MCMC Methods for Drawing Random Samples From the Discrete-Grains Model of a Magnetic Medium

Abstract

The discrete-grains model is a simple model for the distribution of “grains” on a magnetic medium. In this model, grains on the medium are taken to be one of four basic rectangular shapes (“tiles”)— $1 \times 1$ , $1 \times 2$ , $2 \times 1$ , and $2 \,\, \times \,\, 2$ . The magnetic medium is then modeled as an $N \times N$ square tiled by these four basic tiles. In this paper, we present Markov chain Monte Carlo (MCMC) methods for generating random tilings of an $N \times N$ square consisting only of the four basic tiles. To be precise, given a target probability distribution, e.g., the uniform distribution, on the space, $ {\mathcal S}_{N}$ , of all possible such tilings, we make use of the Metropolis–Hastings algorithm to design an MCMC method for sampling from a probability distribution arbitrarily close, in total variation distance, to the target distribution. We further extend this approach to enable sampling from probability distributions on certain subsets of $ {\mathcal S}_{N}$ , namely, those consisting of tilings, in which each kind of tile occurs a fixed number of times. We finally present some bounds and conjectures on the mixing times of the underlying Markov chains, which provide estimates of the amount of time taken by the MCMC methods to generate a random tiling sampled from the target probability distribution.