Percolation by cumulative merging and phase transition for the contact process on random graphs

Laurent Ménard, Arvind Singh

Published 2015-02-24Version 1

Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging smaller clusters and cumulating their weights. For several classical random weighted graphs, we show that there exists a phase transition regarding the existence of an infinite cluster. The motivation for introducing this partition arises from a connection with the contact process as it roughly describes the geometry of the sets where the process survives for a long time. We give a sufficient condition on a graph to ensure that the contact process has a non trivial phase transition in terms of the existence of an infinite cluster. As an application, we prove that the contact process admits a sub-critical phase on d-dimensional random geometric graphs and on random Delaunay triangulations. To the best of our knowledge, these are the first examples of graphs with unbounded degrees where the critical parameter is shown to be strictly positive.