Two-dimensional Rademacher walk

TL;DR

This work analyzes recurrence versus transience for a generalized two-dimensional Rademacher walk on $\mathbb{Z}^2$ with deterministic step sizes $a_n$ and directions chosen uniformly among $\pm\mathbf{e}_1, \pm\mathbf{e}_2$, i.e., $S_n=\sum_{i=1}^n a_i\xi_i$. It extends the one-dimensional construction of Bhattacharya and Volkov to higher dimensions and derives growth conditions on $(a_n)$ that yield transience or recurrence, employing anti-concentration tools for sums of weighted Rademacher variables, modulo-bound arguments, and coordinate-wise decompositions. The results include transience for monotone or suitably good non-integer sequences with $a_n\ge(\log n)^{1/2+\varepsilon}$ and a constructive recurrence criterion via good sets of integers, plus an explicit block-based recurrent sequence. Together, these findings illuminate how the growth rate and arithmetic structure of $a_n$ govern long-term behavior in 2D non-homogeneous random walks, extending 1D BV-type results to a richer higher-dimensional setting.

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.

Figures (1)

• Figure 1: The disjoint union of unit squares in case $K=2$; the left-bottom corners of each square are the elements of $B$.

Theorems & Definitions (24)

• Definition 1
• Theorem 1
• Lemma 1
• proof
• Corollary 1
• proof
• proof : Proof of Theorem \ref{['t1']}
• Definition 2
• Definition 3
• Remark 1