### 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) |
---|---|

Pages (from-to) | 161-207 |

Number of pages | 47 |

Journal | Communications in Mathematical Physics |

Volume | 295 |

Issue number | 1 |

DOIs | |

State | 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

**Mixing time of critical ising model on trees is polynomial in the height.** / Ding, Jian; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 295, no. 1, pp. 161-207. https://doi.org/10.1007/s00220-009-0978-y

}

TY - JOUR

T1 - Mixing time of critical ising model on trees is polynomial in the height

AU - Ding, Jian

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2010/2

Y1 - 2010/2

N2 - 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)].

AB - 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)].

UR - http://www.scopus.com/inward/record.url?scp=76349103260&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76349103260&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0978-y

DO - 10.1007/s00220-009-0978-y

M3 - Article

VL - 295

SP - 161

EP - 207

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -