Uniqueness of infinite open cluster in bond percolation in ℤ^d

2016/03/09 by Ganesan, Ghurumuruhan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1603.02871

Abstract

In this paper, we alternately obtain the almost sure uniqueness of the infinite open cluster in bond percolation in ℤ^d, d ≥ 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 ∈ \0,1,2\ a.s. and then use a pivotal edges argument to obtain that N ∈ \0,1\ a.s.

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