Inhomogeneous Long-Range Percolation on the Hierarchical Lattice.

We study a model for inhomogeneous long-range percolation on the hierarchical lattice Ω N of order N with an ultrametric d . Each vertex x ∈ Ω N is assigned a nonnegative weight W x , where ( W x ) x∈Ω N 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{-α W x W y /β d ( x, y ) }. Conditions on the weight distribution and the parameter β are formulated for the existence of a critical percolation value α c ∈ (0, ∞) such that the resulting graph contains an infinite component when α > α c and no infinite component when α < α c . Numerical simulations are also provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]