Some locally self-interacting walks on the integers

TL;DR

This paper investigates self-interacting random walks on integers influenced by their local history, exploring various behaviors including confinement and asymptotic movement patterns like square root or logarithmic time scales.

Contribution

It introduces and analyzes new models of locally self-interacting walks, providing insights into their long-term behaviors and possible confinement or deterministic large-scale movement.

Findings

Walks can be confined to large intervals.

Certain asymmetric drifts lead to predictable large-scale behavior.

Walks can behave like square root or logarithm of time asymptotically.

Abstract

We study certain self-interacting walks on the set of integers, that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighbourhood of their current position (in the present paper, typically, we will discuss the case where one considers the neighbouring edges and the next-to-neighbouring edges). We survey a variety of possible behaviours, including some where the walk is eventually confined to an interval of large length. We also focus on certain "asymmetric" drifts, where we prove that with positive probability, the walks behave deterministically on large scale and move like a constant times the square root of time, or like a constant times the logarithm of time.