Elephants explore in spirals sometimes

TL;DR

This paper studies a planar Elephant Random Walk with reinforcement, demonstrating that it asymptotically follows a randomly rotated logarithmic spiral with Gaussian fluctuations, supported by a central limit theorem.

Contribution

It introduces a new model of reinforced random walk in the plane and proves a central limit theorem describing its asymptotic spiral behavior.

Findings

The process follows a logarithmic spiral at large times.

Gaussian fluctuations characterize the process around the spiral.

The model extends understanding of reinforced stochastic processes in continuous space.

Abstract

We consider in this article an Elephant Random Walk evolving in the plane. Specifically, this is a reinforced stochastic process in which the $n$th step is given by a random rotation of one of the previous steps chosen uniformly at random. We obtain a central limit theorem for this process, which shows that the process follows a randomly rotated logarithmic spiral at large times, with Gaussian fluctuations.