On a class of random walks with reinforced memory

TL;DR

This paper studies reinforced random walks with memory, extending the Elephant Random Walk by adding stronger reinforcement and analyzing their long-term behavior, phase transitions, and connections to stable processes.

Contribution

It introduces a stronger reinforcement mechanism for random walks with memory and analyzes their asymptotic behavior using urns, branching processes, and stable processes.

Findings

Identification of phase transitions between subcritical and supercritical regimes

Analysis of limit behavior of reinforced Elephant Random Walks

Extension to alpha-stable Shark Random Swims

Abstract

This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Sch\"utz and Trimper [2004] with a stronger reinforcement mechanism, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability $p$. With probability $1-p$, the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor $b\geq 0$, making it likelier to repeat the step again in the future. A combination of techniques from the theory of urns, branching processes and $\alpha$-stable processes enables us to discuss the limit behavior of reinforced versions of both the Elephant Random Walk and its $\alpha$-stable counterpart, the so-called Shark Random Swim introduced by Businger [2018]. We establish phase transitions, separating subcritical from supercritical regimes.