Uniqueness of infinite open cluster in bond percolation in $\mathbb{Z}^d$

TL;DR

This paper proves the almost sure uniqueness of the infinite open cluster in bond percolation on $ ext{Z}^d$ for $d \\geq 2$ without relying on ergodicity, refining previous results on cluster count.

Contribution

It introduces a new approach to establish the uniqueness of the infinite cluster without ergodicity assumptions, extending percolation theory.

Findings

Almost sure uniqueness of the infinite open cluster in $ ext{Z}^d$ for $d \\geq 2$

The number of infinite clusters is 0 or 1 almost surely

The method avoids ergodicity of translation action

Abstract

In this paper, we alternately obtain the almost sure uniqueness of the infinite open cluster in bond percolation in $\mathbb{Z}^d, d \geq 2,$ without requiring the use of ergodicity of translation action. If $N$ denotes the number of infinite open clusters, we use the Burton-Keane argument to get $N \in \{0,1,2\}$ a.s. and then use a pivotal edges argument to obtain that $N \in \{0,1\}$ a.s.