Two-dimensional Rademacher walk

TL;DR

This paper extends the study of Rademacher random walks from one dimension to two dimensions, analyzing conditions for recurrence or transience based on step sizes and directions.

Contribution

It generalizes the Rademacher walk to two dimensions and provides criteria for recurrence or transience depending on step sequences.

Findings

Conditions for recurrence and transience are established.

The walk's behavior depends on the sequence of step sizes.

Extension of previous one-dimensional results to two dimensions.

Abstract

We study a generalisation of the one-dimensional Rademacher random walk introduced in Bhattacharya and Volkov (2023) to $\mathbb{Z}^2$ (for $d\ge 3$, the Rademacher random walk is always transient, as follows from Theorem 8.8 in Englander and Volkov (2025)). This walk is defined as the sum of a sequence of independent steps, where each step goes in one of the four possible directions with equal probability, and the size of the $n$th step is $a_n$ where $\{a_n\}$ is a given sequence of positive integers. We establish some general conditions under which the walk is recurrent or transient.