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A SPATIO-TEMPORAL POINT PROCESS MODEL FOR PARTICLE GROWTH
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 12019 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 56, no 1, p. 23-38Article in journal (Refereed) Published
Abstract [en]

A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson-Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

Place, publisher, year, edition, pages
2019. Vol. 56, no 1, p. 23-38
Keywords [en]
Hoppe-type urn scheme, marked point process, mass distribution, Poisson-Dirichlet distribution, spatio-temporal process, Yule process
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-172075DOI: 10.1017/jpr.2019.3ISI: 000475365600002OAI: oai:DiVA.org:su-172075DiVA, id: diva2:1344769
Available from: 2019-08-22 Created: 2019-08-22 Last updated: 2019-08-22Bibliographically approved

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