Local Convergence of Random Graph Colorings

Let G = G ( n , m ) be a random graph whose average degree d = 2 m / n is below the k -colorability threshold. If we sample a k -coloring σ of G uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called condensation threshold d k ,cond , the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007]. We prove this conjecture for k exceeding a certain constant k 0 . More generally, we investigate the joint distribution of the k -colorings that σ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem.