Random walks with different directions

Abstract

As an extension of Polya's classical result on random walks on the square grids (Z d ), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most n −d/2−d/(d−2)+o(1) , which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is n −ω(1) , which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order n −4+o(1) . We find a new conjecture concerning incidences between spheres and points in R 3 , which, if holds, would improve the bound to n −9/2+o(1) , which is consistent to the d ≥ 4 case. This conjecture resembles Szemerédi-Trotter type results and is of independent interest.