Abstract
For a symmetric random walk in $Z^2$ 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 Erdös-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.
Highlights
The paper (4) proved a conjecture of Erdos and Taylor concerning the number L∗n of visits to the most visited site for simple random walk in Z2 up to step n
The approach in that paper was to first prove an analogous result for planar Brownian motion and to use strong approximation. This approach applies to other random walks, but only if they have moments of all orders
When we turn to random walks with jumps, this tree structure is no longer automatic, since the walk may jump across disks
Summary
The paper (4) proved a conjecture of Erdos and Taylor concerning the number L∗n of visits to the most visited site for simple random walk in Z2 up to step n. The approach in that paper was to first prove an analogous result for planar Brownian motion and to use strong approximation This approach applies to other random walks, but only if they have moments of all orders. We develop Harnack inequalities extending those of (10) and we develop estimates for Green’s functions for random walks killed on entering a disk. Let Lxn denote the number of times that x ∈ Z2 is visited by the random walk in Z2 up to step n and set L∗n = maxx∈Z2 Lxn. Theorem 1.1. Let {Xj ; j ≥ 1} be a symmetric strongly aperiodic random walk in Z2 with X1 having the identity as the covariance matrix and satisfying Condition A and (1.2).
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