Localisation in a growth model with interaction. Arbitrary graphs

Menshikov, Mikhail; Shcherbakov, Vadim

doi:10.30757/alea.v17-19

Localisation in a growth model with interaction. Arbitrary graphs

Menshikov, Mikhail; Shcherbakov, Vadim

Authors

Professor Mikhail Menshikov mikhail.menshikov@durham.ac.uk
Professor

Vadim Shcherbakov

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