Phase coexistence for q-colorings of high-dimensional lattices

Let ${\mathbb Z}^d$ be the nearest-neighbor graph on the $d$-dimensional integer lattice. For an integer $q\geq 3$, consider the $q$-coloring model, whose Gibbs measures are infinite-volume measures on proper $q$-colorings, and call a Gibbs measure of maximal entropy if it has maximal entropy among Gibbs measures. Phase-coexistence conjecture for q-colorings. For all $q>3$ and all sufficiently large $d=d(q)$, there is more than one Gibbs measure of maximal entropy for the $q$-coloring model on ${\mathbb Z}^d$. This extends the established $q=3$ phase-coexistence result to every fixed number of colors greater than three; the paper states that its methods do not currently extend beyond $q=3$.