Simulation reductions for the Ising model

TL;DR

This paper establishes linear time simulation reductions between two versions of the Ising model, enabling perfect simulation and introducing a new Markov chain approach.

Contribution

It provides the first direct simulation reductions between the Ising spins and subgraphs worlds, solving a longstanding open problem.

Findings

First perfect simulation method for the subgraphs world

New Swendsen-Wang style Markov chain for the Ising model

Linear time simulation reductions established

Abstract

Polynomial time reductions between problems have long been used to delineate problem classes. Simulation reductions also exist, where an oracle for simulation from some probability distribution can be employed together with an oracle for Bernoulli draws in order to obtain a draw from a different distribution. Here linear time simulation reductions are given for: the Ising spins world to the Ising subgraphs world and the Ising subgraphs world to the Ising spins world. This answers a long standing question of whether such a direct relationship between these two versions of the Ising model existed. Moreover, these reductions result in the first method for perfect simulation from the subgraphs world and a new Swendsen-Wang style Markov chain for the Ising model. The method used is to write the desired distribution with set parameters as a mixture of distributions where the parameters are at their extreme values.