Thesis title: 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 configuration 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 $ \mathcal I $; if all vertices fixate the model is of type $ \mathcal F $, and if there are vertices that fixate and vertices that do not, the model is called of type $ \mathcal 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 $ \mathcal I $. This property requires that each finite 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, quasi-transitive, invariant under rotations and translations, then a strenghening of the shrink property, called the planar shrink property, implies that the model is of type $ \mathcal 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 $ \mathcal I $.