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Localisation in a growth model with interaction : arbitrary graphs.

Menshikov, Mikhail and Shcherbakov, Vadim (2020) 'Localisation in a growth model with interaction : arbitrary graphs.', Latin American journal of probability and mathematical statistics., 17 (1). pp. 473-489.


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.

Item Type:Article
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Date accepted:09 April 2020
Date deposited:17 June 2020
Date of first online publication:2020
Date first made open access:17 June 2020

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