Stratified sampling for the Ising model: A graph-theoretic approach.

We present a new approach to a classical problem in statistical physics: estimating the partition function and thermodynamic quantities of the ferromagnetic Ising model. The standard approach to this problem is to use Markov chain Monte Carlo methods that are based on the classic work of Metropolis et al. (1953). Although great improvements to these original ideas have been made, there remains scope for improvement. The first polynomial time algorithm for the estimation of the partition function was developed by Jerrum and Sinclair (1993), who reduced the problem to counting subgraphs via the high-temperature expansion. However, the polynomial bound achieved has large degree and so yields an algorithm that is too slow for practical use. Our approach, which also uses the high-temperature expansion, yields a broad class of Monte Carlo algorithms that are not based on the work of Metropolis et al., but instead use heuristic sampling techniques. In particular, we estimate coefficients of a polynomial that, once obtained, can be used to determine the quantities of interest at all temperatures simultaneously. This class of algorithms can be applied to any underlying graph, with or without an external field. These algorithms are also highly parallelizable, which, among other features, makes their implementation possible in practice.