In this talk, we investigate the exact long time asymptotics of the heat kernel  on abelian covering spaces over Riemannian manifolds with Dirichlet or Neumann boundary conditions. As an application, we establish a Gaussian type central limit theorem for the winding number of reflected Brownian motion in a planar domain with multiple holes. In particular, this is a refinement of a classical result by Toby and Werner, in which a law of large numbers was obtained. This is a recent joint work with Gautam Iyer.

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