Pieter Trapman,
Department of Mathematics,
Stockholm University

Title: 
Long-range percolation on the hierarchical lattice

Abstract: 
The hierarchical lattice of order N, may be seen as the leaves
of an infinite regular N-tree, in which the distance between two
vertices is the distance to their most recent common ancestor in the tree.

We create a random graph by independent long-range percolation on the
hierarchical lattice of order N:
The probability that a pair of vertices at (hierarchical) distance R
share an edge depends only on R and is exponentially decaying in R,
furthermore the presence of absence of different edges are independent.

We give criteria for percolation (the presence of an infinite cluster)
and we show that in the supercritical parameter domain, the infinite
component is unique. Furthermore, we show the percolation probability
(the density of the infinite cluster) is continuous in the model
parameters, in particular, there is no percolation at criticality.

Joint work with Slavik Koval and Ronald Meester