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| ≤ x ε} 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.