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.