Eden growth model

Example

The Eden growth model on $\Z^d$ is a random growing sequence of finite connected sets $A_0 \subset A_1 \subset \cdots$. Start with $A_0 = \set{0}$. Given $A_n$, let $\partial A_n$ be the set of points of $\Z^d \setminus A_n$ adjacent to some point of $A_n$. Choose $y$ uniformly at random in $\partial A_n$, independently of the past, and let $A_{n+1} = A_n \cup \set{y}$.

Check of the first two steps in $\Z^2$: $\partial A_0$ is the set of the $4$ neighbours of $0$, so $A_1 = \set{0, e}$ for a uniform neighbour $e$. Then $\partial A_1$ has $6$ points: the $3$ further neighbours of $0$ and the $3$ further neighbours of $e$. So each of these $6$ points is added at step $2$ with probability $1/6$.