Heshan Aravinda, Yevgeniy Kovchegov, Peter T. Otto, Amites Sarkar
{"title":"Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes","authors":"Heshan Aravinda, Yevgeniy Kovchegov, Peter T. Otto, Amites Sarkar","doi":"arxiv-2409.06910","DOIUrl":null,"url":null,"abstract":"Random coalescent processes and branching processes are two fundamental\nconstructs in the field of stochastic processes, each with a rich history and a\nwide range of applications. Though developed within distinct contexts, in this\nnote we present a novel connection between a multi-type (vector) multiplicative\ncoalescent process and a multi-type branching process with Poisson offspring\ndistributions. More specifically, we show that the equations that govern the\nphenomenon of gelation in the vector multiplicative coalescent process are\nequivalent to the set of equations that yield the extinction probabilities of\nthe corresponding multi-type Poisson branching process. We then leverage this\nconnection with two applications, one in each direction. The first is a new\nquick proof of gelation in the vector multiplicative coalescent process using a\nwell known result of branching processes, and the second is a new series\nexpression for the extinction probabilities of the multi-type Poisson branching\nprocess using results derived from the theory of vector multiplicative\ncoalescence. While the correspondence is fairly straightforward, it illuminates\na deep connection between these two paradigms which we hope will continue to\nreveal new insights and potential for cross-disciplinary research.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06910","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Random coalescent processes and branching processes are two fundamental
constructs in the field of stochastic processes, each with a rich history and a
wide range of applications. Though developed within distinct contexts, in this
note we present a novel connection between a multi-type (vector) multiplicative
coalescent process and a multi-type branching process with Poisson offspring
distributions. More specifically, we show that the equations that govern the
phenomenon of gelation in the vector multiplicative coalescent process are
equivalent to the set of equations that yield the extinction probabilities of
the corresponding multi-type Poisson branching process. We then leverage this
connection with two applications, one in each direction. The first is a new
quick proof of gelation in the vector multiplicative coalescent process using a
well known result of branching processes, and the second is a new series
expression for the extinction probabilities of the multi-type Poisson branching
process using results derived from the theory of vector multiplicative
coalescence. While the correspondence is fairly straightforward, it illuminates
a deep connection between these two paradigms which we hope will continue to
reveal new insights and potential for cross-disciplinary research.