Contact process

Example

Let $d \ge 1$ and $\lambda > 0$. For $x \in \Z^d$ and $\eta \in \set{0, 1}^{\Z^d}$, let $$c(x, \eta) = \begin{cases} 1 & \text{if } \eta(x) = 1, \\ \lambda \, \#\set{y \in \Z^d : \abs{y - x} = 1, \ \eta(y) = 1} & \text{if } \eta(x) = 0. \end{cases}$$ The contact process on $\Z^d$ with infection rate $\lambda$ is the spin system on $\set{0, 1}^{\Z^d}$ with these flip rates. Here $\eta(x) = 1$ means that site $x$ is infected. An infected site recovers at rate $1$, and a healthy site is infected at rate $\lambda$ times its number of infected neighbours. The rates are bounded by $\max(1, 2d\lambda)$ and each depends on $2d + 1$ sites, so it is an interacting particle system.