Self-repelling random walk with directed edges on Z

TL;DR

This paper studies a self-repelling random walk on the integer lattice with directed edges, revealing unique long-term behavior and proving limit theorems using a Ray-Knight-type approach, supported by simulations.

Contribution

It introduces a novel variant of self-repelling walk with directed edges and establishes its asymptotic properties, differing from unoriented cases.

Findings

Different asymptotic scaling from unoriented edge models

Proved limit theorems for local time and position

Simulations demonstrate unusual scaling behavior

Abstract

We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray-Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behaviour of the walk considered.