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We show that for all sufficiently large d, the uniform proper 3-coloringmodel (in physics called the 3-state antiferromagnetic Potts model at zerotemperature) on Z^d admits multiple maximal-entropy Gibbs measures. This is aconsequence of the following combinatorial result: if a proper 3-coloring ischosen uniformly from a box in Z^d, conditioned on color 0 being given to allthe vertices on the boundary of the box which are at an odd distance from afixed vertex v in the box, then the probability that v gets color 0 isexponentially small in d.  The proof proceeds through an analysis of a certain type of cutset separatingv from the boundary of the box, and builds on techniques developed by Galvinand 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 forsampling proper 3-colorings of the discrete torus Z^d_n. We show that there isa constant \rho \approx 0.22 such that for all even n \geq 4 and d sufficientlylarge, if M is a Markov chain on the set of proper 3-colorings of Z^d_n thatupdates the color of at most \rho n^d vertices at each step and whosestationary distribution is uniform, then the mixing time of M (the time takenfor M to reach a distribution that is close to uniform, starting from anarbitrary coloring) is essentially exponential in n^{d-1}.

Phase coexistence and torpid mixing in the 3-coloring model on Z^d