### Abstract

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β
_{c}, the inverse-gap is O(1) for β < β
_{c}, polynomial in the surface area for β = β
_{c} and exponential in it for β > β
_{c}. This has been proved for ℤ
^{2} except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β
_{c} and exponential for β > β
_{c} were established, where β
_{c} is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β
_{c}, the inverse-gap and mixing-time are both exp[Θ((β - β
_{c})h)].

Original language | English (US) |
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Pages (from-to) | 161-207 |

Number of pages | 47 |

Journal | Communications in Mathematical Physics |

Volume | 295 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2010 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*295*(1), 161-207. https://doi.org/10.1007/s00220-009-0978-y