Anisotropic oriented percolation in high dimensions

TL;DR

This paper investigates anisotropic oriented percolation in high-dimensional lattices, establishing that local conditions for phase transition align with mean-field predictions when certain probability sums exceed one.

Contribution

It demonstrates that in high dimensions, the local probability sum determines percolation, linking local conditions to mean-field theory in anisotropic oriented percolation.

Findings

Percolation occurs when the sum of local probabilities exceeds one.

Phase transition conditions are closely related to mean-field predictions.

Results hold for dimensions d ≥ 4 with controlled probabilities.

Abstract

In this paper we study anisotropic oriented percolation on $\mathbb{Z}^d$ for $d\geq 4$ and show that the local condition for phase transition is closely related to the mean-field condition. More precisely, we show that if the sum of the local probabilities is strictly greater than one and each probability is not too large, then percolation occurs.