Percolation in an ultrametric space

Abstract

We study percolation in the hierarchical lattice of order $N$ where the probability of connection between two points separated by distance $k$ is of the form $c_{k}/N^{k(1+\delta}, \delta \gr -1$. Since the distance is an ultrametric, there are significant differences with percolation in the Euclidean lattice. We consider three regimes: $\delta \lt 1$, where percolation occurs, $\delta \gt 1$, where it does not occur, and $\delta = 1$ which is the critical case corresponding to the phase transition. In the critical case we use an approach in the spirit of the renormalization group method of statistical physics, and connectivity results of Erd˝os-Rényi random graphs play a key role. We find sufficient conditions on ck such that percolation occurs, or that it does not occur. An intermediate situation called pre-percolation, which is necessary for percolation, is also considered. In the cases of percolation we prove uniqueness of the constructed percolation clusters. In a previous paper we studied percolation in the $N \rightarrow \infy$ limit (mean field percolation), which provided a simplification that allowed finding a necessary and sufficient condition for percolation. For fixed $N$ there are open questions, in particular regarding the behaviour at the critical values of parameters in the definition of $c_{k}$ Those questions suggest the need to study ultrametric random graphs.