Localisation in a growth model with interaction

TL;DR

This paper studies a growth model on a cycle graph where particles are sequentially allocated with interaction-based reinforcement, showing that the process eventually localizes at one or two neighboring sites with probability one.

Contribution

It introduces a graph-based reinforced urn model for cooperative adsorption and proves almost sure localization at one or two sites.

Findings

Growth localizes at a single or neighboring sites almost surely.

The model extends reinforced urn concepts to graph interactions.

Results have implications for cooperative adsorption processes.

Abstract

This paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite cycle graph. The model can be regarded as a reinforced urn model with graph-based interactions. It is motivated by cooperative sequential adsorption, where adsorption rates at a site depend on the configuration of existing particles in the neighbourhood of that site. Our main result is that, with probability one, the growth process will eventually localise either at a single site, or at a pair of neighbouring sites.