Simple random walks in the plane

Random Walks in the Plane appear very naturally, when two players flip a coin. The player who wins receives one Euro from the other, and the total gain of each player is a « random walk », given by a sequence of independent variables with same law. Conceptually speaking, this is very simple.

But such an experiment, if repeated twice, does not lead to the same results, to the same fortune, which contradicts the assumption of « repeatability » which is usually required for any experiment in Physics.

When we study the behavior of the gains, we find that (despite what most people think), they do not tend to an equilibrium ; on the contrary, Nature proceeds by large oscillations.

In this book, we introduce completely new methods for the study of these sums : they are « energy based », which means that, instead of probabilistic statements, we use tools derived from classical analysis, operator theory, special functions and trigonometry. They prove much more efficient and provide sharp quantitative results.