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.: Proc. National Academy of Sciences 2007]. We prove this conjecture for k exceeding a certain constant k0. 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.
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An extended abstract version of this work appeared in the proceedings of RANDOM 2015.
The research leading to these results has received funding from the European Re-search Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 278857-PTCC.
Research is supported by ARC GaTech.
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Coja-Oghlan, A., Efthymiou, C. & Jaafari, N. Local Convergence of Random Graph Colorings. Combinatorica 38, 341–380 (2018). https://doi.org/10.1007/s00493-016-3394-x
Mathematics Subject Classification (2000)