A Eulerian approach to the analysis of rendez-vous algorithms

In this paper we analyze rendez-vous algorithms in the situation when agents can only exchange information below a given distance threshold R. We study the system under an Eulerian point of view considering (possibly continuous) probability distributions of agents and we present convergence results both in discrete and in continuous time. The limit distribution is always necessarily a convex combination of delta functions at least R far apart from each other: in other terms these algorithms are locally aggregating. Numerical simulations seem to show that starting from continuous distributions, in general these algorithms do not converge to a unique delta (rendez-vous) in agreement with previous literature on this subject.