10 - Interacting particle systems

Summary

The contact, voter, and exclusion models are Markov processes in continuous time with state space {0, 1}^V for some countable set V. In the voter model, each element of V may be in either of two states, and its state flips at a rate that is a weighted average of the states of the other elements. Its analysis hinges on the recurrence or transience of an associated Markov chain. When V = ℤ² and the model is generated by simple random walk, the only invariant measures are the two point masses on the (two) states representing unanimity. The picture is more complicated when d ≥ 3. In the exclusion model, a set of particles moves about V according to a ‘symmetric’ Markov chain, subject to exclusion. When V = ℤ^d and the Markov chain is translation-invariant, the product measures are invariant for this process, and furthermore these are exactly the extremal invariant measures. The chapter closes with a brief account of the stochastic Ising model.

Introductory remarks

There are many beautiful problems of physical type that may be modelled as Markov processes on the compact state space = {0, 1}^V for some countable set V. Amongst the most studied to date by probabilists are the contact, voter, and exclusion models, and the stochastic Ising model.