Probability and Statistics seminar. "Long term behaviour of a growth model with graph based interaction.". Vadim Shcherbakov, University of London, UK.
Kontakt: stanislav [dot] volkov [at] matstat [dot] lu [dot] se
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This talk 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 the vertex. When this function depends only on the number of particles at a vertex (i.e. in the absence of interaction), the model becomes a special case of the generalized Pólya urn model with exponential reinforcement. It is known that in this special no interaction case there are two possible limit behaviours depending on a model parameter. In the case of positive reinforcement (feedback) all but finitely many particles are allocated at a single random vertex almost surely. In the case of negative reinforcement each vertex receives infinitely many particles. It will be demonstrated by examples how the interactions affects the long-term behaviour of the model. The talk is based on a series of results obtained in collaboration with Mikhail Menshikov and Stanislav Volkov.