LISA: Locally Interacting Sequential Adsorbtion

TL;DR

This paper introduces a class of sequential point processes where each new point's placement depends on local interactions, analyzing their boundedness and convergence, with implications for models like Polya urns and cooperative adsorption.

Contribution

It develops a unified framework for locally interacting sequential adsorption models, analyzing their boundedness and convergence properties, extending previous specific models.

Findings

Limiting measure is generally random when it exists.

Many processes in the class are almost surely bounded.

Convergence properties depend on local interaction rules.

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.