arXiv subjects

Thomas Beekenkamp

Publications and source records attributed to Thomas Beekenkamp.

4 recordsLinked to original sources

arXiv2021

Sharpness of the Phase Transition for the Orthant Model

The orthant model is a directed percolation model on $\mathbb{Z}^d$, 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 ballisiticity of the random walk on this cluster.

math.PR↗

arXiv2020

Sharpness of the Phase Transition for the Corrupted Compass Model on Transitive Graphs

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. 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.

math.PR↗

arXiv2020

Sharpness of the Percolation Phase Transition for the Contact Process on $\mathbb{Z}^d$

We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raoufi and Tassion in which they prove similar sharpness results for a variety of models.

math.PR↗

arXiv2018

Sharpness for Inhomogeneous Percolation on Quasi-Transitive Graphs

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 Duminil-Copin and Tassion [Comm. Math. Phys., 2016], and the result generalizes previous results of Antunović and Veselić [J. Stat. Phys., 2008] and Menshikov [Dokl. Akad. Nauk 1986].

math.PR↗