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We study the geometric bound introduced by Diaconis and Stroock $(1991)$ ofthe Gibbs sampler for the two-dimensional Ising model with free boundarycondition. The obtained result generalizes the method proposed by Shiu and Chen$(2015)$ from dimension one to dimension two. Furthermore we observe that thenew bound improves the result given by Ingrassia $(1994)$.

The Convergence rate of the Gibbs sampler for the $2-$D Ising model via a geometric bound