Work

A Hybrid Model of Elephant and Moran Random Walks: Exact Distribution and Symmetry Properties

Year: 2025

Type: article

Abstract: This work introduces a hybrid memory-based random walk model that combines the Elephant Random Walk with a modified Moran Random Walk. The model introduces a sequence of independent and identically distributed random variables with mean 1, representing step sizes. A particle starts at the origin and moves upward with probability r or remains stationary with probability 1−r. From the second step onward, the particle decides its next action based on its previous movement, repeating it with probability p or taking the opposite action with probability 1−p. The novelty of our approach lies in integrating a short-memory mechanism with variable step sizes, which allows us to derive exact distributions, recurrence relations, and central limit theorems. Our main contributions include (i) establishing explicit expressions for the moment-generating function and the exact distribution of the process, (ii) analyzing the number of stops through a symmetry phenomenon between repetition and inversion, and (iii) providing asymptotic results supported by simulations.

Source: Symmetry

DOI: 10.3390/sym17101709

Authors: Rafik Aguech, Mohamed Abdelkader

Institution: King Saud University

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funder: King Saud University