This simulation is inspired by an exercise. (MP4 version is available here.)

Consider the infinite square lattice in d = 2, and associate to each edge an i.i.d. Uniform[0,1] random variable called the resistance of the edge. The growth process (S(t); t ≥ 0) is constructed as follows:

• Initially, the set S(0) contains only the origin.

• At step n, consider the boundary of S(n-1), which is the set of points that are not in S(n-1) but adjacent to some points in S(n-1). Then S(n) is defined as the union of S(n-1) and the boundary point p(n) which is connected to S(n-1) by an edge with the lowest resistance R(n).

(One can check that limsup R(n) =1/2,  the critical probability of percolation on the square lattice.)