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

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