Then Shiu and Chen (2015) used this result to study the convergence rate of the Gibbs sampler for the 1?D Ising model. In the first part of this thesis, we generalize the result of Shiu and Chen to the case of the 1?D Potts model with three and more spin states (see Theorem 2.2.1 and Theorem 2.2.2). In the second part we generalize their method to the case of the 2?D Ising model with two spin states (see Theorem 3.2.3), 1991. ,
, Note that in the case of the 2?D Ising model (see Theorem 3.2.3) we did not succeed to compute the Diaconis and Stroock bound as explicitly in Theorem 3.2.3 as in the one dimensional case. We only obtained an upper bound for the bound they give for ? 1 . This raises the following question: Is there a way to use some symmetry argument, like in the one dimensional case, to obtain an exact expression for the Diaconis Stroock bound?, We study in the second and third chapter of this thesis the rate of convergence of the Gibbs sampler algorithm for some model from statistical mechanics
The difficulty is to understand the Cheeger constant at finite temperature for Ising model and Potts model. Yet another method is to use coupling to bound the convergence time ,
We think that we can use this approach in the case of the Potts model ,
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