Bernoulli hyper-edge percolation on Zd

TL;DR

This paper studies Bernoulli hyper-edge percolation on integer lattices, exploring phase transitions, cluster uniqueness, and supercritical behavior in a generalized percolation model involving hyper-edges connecting multiple vertices.

Contribution

It introduces conditions for phase transitions, cluster uniqueness, and Grimmett-Marstrand type results in hyper-edge percolation, extending classical percolation theory to hyper-graphs.

Findings

Conditions for non-trivial phase transitions identified.

Criteria for the uniqueness of the infinite cluster established.

Supercritical regime behavior characterized with Grimmett-Marstrand type results.

Abstract

We consider Bernoulli hyper-edge percolation on $\mathbb{Z}^d$. This model is a generalization of Bernoulli bond percolation. An edge connects exactly two vertices and a hyper-edge connects more than two vertices. As in the classical Bernoulli bond percolation, we open hyper-edges independently in a homogeneous manner with certain probabilities parameterized by a parameter $u\in[0,1]$. We discuss conditions for non-trivial phase transitions when $u$ varies. We discuss the conditions for the uniqueness of the infinite cluster. Also, we provide conditions under which the Grimmett-Marstrand type theorem holds in the supercritical regime.