Diffusion-limited aggregation

Example

Diffusion-limited aggregation (DLA) on $\Z^2$ 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 $\Z^2 \setminus A_n$ adjacent to $A_n$. Release a simple random walk "from infinity", and let $A_{n+1} = A_n \cup \set{y}$, where $y$ is the first point of $\partial A_n$ that the walk visits. Precisely, $y$ is chosen with the harmonic measure of $\partial A_n$ from infinity. This is the limit, as $\abs{z} \to \infty$, of the law of the first point of $\partial A_n$ hit by a simple random walk started at $z$.

Check for the first step: $\partial A_0$ consists of the four neighbours of $0$. By the symmetry of $\Z^2$ under rotation by a right angle, each of them is hit first with probability $1/4$. So $A_1$ is $\set{0, e}$ for a uniformly random neighbour $e$ of $0$. From then on the probabilities are no longer uniform. Points at the tips of $A_n$ are much more likely to be hit than points in its fjords.