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

TL;DR

This paper analyzes the convergence rate of the Gibbs sampler for the 2D Ising model using a geometric bound, extending previous methods from 1D to 2D and improving existing bounds.

Contribution

It generalizes the geometric bound approach for the Gibbs sampler from one dimension to two dimensions in the Ising model, providing tighter convergence bounds.

Findings

The new geometric bound improves previous convergence estimates.

Extension of the method from 1D to 2D Ising model.

Enhanced understanding of the Gibbs sampler's convergence in 2D.

Abstract

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