Transience of conditioned walks on the plane: encounters and speed of escape

TL;DR

This paper studies a conditioned two-dimensional random walk that never hits the origin, analyzing its escape speed, minimum distance behavior, and recurrence properties of independent copies, revealing complex transience and encounter phenomena.

Contribution

It introduces a detailed analysis of the conditioned walk's behavior, including its transience, minimum distance dynamics, and infinite encounters between independent copies.

Findings

The conditioned walk is transient but still exhibits infinite meetings with independent copies.

The future minimum distance to the origin has specific asymptotic behavior.

Independent conditioned walks meet infinitely often despite transience.

Abstract

We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is,formally speaking, the Doob's $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.