Winding of a two-dimensional Brownian particle in a random environment

We study analytically the average probability for a Brownian particle to wind n times around a removed area of finite size in a 2D plane with randomly distributed traps. Such a model describes, for example, the Abrikosov vortex entanglement around a cylindrical cavity in a superconductor with repulsive columnar defects. The problem amounts to the quantum mechanics of a particle moving in a plane with point-like random scatterers pierced by a solenoid. It is shown that at large times t the asymptotic winding angle distribution, which is determined by a 'Lifshitz tail' in the density of states of such a particle, is Gaussian with the scaling variable x = n/(t(1/4)ln(1/2) t).