Random search in a multi-target environment
Bristol Centre for Complexity Sciences, Department of Engineering Mathematics, Queen’s Building, Bristol BS8 1TR
Foraging, rescue operations, environmental mapping and disease spread are examples where animals, humans, robots or pathogens aim to find either of a set of potential targets in space, being their static or mobile. In the absence of cues from the environment a random search may represent an effective way to find such targets. When the environment is discrete and confined, past studies has focused mainly on estimating the mean first-passage time of a random walker to one or a small set of targets. The lack of an exact analytic expression for the spatial occupation probability, or propagator, of a discrete random walker has represented one of the limitations to develop a general theory of multi-target search in confined space. Here I present such a theory by deriving the exact propagator, obtaining expressions for the first-passage probability and the mean first-passage time to any number of targets, as well as the encounter and transmission probability of pairs of walkers in any dimensions.