The contact process: recent results on finite-volume phase transitions

The contact process is a model for the spread of an infection in a population. At any given point in time, vertices of a graph (interpreted as individuals) can be either healthy or infected; infected individuals recover at rate 1 and transmit the infection to neighbours at rate lambda. On finite graphs, the infection eventually disappears with probability one. In many cases, the time it takes for this to occur depends sensitively on the value of the parameter lambda, and this finite-volume phase transition can be linked to a phase transition of the contact process on a related infinite graph. We will survey recent works in this direction, including some general results which hold on large classes of graphs, as well as results on specific graphs, such as finite d-ary trees and the random graph known as the configuration model.

This talk includes joint work with M. Cranston, T. Mountford, J.C. Mourrat, Bruno Schapira and Q. Yao.