Subgraphs-world Process in the Ising Model

The physics of phase transitions can be modeled by an arrangement of sites in a d-dimensional lattice known as the Ising model, in which each site is assigned either a positive or negative spin. In this paper, we will first provide an introduction to the Ising model. We will define the partition function of an Ising system, and present the problem of its computation. In 1993, Jerrum and Sinclair developed a revolutionary new approximation algorithm for the partition function of an arbitrary ferromagnetic Ising system. The key innovation Jerrum and Sinclair presented is a new Markov chain known as the Subgraphs-world model from which we can sample random states of the Ising system. The Subgraphsworld chain proves to be rapidly mixing, which allows us to efficiently approximate the partition function through sampling. We will introduce the chain, and show that it is rapidly mixing.