Anisotropic non-oriented bond percolation in high dimensions

TL;DR

This paper investigates anisotropic bond percolation on high-dimensional lattices, establishing conditions under which percolation occurs based on directional parameters using a dynamical coupling method.

Contribution

It introduces a new condition for percolation in high dimensions for anisotropic models and employs a dynamical coupling approach to analyze different dimensional models.

Findings

Percolation occurs when the sum of directional parameters exceeds 1/2 in high dimensions.

A dynamical coupling method effectively compares models across different dimensions.

Provides conditions for anisotropic percolation in high-dimensional lattices.

Abstract

We consider inhomogeneous non-oriented Bernoulli bond percolation on $\mathbb{Z}^d$, where each edge has a parameter depending on its direction. We prove that, under certain conditions, if the sum of the parameters is strictly greater than 1/2, we have percolation in sufficiently high dimensions. The main tool is a dynamical coupling between models for different dimensions with different sets of parameters.