Ideas Seminar: Ted Cox, Syracuse University

Title: Complete convergence for the q-voter model in two dimensions

Abstract:The q-voter model is a spin-flip system in which the rate of flipping to type i is given by the q th power of the proportion of nearest neighbours in type i for i = 0, 1. If q = 1 it reduces to the classical voter model. We show that in the critical 2-dimensional case, for q < 1 and close enough to 1, for any initial state as t → ∞ the system converges weakly to a mixture of all 0’s, all 1’s, and a unique invariant law which contains infinitely many sites of both types. This kind of asymptotic behavior is quite different from that of the 2-dimensional voter model itself, which undergoes clustering, and converges to a mixture of all 0’s and all 1’s. In this talk we will discuss some of the novel features in the proof of our complete convergence theorem.