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arXiv · 2101.11308

Sharpness of the Phase Transition for the Orthant Model

Thomas Beekenkamp
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Abstract

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.

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Thomas Beekenkamp. 2021-01-27. Sharpness of the Phase Transition for the Orthant Model. https://doi.org/10.1007/s11040-021-09408-z

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