Elephant Random Walks on Coverings of Dipole Graphs

TL;DR

This paper introduces and analyzes elephant random walks on bipartite periodic lattices derived from dipole graphs, using urn and martingale techniques to establish limit theorems and asymptotic behaviors.

Contribution

It extends ERW analysis to complex geometric structures like coverings of dipole graphs, providing new asymptotic results and limit laws for these models.

Findings

Established strong law of large numbers for the counting process.

Derived non-standard strong laws and CLTs for the position process.

Identified different regimes (diffusive, critical, superdiffusive) with corresponding scaling limits.

Abstract

In the present paper, we introduce and analyze elephant random walks (ERWs) on bipartite periodic lattices arising as coverings of dipole graphs. We focus on lattices whose admissible step directions in the two parts of the bipartition are negatives of each other and disjoint. On such graphs, we define an ERW in which each step is chosen by referring to the entire history of the walk. The ERW on the hexagonal lattice is a prototypical example of our model. The definition and asymptotic analysis of such ERWs are not straightforward because both depend strongly on the underlying geometric structure. Our analysis is based on a combination of the P\'olya-type urn techniques and the martingale approach, two standard methods for analyzing ERWs. We find that the counting process of the ERW forms a P\'olya-type urn with two-periodic generating matrices. By analyzing for such urn models, we show the strong law of large numbers for the counting process. Combining the result for the counting process with the martingale approach, we derive non-standard strong laws of large numbers and central limit theorems for the position process of the ERW in the diffusive and critical regimes, as well as almost sure and $L^2$ scaling limits in the superdiffusive regime.