Nearest-neighbor walks with low predictability profile and percolation in 2+ε dimensions

A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on Z³, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in Z³ including, for any ε ε (0, 1), the wedge Wε = (x, y, z) ε Z³: x ≥ 0, |z| ≤ xε which can be thought of as representing a (2 + ε)-dimensional lattice. Our proof builds on recent work of Benjamini, Pemantle and Peres, and involves the construction of finite-energy flows using nearest-neighbor walks on Z with low predictability profile. Along the way, we obtain some new results on attainable decay rates for predictability profiles of nearest-neighbor walks.