Continuity of a percolation function on the hierarchical group

Abstract

In this paper, we consider a long-range percolation model on the hierarchical group Ω_N Two nodes in Ω_N 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 ∊ Ω_N. We show that θ(α, β) is continuous with respect to both α and β.

Keywords:

• percolation
• component
• hierarchical group

MSC 2000:

• 60K35
• 60C05
• 05C80