Localisation in a growth model with interaction. Arbitrary graphs

Menshikov, Mikhail; Shcherbakov, Vadim

Abstract

This paper concerns the long term behaviour of a growth model describing a random sequential allocation 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 Pólya 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 maximal clique.

Citation

Menshikov, M., & Shcherbakov, V. (2020). Localisation in a growth model with interaction. Arbitrary graphs. Alea (2006. Online), 17(1), 473-489. https://doi.org/10.30757/alea.v17-19

• Journal Article Type: Article
• Acceptance Date: Apr 9, 2020
• Publication Date: 2020
• Deposit Date: Jun 17, 2020
• Publicly Available Date: Jun 17, 2020
• Journal: Alea (2006)
• Publisher: Instituto Nacional de Matemática Pura e Aplicada
• Peer Reviewed: Peer Reviewed
• Volume: 17
• Issue: 1
• Pages: 473-489
• DOI: https://doi.org/10.30757/alea.v17-19