On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

The second largest eigenvalue in absolute value determines the rate of convergence of the Markov chain Monte Carlo methods. In this paper we consider the Gibbs sampler for the 1-D Ising model. We apply the geometric bound by Diaconis and Stroock (1991) to calculate an upper bound of the second largest eigenvalue, which we show is also a bound of the second largest eigenvalue in absolute value. Based on this upper bound, we derive that the convergence time is O(n2)O(n2), where nn is the number of sites. Our result includes a constant of proportionality, which enables us to give a precise bound of the convergence time. The results presented in this paper provide the lowest bound compared to those with a constant of proportionality in the literature.