Two-Dimensional Rademacher Walk

Abstract We study a generalisation of the one-dimensional Rademacher random walk introduced in Bhattacharya and Volkov (ALEA Lat. Am. J. Probab. Math. Stat. 20(1):33–51, 2023) to $$\mathbb {Z}^2$$ Z 2 (for $$d\ge 3$$ d ≥ 3 , the Rademacher random walk is always transient, as follows from Theorem 8.8 in Engländer and Volkov (Coin Turning, Random Walks and Inhomogeneous Markov Chains, World Scientific and Volkov, Singapore, 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 $$a_n$$ a n where $$\{a_n\}$$ { a n } is a given sequence of positive integers. We establish some general conditions under which the walk is recurrent or transient.