Exploration and Trapping of Mortal Random Walkers

TL;DR

This paper investigates the exploration and trapping behavior of mortal random walkers that can evanesce, providing analytical results on how finite lifetime affects key properties like site visitation and target survival.

Contribution

It offers the first analytical treatment of evanescent random walkers, revealing how their finite lifetime influences exploration and trapping phenomena.

Findings

Finite lifetime can limit the number of visited sites.

Evanescence can prevent the target's certain death.

Analytical expressions for survival probabilities are derived.

Abstract

Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random walk properties may be profound. For instance, whereas the average number of distinct sites visited by an immortal walker grows with time without bound, that of a mortal walker may, depending on dimensionality and rate of evanescence, remain finite or keep growing with the passage of time. This number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps. If the traps are immortal, the survival probability will vanish with increasing time. However, if the traps are evanescent, the target may be spared a certain death. We analytically calculate a number of basic and broadly used quantities for evanescent random walkers.