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**Exact thresholds for Ising-Gibbs samplers on general graphs.**
*(English)*
Zbl 1270.60113

Gibbs sampling is a standard model for the temporal evolution of spin systems and a technique for sampling high-dimensional distributions. The study of convergence rate of Gibbs samplers, typically studied on lattices, for computer science purposes and spin-glasses theory, has been extended to general graphs of bounded degrees.

In the present paper, the Gibbs sampling is considered for the ferromagnetic Ising model on general graphs. Criteria are provided for a rapid convergence of the pertinent dynamics, for any graph and any external field. Specifically, the rapid mixing is quantified and tight sufficient conditions for mixing to occur in spin systems, are given. Lower bounds are established on the spectral gap of the continuous time Glauber dynamics which are independent of the size of the graph. The proofs are short, however, it is worth to mention that the authors’ method is fundamentally different from previous approaches in this area. The main novelty is a new application of the censoring technique, developed for the study of Markov chains.

In the present paper, the Gibbs sampling is considered for the ferromagnetic Ising model on general graphs. Criteria are provided for a rapid convergence of the pertinent dynamics, for any graph and any external field. Specifically, the rapid mixing is quantified and tight sufficient conditions for mixing to occur in spin systems, are given. Lower bounds are established on the spectral gap of the continuous time Glauber dynamics which are independent of the size of the graph. The proofs are short, however, it is worth to mention that the authors’ method is fundamentally different from previous approaches in this area. The main novelty is a new application of the censoring technique, developed for the study of Markov chains.

Reviewer: Piotr Garbaczewski (Opole)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |

### Keywords:

Ising model; Glauber dynamics; phase transition; Gibbs sampling; random graphs; trees; convergence rate; tightness; rapid mixing
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\textit{E. Mossel} and \textit{A. Sly}, Ann. Probab. 41, No. 1, 294--328 (2013; Zbl 1270.60113)

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