Slow Encounters of Particle Pairs in Branched Structures

TL;DR

This paper investigates how inhomogeneous structures like finite combs affect the meeting times of two random walkers, revealing that topological complexity can significantly delay encounters compared to single-particle expectations.

Contribution

It demonstrates through numerical simulations that finite inhomogeneous structures can cause polynomially larger mean encounter times for two particles, highlighting effects not seen in homogeneous structures.

Findings

Encounter times can be polynomially larger than single-particle expectations.

Topological inhomogeneity significantly influences two-particle meeting dynamics.

Finite comb structures exhibit delayed encounters due to inhomogeneity.

Abstract

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comb-like structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one particle problem.