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December 2012 Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
Prasad Tetali, Juan C. Vera, Eric Vigoda, Linji Yang
Ann. Appl. Probab. 22(6): 2210-2239 (December 2012). DOI: 10.1214/11-AAP833


We prove that the mixing time of the Glauber dynamics for random $k$-colorings of the complete tree with branching factor $b$ undergoes a phase transition at $k=b(1+o_{b}(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with $n$ vertices for $k=Cb/\ln{b}$ colors with constant $C$. For $C\geq1$ we prove the mixing time is $O(n^{1+o_{b}(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_{b}(1)}\ln{n})$ and $\Omega(n^{1/C-o_{b}(1)})$. The critical point $C=1$ is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.


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Prasad Tetali. Juan C. Vera. Eric Vigoda. Linji Yang. "Phase transition for the mixing time of the Glauber dynamics for coloring regular trees." Ann. Appl. Probab. 22 (6) 2210 - 2239, December 2012.


Published: December 2012
First available in Project Euclid: 23 November 2012

zbMATH: 1266.82043
MathSciNet: MR3024967
Digital Object Identifier: 10.1214/11-AAP833

Primary: 60J10

Keywords: Glauber dynamics , graph colorings , Markov chain Monte Carlo , mixing time , phase transition

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 6 • December 2012
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