On the winding number problem with finite steps

Abstract

The winding number problem (Lévy (1940)) concerns the net angle through which the route of a random walk winds about the origin. We consider the problem of finding the winding number for a walk with finite step sizes; the eigenfunction method (Roberts and Ursell (1960)) is shown to be inapplicable because the probability distribution for a sequence of steps of different length depends on the order in which those steps are taken. In the diffusion limit, however, commutivity is restored. We derive the winding number distribution for a diffusion process, starting from a point displaced from the origin, and consider its asymptotic form. An important difference between the finite step and diffusion distributions is that the former possesses finite moments while the latter does not. We compute numerically the finite step distributions for 20000 particles undergoing N = 100000 steps, and compare the results with the diffusion distribution. Even for small winding numbers, perceptible differences between the two distributions appear even for N as large as 100000.