Multi-Dimensional Elephant Random Walk with Coupled Memory
Vitor M. Marquioni

TL;DR
This paper introduces a multi-dimensional coupled memory model of the elephant random walk, revealing new diffusion behaviors and extending the ERW to higher dimensions with coupled directional dependencies.
Contribution
It proposes a novel multi-dimensional ERW with coupled memory, demonstrating new diffusion regimes and generalizing the model to N-dimensions.
Findings
Existence of following and opposing movement behaviors.
Discovery of two super-diffusion regimes.
Extension of ERW to N-dimensional coupled systems.
Abstract
The elephant random walk (ERW) is a microscopic, one-dimensional, discrete-time, non-Markovian random walk, which can lead to anomalous diffusion due to memory effects. In this study, I propose a multi-dimensional generalization in which the probability of taking a step in a certain direction depends on the previous steps in other directions. The original model is generalized in a straightforward manner by introducing coefficients that couple the probability of moving in one direction with the previous steps in all directions. I motivate the model by first introducing a two-elephant system and then elucidating it with a specific coupling. With the explicit calculation of the first moments, I show the existence of two newsworthy relative movement behaviours: one in which one elephant follows the other, and another in which they go in opposite directions. With the aid of a Fokker-Planck…
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