Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

P. Tetali, J.C. Vera, E. Vigoda, L. Yang

Research output: Contribution to journalArticleScientificpeer-review

Abstract

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+ob(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 ≥1 we prove the mixing time is O(n1+ob(1)ln n). On the other side, for C<1 the mixing time experiences a slowing down; in particular, we prove it is O(n1/C+ob(1)ln n) and Ω(n1/C-ob(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.
Original languageEnglish
Pages (from-to)2210-2239
JournalAnnals of Applied Probability
Volume22
Issue number6
DOIs
Publication statusPublished - 2012

Fingerprint

Glauber Dynamics
Mixing Time
Colouring
Phase Transition
Local Algorithms
Sharp Bound
Branching
Critical point
First-order
Phase transition

Cite this

@article{59d09225aaf8417bbcfa76c594b4fdec,
title = "Phase transition for the mixing time of the Glauber dynamics for coloring regular trees",
abstract = "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+ob(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 ≥1 we prove the mixing time is O(n1+ob(1)ln n). On the other side, for C<1 the mixing time experiences a slowing down; in particular, we prove it is O(n1/C+ob(1)ln n) and Ω(n1/C-ob(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.",
author = "P. Tetali and J.C. Vera and E. Vigoda and L. Yang",
year = "2012",
doi = "10.1214/11-aap833",
language = "English",
volume = "22",
pages = "2210--2239",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "6",

}

Phase transition for the mixing time of the Glauber dynamics for coloring regular trees. / Tetali, P.; Vera, J.C.; Vigoda, E.; Yang, L.

In: Annals of Applied Probability, Vol. 22, No. 6, 2012, p. 2210-2239.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

AU - Tetali, P.

AU - Vera, J.C.

AU - Vigoda, E.

AU - Yang, L.

PY - 2012

Y1 - 2012

N2 - 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+ob(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 ≥1 we prove the mixing time is O(n1+ob(1)ln n). On the other side, for C<1 the mixing time experiences a slowing down; in particular, we prove it is O(n1/C+ob(1)ln n) and Ω(n1/C-ob(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.

AB - 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+ob(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 ≥1 we prove the mixing time is O(n1+ob(1)ln n). On the other side, for C<1 the mixing time experiences a slowing down; in particular, we prove it is O(n1/C+ob(1)ln n) and Ω(n1/C-ob(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.

U2 - 10.1214/11-aap833

DO - 10.1214/11-aap833

M3 - Article

VL - 22

SP - 2210

EP - 2239

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 6

ER -