The randomly oriented Manhattan lattice in 2D is transient

Abstract

Independently orient each horizontal and vertical line of ${\mathbb{Z}}^2$ by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. Redner (1989) introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like $n^{2/3}$. We prove that the walk is transient almost surely. The proof is a short variational argument.

16 pages, 2 figures. Lean formalization: https://github.com/nitromannitol/Manhattan-Transience