The infinite two-sided loop-erased random walk

TL;DR

This paper establishes the existence of a two-sided loop-erased random walk (LERW) in multidimensional integer lattices, providing a new perspective on LERW as viewed from a midpoint, expanding understanding of its structure.

Contribution

It introduces the concept of a two-sided LERW and proves its existence, offering a novel way to analyze LERW from a central viewpoint.

Findings

Existence of the two-sided LERW proved

Provides a new framework for understanding LERW from the middle point

Extends the theory of LERW in higher dimensions

Abstract

The loop-erased random walk (LERW) in $ \Z^d, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the "middle" of the path.