A Note on the Perturbation of Mixed Percolation on the Hierarchical Group

We study mixed percolation on the hierarchical group of order N where each node is open withprobability 1-g, 0 ≤ g ≤ 1, and the probability of connection between two open nodes separatedby a distance k is of the form 1-exp(-αβ-k), α ≥ 0, and β > 0. The parameters α and γ are thepercolation parameters, while b describes the long-range nature of the model. In terms of parametersα,β, and γ, we show some perturbation results for the percolation function θ(α,β, γ), which is theprobability of existing an infinite component containing a prescribed node.