Superdiffusivity of random walks on the three-dimensional randomly oriented Manhattan lattice

Abstract

We study the superdiffusive behavior of random walks on the randomly oriented Manhattan lattice, i.e., the $d$-dimensional integer lattice ${\mathbb{Z}}^d$ where each axis-aligned line is independently assigned a random direction (forward or backward) with equal probability. The walker takes nearest-neighbor steps, choosing an axis randomly and moving along the assigned direction of that axis's line, with equal probabilities for each axis. We show that, in the critical dimension $d=3$, the diffusion coefficient of the random walk diverges in the Tauberian sense as $\sqrt{\log t}$ with a multiplicative correction $(\log \log t{)}^{\pm (2+\epsilon )}$ as time $t\to \infty$. This gives an answer to a conjecture by Ledger, Tóth and Valkó (2018).

46 pages, 1 figure