Random Recursive Trees and the Elephant Random Walk

TL;DR

This paper explores the connection between elephant random walks, a model of anomalous diffusion with infinite memory, and bond percolation on random recursive trees, providing exact calculations and introducing a new variant called the skew elephant random walk.

Contribution

It establishes a novel coupling between elephant random walks and percolation on recursive trees, enabling transfer of results and introducing the skew elephant random walk model.

Findings

Exact expressions for root cluster size moments

Results on the number of nodes in child clusters

Introduction of the skew elephant random walk model

Abstract

One class of random walks with infinite memory, so called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce a new model, the skew elephant random walk and calculate the first and second moment of this process.