Random walk with multiple memory channels: a new paradigm

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.