### Abstract

*k*-colorings of the complete tree with branching factor

*b*undergoes a phase transition at

*k*=

*b*(1+

*o*(1))/ln

_{b}*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*(

*n*

^{1+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*(

*n*

^{1/C+ob(1)}ln

*n*) and Ω(

*n*

^{1/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 language | English |
---|---|

Pages (from-to) | 2210-2239 |

Journal | Annals of Applied Probability |

Volume | 22 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2012 |

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### Cite this

*Annals of Applied Probability*,

*22*(6), 2210-2239. https://doi.org/10.1214/11-aap833

}

*Annals of Applied Probability*, vol. 22, no. 6, pp. 2210-2239. https://doi.org/10.1214/11-aap833

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

Research output: Contribution to journal › Article › Scientific › peer-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 -