The random walk winding number problem: convergence to a diffusion process with excluded area

The winding number of a random walk in the plane is the net angle through which the walker encircles the origin. The author discusses the moments and asymptotic form of the winding number distribution for walks of constant step size l. The author shows that the root mean square winding number theta _rms grows logarithmically with the number of steps N as N to infinity , in contrast with the N^1/2 growth of the radial distribution. The corresponding diffusion process, however, has infinite theta _rms, and does not provide a good approximation for the random walk distribution. Instead, a diffusion process considered recently by Rudnick and Hu (1987), where the area surrounding the origin has been removed out to a radius R, provides the correct asymptotic distribution. The author finds that the optimum radius for convergence of the finite step and diffusion distributions is precisely R=e^-2l.