Random walks with different directions: Drunkards beware !
Sim\~ao Herdade, Van Vu

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 (), 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 steps is at most , 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 , which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order . We discover a new conjecture concerning incidences between spheres and points in , which, if holds, would improve the bound to , which is consistent % with the case. to the case. This conjecture resembles Szemer\'edi-Trotter type results and is of independent interest.
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Taxonomy
TopicsAlgorithms and Data Compression · Stochastic processes and statistical mechanics · Data Management and Algorithms
