LISA: Locally Interacting Sequential Adsorbtion

Abstract

We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This class covers, in particular, generalised Polya urn scheme, Dubbins--Freedman random measures and cooperative sequential adsorption models studied previously. Specifically, we address models where the distribution of a newly added particle is determined by the distance to the closest particle from the chosen one. We address boundedness of the processes and convergence properties of the corresponding sample measure. We show that in general the limiting measure is random when exists and that this is the case for a wide class of almost surely bounded processes.