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We consider the two-dimensional simple random walk conditioned on neverhitting the origin, which is,formally speaking, the Doob's $h$-transform of thesimple random walk with respect to the potential kernel. We then study thebehavior of the future minimum distance of the walk to the origin, and alsoprove that two independent copies of the conditioned walk, although bothtransient, will nevertheless meet infinitely many times a.s.

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