Random walk with multiple memory channels: a new paradigm

TL;DR

This paper introduces a novel one-dimensional random walk model with multiple memory channels, analyzing its diffusive behaviors, and establishing a connection to Pólya urn models for applications in population dynamics.

Contribution

The paper proposes the RW$n$MC model with multiple memory channels and provides exact calculations for the RW2MC, revealing diverse diffusive behaviors and linking it to Pólya urn models.

Findings

Exact mean and variance for RW2MC calculated.

Model exhibits diffusive and superdiffusive regimes.

Connection established between RW$n$MC and Pólya urn models.

Abstract

A new class of one-dimensional, discrete time random walk model with memory, termed "Random walk with $n$ memory channels" (RW$n$MC) is proposed. In this model the information of $n$ ($n\in \mathbb{Z}$) previous steps from the walker's entire history are needed to decide future step. Exact calculation of the mean and variance of position of the RW2MC ($n=2$) has been done which shows that it can lead to asymptotic diffusive and superdiffusive behavior in different parameter regimes. A connection between RW$n$MC and P\'olya type urn model evolving by $n$ drawings has also been reported. This connection for the RW2MC is discussed in detail which suggests the applicability of RW$2$MC in many population dynamics model with multiple competing species.