Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynam-ics for sampling k-colorings of a sparse random graph G(n, d/n) for constant d. The best knownrapid mixing results for general graphs are in terms of the maximum degree ∆ of the input graphG and hold when k > 11∆/6 for all G. Improved results hold when k > α∆ for graphs withgirth ≥ 5 and ∆ sufficiently large where α ≈ 1.7632 . . . is the root of α = exp(1/α); further im-provements on the constant α hold with stronger girth and maximum degree assumptions. Forsparse random graphs the maximum degree is a function of n and the goal is to obtain results interms of the expected degree d. The following rapid mixing results for G(n, d/n) hold with highprobability over the choice of the random graph for sufficiently large constant d. Mossel and Sly(2009) proved rapid mixing for constant k, and Efthymiou (2014) improved this to k linear in d.The condition was improved to k > 3d by Yin and Zhang (2016) using non-MCMC methods.Here we prove rapid mixing when k > αd where α ≈ 1.7632 . . . is the same constant as above.Moreover we obtain O(n) mixing time of the Glauber dynamics, while in previous rapid mixingresults the exponent was an increasing function in d. As in previous results for random graphsour proof analyzes an appropriately defined block dynamics to “hide” high-degree vertices. Onenew aspect in our improved approach is utilizing so-called local uniformity properties for theanalysis of block dynamics. To analyze the “burn-in” phase we prove a concentration inequalityfor the number of disagreements propagating in large blocks. ∗Goethe University, Frankfurt am Main, Germany. Email: [email protected]. Research supported by DFGgrant EF 103/11.†University of New Mexico, USA. Email: [email protected].‡University of Rochester, USA. Email: [email protected]. Research supported in part by NSF grantCCF-1318374.§Georgia Institute of Technology, USA. Email: [email protected]. Research supported in part by NSF grantsCCF-1617306 and CCF-1563838.arXiv:1707.03796v1[cs.DM]12Jul2017