Frequent points for random walks in two dimensions. (English) Zbl 1127.60042

Bass, Richard F.; Rosen, Jay

Electron. J. Probab. 12, 1-46 (2007).

Summary: For a symmetric random walk in \(\mathbb Z^{2}\) which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erdős-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time \(n\). Among the tools we use are Harnack inequalities and Green’s function estimates for random walks with unbounded jumps; some of these are of independent interest.

MSC:

60G50	Sums of independent random variables; random walks
60F15	Strong limit theorems

Keywords:

frequent points; Harnack inequalities; Green’s function