Two-sided random walks conditioned to have no intersections

TL;DR

This paper constructs and analyzes two-sided random walk paths in two and three dimensions conditioned on never intersecting, providing insights into their behavior and properties.

Contribution

It introduces a novel construction of two-sided random walks conditioned to avoid intersections, extending understanding of non-intersecting path behavior in lattice models.

Findings

Construction of conditioned two-sided random walks in 2D and 3D.

Analysis of properties of these non-intersecting paths.

Insights into the structure and probability measures of such conditioned walks.

Abstract

Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty) \cap S^{2}[1, \infty) = \emptyset$.