A spatio-temporal point process model for particle growth. (March 2019)

• Record Type: Journal Article
• Authors: Hössjer, Ola
• Is Part Of: Journal of applied probability. Volume 56:Number 1(2019)
• Page Start: 23
• Page End: 38
• Publication Date: 2019-03
• Subjects: Hoppe-type urn scheme, -- marked point process, -- mass distribution, -- Poisson–Dirichlet distribution, -- spatio-temporal process, -- Yule process; Primary 60G55, -- 60J05, -- Secondary 60J25, -- 60J80; 519.2
• DOI: 10.1017/jpr.2019.3
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: 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.