Universiteit Utrecht

Department of Mathematics

The growth of a long-range percolation cluster on Zd and an epidemiological application

Pieter Trapman

9 June 2008

We consider long range percolation on Zd in which the measure on the configuration of edges is a product measure and the probability that two vertices at distance r share an edge is given by the connection function p(r). We investigate the growth of the size of the k-ball around the origin, |Bk|, i.e. the number of vertices that are within graph-distance k of the origin, for k → ∞ for different non-increasing p(r). We show that conditioned on the origin being in the infinite component, three regimes of non-increasing p(r) exist in which respectively

Particularly, we investigate the dependence of the rate of growth of the number of infectious individuals in a spatial epidemic and the distribution of long-range contacts. We show that it is possible to construct a distribution of long range contacts, such that the number of infectious individuals stays above an exponentially growing function.

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Charlene Kalle