Long-range contact process and percolation on a random lattice

Abstract

We study the phase transition phenomena for long-range oriented percolation and contact process. We study a contact process in which the range of each vertex are independent, updated dynamically and given by some distribution N. We also study an analogous oriented percolation model on the hyper-cubic lattice, here there is a special direction where long-range oriented bonds are allowed; the range of all vertices are given by an i.i.d. sequence of random variables with common distribution N. For both models, we prove some results about the existence of a phase transition in terms of the distribution N.