Universal exploration dynamics of random walks.
basic_science · Level V
Where this comes from
- Record sourced from PubMed, PMID 36739291.
- Also identified by DOI 10.1038/s41467-023-36233-5 and PMC identifier 9899275.
- Licence recorded as CC BY.
- The licence permits redistribution, so the abstract is shown in full and the full text is available from the publisher.
Abstract
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. We introduce a more fundamental quantity, the time τ<sub>n</sub> required by a random walk to find a site that it never visited previously when the walk has already visited n distinct sites, which encompasses the full dynamics about the visitation statistics. To study it, we develop a theoretical approach that relies on a mapping with a trapping problem, in which the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk, the distribution of the τ<sub>n</sub> can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes.