The two-particle problem in comb-like structures

TL;DR

This paper investigates how the structure of comb-like networks influences the encounter probabilities of two random walkers, revealing conditions under which they may never meet and how diffusivity and network shortcuts affect these outcomes.

Contribution

It introduces an analytical framework for understanding two-particle encounters on comb-like structures, including cases with different diffusivities and complex topologies.

Findings

Encounter probability can be zero on infinite combs when both walkers move.

Single walkers will visit all sites, but two walkers may never meet.

Network shortcuts significantly influence encounter outcomes.

Abstract

Encounters between walkers performing a random motion on an appropriate structure can describe a wide variety of natural phenomena ranging from pharmacokinetics to foraging. On homogeneous structures the asymptotic encounter probability between two walkers is (qualitatively) independent of whether both walkers are moving or one is kept fixed. On infinite comb-like structures this is no longer the case and here we deepen the mechanisms underlying the emergence of a finite probability that two random walkers will never meet, while one single random walker is certain to visit any site. In particular, we introduce an analytical approach to address this problem and even more general problems such as the case of two walkers with different diffusivity, particles walking on a finite comb and on arbitrary bundled structures, possibly in the presence of loops. Our investigations are both analytical and numerical and highlight that, in general, the outcome of a reaction involving two reactants on a comb-like architecture can be strongly different according to whether both reactants are moving (no matter their relative diffusivities) or only one, and according to the density of short-cuts among the branches.