Sampling Random Colorings of Sparse Random Graphs
Charilaos Efthymiou, Thomas P. Hayes, Daniel Stefankovic, Eric Vigoda

TL;DR
This paper proves rapid mixing of Glauber dynamics for sampling proper colorings in sparse random graphs, improving bounds on the number of colors needed and analyzing the mixing time with new probabilistic techniques.
Contribution
It establishes the first rapid mixing results for $G(n,d/n)$ with $k> ext{approximately }1.7632d$, and introduces a novel analysis using local uniformity and concentration inequalities.
Findings
Rapid mixing for $k> ext{approximately }1.7632d$ in sparse random graphs
Achieves $O(n^{3})$ mixing time, improving previous bounds
Utilizes local uniformity properties and concentration inequalities in analysis
Abstract
We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling -colorings of a sparse random graph for constant . The best known rapid mixing results for general graphs are in terms of the maximum degree of the input graph and hold when for all . Improved results hold when for graphs with girth and sufficiently large where is the root of ; further improvements on the constant hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of and the goal is to obtain results in terms of the expected degree . The following rapid mixing results for hold with high probability over the choice of the random graph for sufficiently large…
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