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Combinatorica

, Volume 38, Issue 2, pp 341–380 | Cite as

Local Convergence of Random Graph Colorings

  • Amin Coja-Oghlan
  • Charilaos Efthymiou
  • Nor Jaafari
Original Paper

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 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.

Mathematics Subject Classification (2000)

05C80 05C15 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Charilaos Efthymiou
    • 1
  • Nor Jaafari
    • 1
  1. 1.Mathematics InstituteGoethe UniversityFrankfurtGermany

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