The Potts model was introduced in the 1950's as a model of ferromagnetism where interacting particles take q ≥ 2 possible states (q = 2 being the Ising model). The random-cluster (FK) model is a dependent percolation model indexed by a real-valued q ∈ (0, ∞) (q = 1 being independent percolation); it is central to the analysis of the Potts models, as at integer q it encodes the correlations of the corresponding Potts model. In 2D, the models have a sharp transition between a high-temperature phase and a low-temperature phase, via rich behavior at a critical point: at criticality, when q ≤ 4, they are conjectured to have a conformally invariant scaling limit with random fractal interfaces (SLE's), while when q > 4 they are characterized by phase coexistence and Brownian interfaces.

We are interested in understanding how long canonical Markov chains like the Glauber dynamics for the Potts and FK models, as well as cluster dynamics such as Swendsen–Wang, take to approximate the equilibrium measure on, say, an n × n box. These dynamics are well-known to equilibrate rapidly at all high temperatures, but are believed to undergo critical slowdowns, depending on the order of the phase transition: this was only known for q = 2 where they mix in polynomial time (in n), and for sufficiently large q, where they take exponentially long to equilibrate.

We begin by showing that the following holds for the mixing times (as well as inverse spectral gaps) of Potts Glauber dynamics on the n × n torus, Z/nZ)². The Potts Glauber dynamics at criticality has a mixing time that is polynomial in n when q = 3, and n^{O(log n)} when q = 4; on the other hand, for every q > 4 the mixing time of the critical Potts dynamics grows exponentially in n.

We also fill in the rest of this phase diagram by proving that for every q, as soon as the temperature is sub-critical, the Potts Glauber dynamics becomes exponentially slow.

The FK Glauber dynamics on Z/nZ)² (as well as the related Swendsen–Wang dynamics) are known to be fast mixing O(log n) at all off-critical temperatures. We prove that at the critical point, their mixing time on Z/nZ)² is at most n^{O(log n)} for all q ∈ (1,4] whereas it becomes exponentially slow in n as soon as q > 4.

The analysis of the FK Glauber dynamics and Swendsen–-Wang dynamics is substantially complicated by their non-locality; long-range interactions can be encoded into the FK boundary conditions on an n × n box. We first show that in all off-critical regimes, the mixing time is stable—up to polynomial factors—with respect to choice of realizable FK boundary conditions. We then turn to the critical point, where we show that when q is large, the dynamics are sensitive to the choice of boundary condition. For free or monochromatic boundary conditions and large q, these dynamics at criticality are much faster than on the torus and mix in exp(n^o(1) time. We conclude by investigating this sensitivity at criticality, under boundary conditions that interpolate between free, wired, and periodic.