Path Coalescence in Spatially Correlated Random Walks

TL;DR

This paper investigates how particles undergoing spatially correlated random walks tend to coalesce into fewer trails, revealing a surprising natural phenomenon with diverse real-world applications.

Contribution

It introduces and analyzes the phenomenon of path coalescence in spatially correlated random walks, extending classical random walk models with new insights.

Findings

Trajectories coalesce when displacements are smaller than their correlation length

Quantitative analysis of the coalescence phenomenon

Potential applications in water droplet and animal migration patterns

Abstract

A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in many areas of science and engineering as well as other fields such as finance and the life sciences. This letter describes a phenomenon occurring in a natural extension of this model: we consider the motion of a large number of particles subject to successive random displacements which are correlated in space, but not in time. If these random displacements are smaller than their correlation length, the trajectories coalesce onto a decreasing number of trails. This surprising effect is explained and quantitative results are obtained. Various possible realisations are discussed, ranging from coalescence of the tracks of water droplets blown off a windshield to migration patterns of animals.