Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order N, where any edge of length k is present with probability p(k) = 1 - exp (-beta(-k)alpha), independently of all other edges. For fixed beta, we show that alpha(c)(beta) ( the infimum of those alpha for which an infinite cluster exists a.s.) is non-trivial if and only if N < beta < N-2. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of alpha(c)(beta) as a function of beta. This means that the phase diagram of this model is well understood