Mathematicians crack the code on two-dimensional random walks
Researchers have solved a decades-old puzzle about how particles move randomly across two-dimensional spaces, extending previous work on simpler one-dimensional models. The findings could improve algorithms used in machine learning, network optimization, and Monte Carlo simulations that power financial modeling and drug discovery.
Originaltitel: 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 $$\mathbb {Z}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> (for $$d\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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 $$a_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> where $$\{a_n\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a given sequence of positive integers. We establish some general conditions under which the walk is recurrent or transient.