In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability p, independently for every vertex, every neighbouring edge is opened. We study the size of open clusters in this model. Hirsch et al. [8] have shown that for small p all open clusters are finite almost surely, while for large p, depending on the underlying graph, there exists an infinite open cluster almost surely. We show that the corresponding phase transition is sharp, i.e., in the subcritical regime, all open clusters are exponentially small. Furthermore we prove a mean-field lower bound in the supercritical regime. The proof uses the by now well established method using the OSSS inequality. A second goal of this note is to showcase this method in an uncomplicated setting.
The orthant model is a directed percolation model on $\mathbb {Z}^{d}$
ℤ
d
, in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.
In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive graphs by , and the result generalizes previous results of Antunović and Veselić [2] and Menshikov [7].
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