## deterministic walk

A couple of days ago at lunch I learned about this deterministic “model” for random walks.

The game is as follows: take an infinite square grid, mark an origin facet and place an arrow pointing in one of the four principal directions (say up) on it. Now let a “particle” move from face to face (starting at the origin) with the following rules: On each face with an arrow make a step in direction of that arrow and turn the arrow counterclockwise (mathematicians allways prefer the counterclockwise direction!). If the particle steps on an empty field, place an arrow pointing up there and jump to the origin again.

While this system ist truely deterministic and thus far from beeing a random walk, it still seems to exhibit some of its features. For example, I was told that one can show that the area of visited sites is (in the limit) bounded by a circle (something the above visualization obviously illustrates). Unfortunately I could not find references to this system. So if anyone knows its name, please comment.

(here is a different automaton that exhibits a similar behaviour)