The classical random walk consists of a "walker" who tosses a coin, and decides given the result, whether to take one step to the right or one step to the left. The process repeats itself repeatedly and the result is a random movement, whose analysis has found applications in areas ranging from molecular dynamics to the stock market.
The driving probabilistic force here is coin tossing, a classical example of "positive entropy" randomness. It is natural to ask for the properties of a "random walk" produced by "zero entropy (on the average)" randomness. One of the most interesting mechanisms like that comes from number theory: Pick a point x at random in the unit circle, and rotate it by a fixed angle. Every time the point enters one half of the circle, move one step to the right, and every time it enters the other half of the circle, move one step to the left.
Weizmann scientist and collaborators in the university of Maryland and the CNRS analyzed the probabilistic properties of these walks, and found the laws governing the returns to the point of beginning.