Local convergence of random graph colorings

TL;DR

This paper proves that for random graphs with average degree below a certain threshold, the colors assigned in a random proper coloring become asymptotically independent at large distances, confirming a physics-based prediction.

Contribution

It rigorously establishes the local independence of colorings in random graphs below the condensation threshold for sufficiently large number of colors.

Findings

Colors are asymptotically independent at large distances below the condensation threshold.

The joint distribution of local colorings converges to a product measure.

Implications are discussed for the reconstruction problem.

Abstract

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 $\sigma$ 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 {\em condensation threshold} $d_c(k)$, 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 $\sigma$ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem.