Three lectures on random proper colorings of $\mathbb{Z}^d$

Abstract

A proper q-coloring of a graph is an assignment of one of q colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper q-colorings of a large discrete cube in the integer lattice Z d . Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension d and the number of colors q. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.

Proper -colorings of are Bernoulli

“…For a fixed number of colors q, in high enough dimensions (d ≥ Cq 10 log 2 q suffices), the situation is similar to that of 3-colorings in that the measures obtained from fixed-color boundary conditions are not translation-invariant, but they are invariant to parity-preserving automorphisms [26]. On the other hand, in any given dimension, when the number of colors is large (q > 4d suffices), there is a unique Gibbs measure, the finite-volume measures (with any boundary conditions) converge to this measure, and this measure is translation-invariant and satisfies strong spatial mixing (this all follows from Dobrushin's uniqueness condition; see, for example, [27]). Consequently, this measure is Bernoulli, and in fact, also a finitary factor of an i.i.d.…”