Polynomial time reductions between problems have long been used to delineate problem classes, where an oracle for solving one problem yields a solution to another. Simulation reductions also exist, where an oracle for simulation from a probability distribution is employed in order to obtain draws from another 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.
AMS Subject Classification
68U20 65C05 68Q25
Monte Carlo Simulation reduction Ising model High temperature expansion
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