### Abstract

We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if (d -1) tanh β < 1, then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by β, and arbitrary external fields are bounded by Cnlog n.Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when d tanh β < 1, with high probability over the Erdos-Rényi random graph G(n, d/n), it holds that the mixing time of Gibbs samplers is _{n}1+Θ(1/log log n). Both results are tight, as it is known that the mixing time for random regular and ErdOs-Rényi random graphs is, with high probability, exponential in n when (d - 1) tanhβ >1, and d tanhβ >1, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.

Original language | English (US) |
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Pages (from-to) | 294-328 |

Number of pages | 35 |

Journal | Annals of Probability |

Volume | 41 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2013 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Glauber dynamics
- Ising model
- Phase transition

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## Cite this

*Annals of Probability*,

*41*(1), 294-328. https://doi.org/10.1214/11-AOP737