When we extend the basic CP, a host of intriguing questions arise: How do multi-particle rules and diffusion affect the phase diagram? When should we expect a first-order transition? Can multiple absorbing configurations or conservation laws change the critical behavior? In this chapter we examine a diverse collection of models whose behavior yields some insight into these issues.

Multiparticle rules and diffusion

In the CP described in chapter 6, the elementary events (creation and annihilation) involve single particles. What happens if one of the elementary events involves a cluster of two or more particles? Consider pairwise annihilation: in place of • → ○ (as in the CP), • • → ○ ○. (In other words, a pair of particles at neighboring sites can annihilate one another, but there is no annihilation of isolated particles.) In fact, we have already seen one instance of pairwise annihilation: in the ZGB model (chapter 5), adsorption of O2 destroys a pair of vacancies. The transition to the absorbing O-saturated state corresponds to (and belongs to the same class as) the CP. Another example is a version of the CP (the ‘A2’ model), in which annihilation is pairwise (Dickman 1989a,b). Here again the transition to the absorbing state is continuous, and the critical exponents are the same as for the CP.