A class of one-dimensional, discrete-time random walk models with memory, termed “random walk with $n$ memory channels” ($\mathrm{RW}n\mathrm{MC}$), is proposed. In these models the information of $n$ ($n\in \mathbb{Z}$) previous steps from the walker's entire history is needed to decide a 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 $\mathrm{RW}n\mathrm{MC}$ and a Pólya-type urn model evolving by drawing $n$ balls at a time has also been reported. This connection for the RW2MC is discussed in detail and suggests the applicability of $\mathrm{RW}2\mathrm{MC}$ in many population dynamics models with multiple competing species.

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• Random walks
• Diffusion & random walks
• Non-Markovian processes