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.
I may play with these further, but they make some interesting pattens. We have approximate self-similarity and so these patterns have fractal-like properties. Anyway, enjoy….
These images were created for artistic rather than scientific reasons. That said, random walks are have been applied to many fields including ecology, economics, psychology, computer science, physics, chemistry, and biology.
Probably the most famous application of a random walk is to Brownian motion, which describes the trajectory of a tiny particle diffusing in a fluid. I have no idea if there is anything scientific in these images, but I would not be surprised if for small step sizes we have approximately Brownian motion. However, I would need to think a lot more about this before making concrete statements.