On the local time of the Half-Plane Half-Comb walk

TL;DR

This paper analyzes the Half-Plane Half-Comb walk, a hybrid random walk on a plane with different structures above and below the x-axis, establishing asymptotic return probabilities and laws for local time.

Contribution

It provides the first asymptotic analysis of return probabilities and local time laws for the Half-Plane Half-Comb walk, a novel hybrid lattice model.

Findings

Return probability asymptotically 2/(3 N)

Strong laws for local time established

Limit distribution for local time derived

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). As15 pag a consequence we prove strong laws and a limit distribution for the local time.