Sharpness of the Phase Transition for the Orthant Model

TL;DR

This paper establishes a sharp phase transition threshold for the orthant model on b, showing that above the critical point, the cluster of 0 is contained in a cone with exponentially small shift, leading to a shape theorem and ballisticity.

Contribution

The paper proves a sharp threshold result for the orthant model, demonstrating exponential decay of the shift needed for cluster containment above the critical value.

Findings

Existence of a sharp phase transition at a critical probability p.

Exponential decay of the shift needed to contain the cluster in a cone.

Shape theorem and ballisticity of the random walk above the critical threshold.

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.