Fixation for Two-Dimensional U-Ising and U-Voter Dynamics

摘要

Given a finite family U of finite subsets of Z(d) {0}, the U-voter dynamics in the space of configurations {+, -}(Zd) is defined as follows: every v is an element of Z(d) has an independent exponential random clock, and when the clock at v rings, the vertex v chooses X is an element of U uniformly at random. If the set v + X is entirely in state + (resp. -), then the state of v updates to + (resp. -), otherwise nothing happens. The critical probability p(c)(vot) (Z(d), U) for this model is the infimum over p such that this system almost surely fixates at + when the initial states for the vertices are chosen independently to be + with probability p and to be- with probability 1 - p. We prove that p(c)(vot) (Z(2), U) < 1 for a wide class of families U. We moreover consider the U-Ising dynamics and show that its corresponding critical probability p(c)(Is) (Z(2), U) is also less than 1, for many families U, so that this model exhibits the same phase transition.