Phase transitions for a unidirectional elephant random walk with a power law memory

TL;DR

This paper investigates phase transitions in a unidirectional elephant random walk with power law memory, revealing three distinct behavioral phases depending on memory parameters.

Contribution

It extends the study of elephant random walks by analyzing phase transitions under power law memory distributions for the first time.

Findings

Identifies three phases based on memory parameters.

Characterizes conditions for finite vs. infinite travel.

Determines the speed regimes of the elephant walk.

Abstract

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter $p$ and the power law exponent $\beta$, we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels.