Local Convergence of Random Graph Colorings

Let G = G(n, m) be a random graph whose average degree d = 2m/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 dk,cond, the colors assigned to far away vertices are asymptotically independent [Krzakala et al: PNAS 2007]. We prove this conjecture for k exceeding a certain constant k0. More generally, we determine the joint distribution of the k-colorings that σ induces locally on the bounded-depth neighborhoods of a fixed number of vertices. 1998 ACM Subject Classification G.2 Discrete Mathematics, G.3 Probability and Statistics, G.2.2. Graph Theory, F.2.2, Computations on discrete structures Keywords and phrases Random graph, Galton-Watson tree, phase transitions, graph coloring, Gibbs distribution, convergence Digital Object...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari
• Keywords: Algorithms, APPROX 2015