A uniform spanning tree in a 99x99 square grid is surrounded by a Peano-like curve. The hue denotes the progress along the curve. It has a very interesting scaling limit, i.e., a limiting random curve in a square if we take finer and finer lattice meshes inside the square.
This random curve will be "space-filling", i.e., it will visit every point in the square, even though it never "crosses" itself. It is closely related to SLE (stochastic Loewner evolution). The SLE processes were introduced by Oded Schramm and formed a major part of the work of Schramm, Greg Lawler, and Wendelin Werner that contributed to Werner's earning a Fields Medal. For more info, see http://pages.iu.edu/~rdlyons/rw/rw.html.