Zero-temperature stochastic Ising model on quasi-transitive graphs

In this thesis, we examine the question of fixation for zero-temperature stochastic Ising model on some connected quasi-transitive graphs. The initial spin configuration is distributed according to a Bernoulli product measure with parameter p \in (0; 1). Each vertex, at rate 1, changes its spin value if it disagrees with the majority of its neighbours and determines its spin value by a fair coin toss in case of a tie between the spins of its neighbours. Depending on the graph where the process evolves and the initial density, the behavior of the model can be of three distinct types: if no vertex fixates the model is of type I; if all vertices fixate the model is of type F, and if there are vertices that fixate and vertices that do not, the model is called of type M. We prove that the shrink property for the underlying graph is a necessary condition in order for the zero-temperature Ising model to be of type I. This property requires that each finite set of vertices has at least one vertex whose neighborhood falls mostly outside of this set. Our main result shows that if p = 1=2 and the graph is connected, quasitransitive, invariant under rotations and translations, then a strenghening of the shrink property, called the planar shrink property, implies that the model is of type I. Finally we prove that for one-dimensional translation invariant graphs, the shrink property is a necessary and sufficient condition for the model to be of type I.