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Continuity of a percolation function on the hierarchical group

Abstract In this paper, we consider a long-range percolation model on the hierarchical group ΩN Two nodes in ΩN separated by distance k become connected with probability min {αβ−k, 1}, where α ≥ 0 and β > 0. The percolation function θ(α, β) is defined as the probability of having a infinite component contain the origin 0 ∊ ΩN. We show that θ(α, β) is continuous with respect to both α and β.