Sharp phase transitions in percolation models

In percolation models, vertices or edges are removed from a graph according to a particular probabilistic rule. The connectivity properties of the resulting graph are then of interest. The initial graph is typically taken to be transitive and infinite, for example Zd with nearest neighbour edges. The classical example of such a model is Bernoulli bond percolation, in which an edge is removed from the graph with probability 1− p , and thus kept with probability p , independently for every edge. It is well known that this model exhibits a phase transition: for small values of p , there exist only finite clusters almost surely, while for large values of p , there exists an infinite cluster almost surely. In particular, there exists a critical point p_c ∈ (0,1) at which this transition occurs. Moreover, the phase transition is sharp: for p < p_c , the clusters are exponentially small. This behaviour is not specific to Bernoulli percolation. Rather, it is a common theme in percolation models. Nevertheless, the original proofs of the sharpness of the phase transition were very specific for Bernoulli percolation, and they are not easily applied to models with dependencies. Recently however, a celebrated new proof was given by Duminil-Copin, Raoufi and Tassion, which is far more robust. It makes use of Boolean function theory, in particular the OSSS inequality for decision trees. In this thesis, we will explore this new technique, and apply it to three models: the contact process, the orthant model, and the corrupted compass model. The application of the OSSS method to these models is far from straightforward, and some model-specific hurdles have to be overcome. As an instructive model with dependencies, we will start with the corrupted compass model, since the dependencies are relatively easily controlled in this model. This is in contrast to the contact process, where the dependencies of the model are rather elusive. We give a new proof of the sharpness of the phase transition at λ_c , the phase transition for the survival of the infection. We then investigate the percolation phase transition for the time- t -measure, and show that this transition is sharp as well. Furthermore, we investigate how this might be extended to the upper invariant measure for the contact process. Finally, we will examine the orthant model, which is quite different in nature, since it is a directed model in which there exists an infinite cluster across the entire parameter range. Still, we can speak of a phase transition in this model, and we will prove that it is sharp. The idiosyncratic nature of this model is reflected in the proof