Anisotropic non-oriented bond percolation in high dimensions

TL;DR

The paper addresses anisotropic (inhomogeneous) Bernoulli bond percolation on $\mathbb{Z}^d$, where edge probabilities depend on direction, and investigates when percolation occurs in high dimensions. It introduces a dynamical, monotone coupling between percolation models in consecutive dimensions to relate high-dimensional behavior to lower-dimensional supercritical regimes, and establishes a universal regularity condition under which percolation occurs whenever $\delta = p_1+\cdots+p_d - \tfrac{1}{2} > 0$. The main results show that (i) for some universal $C>0$, if $\delta>0$ and $\max_i p_i \le C\delta^2$, then $\theta_d>0$; and (ii) if $p_1+\cdots+p_d>3\log 2$, then $\theta_d>0$ for all $d\ge2$ without extra regularity. These findings tie the high-dimensional anisotropic threshold to the inhomogeneous Galton–Watson framework and provide practical sufficient conditions for the emergence of an infinite open cluster through a dimension-dynamic coupling strategy.

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.

Theorems & Definitions (14)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 2
• Remark 6
• Proposition 1
• proof