I have again been playing with some random walks, using the same method as here. This time I used 1000000 iterations and added some colour.
Below are random walks, on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.
Once again, these images are rather for artistic purposes than scientific purposes.