LARGE-SCALE BEHAVIOUR AND HYDRODYNAMIC LIMIT OF BETA COALESCENTS

• 成果类型: Article
• 署名作者: Miller, Luke; Pitters, Helmut H.
• 署名单位: University of Oxford; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1782
• 发表日期: 2023
• 页码: 1-27
• 关键词: block counting process; lambda-coalescent; coagulation; models; time

摘要

We quantify the behaviour at large scales of the beta coalescent Pi = {Pi(t), t >= 0} with parameters a, b > 0. Specifically, we study the rescaled block size spectrum of Pi(t) and of its restriction Pi(n)(t) to {1, ... , n}. Our main result is a law of large numbers type of result if Pi comes down from infinity. In the case of Kingman's coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski (Z. Phys. 17 (1916) 557-585). We extend Smoluchowski's result to beta coalescents and show that if Pi comes down from infinity both rescaled spectra n(-1)(c(1)Pi(t tau(n)), ... , c(n)Pi(t tau(n))), and n(-1)(c(1)Pi(n)(t tau(n)), ... , c(n)Pi(n)(t tau(n))), converge to (different) deterministic limits that we compute explicitly in terms of partial Bell polynomials. Here c(i)pi counts the number of blocks of size i in a partition pi, and (tau(n)) is a sequence such that tn tau(n) similar to n(-(1-a)) as n -> infinity. Along the way we study the nontrivial limits of the rescaled block counting processes {n(alpha)#Pi(n)(t tau(n)), t >= 0}, and {n(alpha)#Pi(t tau(n)), t >= 0}, where alpha is an element of [-1,-2/(3 - a)), and tau(n) similar to n(alpha(1-a)) if Pi comes down from infinity.