Random walks with different directions: Drunkards beware !

TL;DR

This paper extends classical random walk results to walks with variable directions, revealing dimension-dependent behaviors and proposing a new incidence conjecture in geometric combinatorics.

Contribution

It provides new bounds on return probabilities for directional random walks in various dimensions and introduces a novel incidence conjecture in 3.

Findings

Dimension 4 and higher have sharp return probability bounds.

In dimensions 2 and 3, the return probabilities are significantly worse than in higher dimensions.

A new incidence conjecture in 3 is proposed, potentially improving bounds in dimension 3.

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 $n^{-\omega (1)}$, which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover 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 % with the $d \ge 4$ case. to the $d \ge 4$ case. This conjecture resembles Szemer\'edi-Trotter type results and is of independent interest.