Inhomogeneous Long-Range Percolation on the Hierarchical Lattice

We study a model for inhomogeneous long-range percolation on the hierarchical lattice ΩN of order N with an ultrametric d. Each vertex x ∈ ΩN is assigned a nonnegative weight Wx, where (Wx)x∈ΩN are i.i.d. random variables. Conditionally on the weights, and given two parameters α ≥ 0, β > 0, the edges are independent and the probability that there is an edge between two vertices x and y is of the form 1 - exp{-αWxWy/βd (x, y)}. Conditions on the weight distribution and the parameter β are formulated for the existence of a critical percolation value αc ∈ (0, ∞) such that the resulting graph contains an infinite component when α > αc and no infinite component when α < αc. Numerical simulations are also provided to validate the theoretical results.