# Localisation in a growth model with interaction. Arbitrary graphs

## TL;DR

This paper studies a growth model on finite graphs where particles are deposited sequentially with probabilities influenced by local interactions, revealing that particles tend to concentrate on cliques in the long run.

## Contribution

It extends existing models by analyzing a log-linear interaction function on arbitrary graphs, showing that particles eventually concentrate on cliques rather than a single vertex.

## Key findings

- Particles almost surely concentrate on a clique in the long term.
- Interaction influences the long-term distribution of particles.
- Special case reduces to a generalized Polya urn model.

## Abstract

This paper concerns the long term behaviour of a growth model describing a random sequential deposition of particles on a finite graph. The probability of allocating a particle at a vertex is proportional to a log-linear function of numbers of existing particles in a neighbourhood of a vertex. When this function depends only on the number of particles in the vertex, the model becomes a special case of the generalised Polya urn model. In this special case all but finitely many particles are allocated at a single random vertex almost surely. In our model interaction leads to the fact that, with probability one, all but finitely many particles are allocated at vertices of a clique.