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.
Serguei Popov. Leonardo T. Rolla. Daniel Ungaretti. "Transience of conditioned walks on the plane: encounters and speed of escape." Electron. J. Probab. 25 1 - 23, 2020. https://doi.org/10.1214/20-EJP458