Simulation reductions for the Ising model

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.