Number of distinct sites visited by a subdiffusive random walker

TL;DR

This paper derives explicit formulas for the asymptotic mean number of distinct sites visited by a subdiffusive random walker in all integer dimensions, filling a gap in the literature for two-dimensional cases.

Contribution

It provides the first explicit derivation of the mean number of distinct sites visited by a subdiffusive walker in two dimensions for all asymptotic waiting time behaviors.

Findings

Explicit formulas for 2D case derived

Includes dominant and subdominant contributions

Results applicable to all integer dimensions

Abstract

The asymptotic mean number of distinct sites visited by a subdiffusive continuous time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableaux of results. In this tableaux we include the dominant as well as subdominant contributions in all integer dimensions. Other quantities that can be calculated from the mean number of distinct sites visited are also discussed.