Phase coexistence and torpid mixing in the 3-coloring model on ${\mathbb Z}^d$

Abstract

We show that for all sufficiently large d, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on Z d admits multiple maximal-entropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3-coloring is chosen uniformly from a box in Z d , conditioned on color 0 being given to all the vertices on the boundary of the box which are at an odd distance from a fixed vertex v in the box, then the probability that v gets color 0 is exponentially small in d.The proof proceeds through an analysis of a certain type of cutset separating v from the boundary of the box, and builds on techniques developed by Galvin and Kahn in their proof of phase transition in the hard-core model on Z d .Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus Z d n . We show that there is a constant ρ ≈ 0.22 such that for all even n ≥ 4 and d sufficiently large, if M is a Markov chain on the set of proper 3-colorings of Z d n that updates the color of at most ρn d vertices at each step and whose stationary distribution is uniform, then the mixing time of M (the time taken for M to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in n d−1 .

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

“…In this work, we prove the high-dimensional case of the Kotecký conjecture, showing that for sufficiently high d there exists a positive temperature below which the model exhibits six BSS phases. Our methods also improve the quantitative sublattice bias estimates obtained in [34] and [18] for the zero-temperature case. and Z τ Λ,β , the partition function of the model, is a normalizing constant.…”

“…Already in [29], Kotecký observed that the problem resists standard Peierls arguments, and suggested looking at correspondences with other models as a possible approach for tackling it. The existence of six BSS phases was recently verified in the zero-temperature case in high dimensions by Peled [34] and, independently, by Galvin, Kahn, Randall and Sorkin [18]. Both groups obtained their results through highly sophisticated contour methods.…”

“…Independently, Galvin and Randall [19] investigated Glauber dynamics on proper 3-colorings in high dimensions under periodic boundary conditions, showed torpid mixing of this Markov chain and obtained a coarse form of rigidity. Together with Kahn and Sorkin [18], they later refined their methods to obtain results parallel to those of [34] directly on proper 3-colorings, although with somewhat weaker bounds. Feldheim and Peled [13] were later able to use topological arguments to obtain a relation between HHFs on the torus Z d /mZ d and proper 3-colorings of the torus, and extend Peled's bounds to periodic boundary conditions.…”

“…[2] where a continuous transition was conjectured), Kotecký conjectured circa 1985 (implied in [29], also mentioned in [18]) that in high dimensions, possibly already in three dimensions, the 3-state AF Potts model indeed undergoes a phase transition, and that at sufficiently low temperature, a configuration typically follows one of six patterns. To understand the nature of these patterns, note first that the graph Z d is bipartite, and that each bipartition class forms a sublattice.…”

“…In the latter case, it is also desirable to understand the structure of a typical ordered configuration. Such a phase transition does not occur in every dimension (e.g., for d = 1), and it is interesting to determine in which dimensions (if any) it does.Following an earlier debate in the physical community (see e.g.[2] where a continuous transition was conjectured), Kotecký conjectured circa 1985 (implied in [29], also mentioned in [18]) that in high dimensions, possibly already in three dimensions, the 3-state AF Potts model indeed undergoes a phase transition, and that at sufficiently low temperature, a configuration typically follows one of six patterns. To understand the nature of these patterns, note first that the graph Z d is bipartite, and that each bipartition class forms a sublattice.…”