On the Local Time of the Half-Plane Half-Comb Walk

Abstract

The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to $$2/(\pi N).$$ 2 / ( π N ) . As a consequence, we prove strong laws and a limit distribution for the local time.