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Elephant Random Walks on Coverings of Dipole Graphs

Nobuaki Naganuma, Kaito Yura

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 Pólya-type urn techniques and the martingale approach, two standard methods for analyzing ERWs. We find that the counting process of the ERW forms a Pólya-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.

Elephant Random Walks on Coverings of Dipole Graphs

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 Pólya-type urn techniques and the martingale approach, two standard methods for analyzing ERWs. We find that the counting process of the ERW forms a Pólya-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 scaling limits in the superdiffusive regime.

Paper Structure

This paper contains 20 sections, 25 theorems, 200 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. A prototypical example
  4. Overview of main results
  5. Organization and notation
  6. Setting of elephant random walks
  7. Periodic realizations and dipole graphs
  8. Elephant random walks
  9. Main results
  10. The urn approach
  11. Conditional expectation of one-step increments
  12. Centered second moment
  13. Some limit theorems with the martingale approach
  14. Construction of martingale
  15. Lemmas
  16. ...and 5 more sections

Key Result

Proposition 2.5

The following holds.

Figures (6)

  • Figure 1: Square
  • Figure 2: Hexagonal
  • Figure 3: Brick-wall
  • Figure 5: Dipole graph with three edges
  • Figure 6: "Square" lattice
  • ...and 1 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Example 2.7
  • Remark 2.9
  • Theorem 2.10
  • ...and 45 more