Number of distinct sites visited by a subdiffusive random walker

Area of Science:

• Statistical Physics
• Random Walks
• Anomalous Diffusion

Background:

• The mean number of distinct sites visited by random walkers is crucial for understanding transport phenomena.
• Previous calculations for subdiffusive continuous-time random walkers were limited to specific dimensions and waiting time distributions.
• The two-dimensional case for subdiffusive walkers remained unaddressed in existing literature.

Purpose of the Study:

• To explicitly derive the asymptotic mean number of distinct sites visited by subdiffusive continuous-time random walkers.
• To extend these calculations to all integer dimensions, addressing a gap in current research.
• To provide a comprehensive tableau of results, including dominant and subdominant contributions.

Main Methods:

• Analytical derivation of the asymptotic behavior of the mean number of distinct sites visited.
• Consideration of two specific asymptotic behaviors for the waiting time distribution between steps.
• Extension of calculations to all integer dimensions (d=1, 2, 3, ...).

Main Results:

• Explicit formulas for the asymptotic mean number of distinct sites visited in all integer dimensions are presented.
• The study provides a complete set of results, including both leading-order and next-to-leading-order terms.
• The derived results formally complete a previously incomplete tableau of theoretical findings.

Conclusions:

• The research successfully calculates previously unobtained results for subdiffusive continuous-time random walkers.
• The explicit derivations in all integer dimensions offer a comprehensive understanding of site-visitation properties.
• The work facilitates the calculation of other related quantities derived from the mean number of distinct sites visited.