The preceding chapter showed how absorbing state transitions arise in catalytic kinetics. Having seen their relevance to nonequilibrium processes, we turn to the simplest example, the contact process (CP) proposed by T.E. Harris (1974) as a toy model of an epidemic (see also §8.2). While this model is not exactly soluble, some important properties have been established rigorously, and its critical parameters are known to high precision from numerical studies. Thus the CP is the ‘Ising model’ of absorbing state transitions, and serves as a natural starting point for developing new methods for nonequilibrium problems. In this chapter we examine the phase diagram and critical behavior of the CP, and use the model to illustrate mean-field and scaling approaches applicable to nonequilibrium phase transitions in general. Closely related models figure in several areas of theoretical physics, notable examples being Reggeon field theory in particle physics (Gribov 1968, Moshe 1978) and directed percolation (Kinzel 1985, Durrett 1988), discussed in §6.6. We close the chapter with an examination of the effect of quenched disorder on the CP.

The model

In the CP each site of a lattice (typically the d-dimensional cubic lattice, Zd) represents an organism that exists in one of two states, healthy or infected. Infected sites are often said to be ‘occupied’ by particles; healthy sites are then ‘vacant.’