Brownian excursions, critical random graphs and the multiplicative coalescent

• 成果类型: Article
• 署名作者: Aldous, D
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 812-854
• 关键词: coagulating systems; fluctuations

摘要

Let(B-t(s), 0 less than or equal to s < infinity) be reflecting inhomogeneous Brownian motion with drift t - s at time s, started with B-t(0) = 0. Consider the random graph G(n, n(-1) + tn(-4/3)), whose largest components have size of order n(2/3) Normalizing by n(-2/3), the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of Bt (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x(i)), and clusters with sizes x(i) and x(j) merge at rate x(i)x(j). The multiplicative coalescent is shown to be a Feller process on l(2). The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time -infinity; the existence of such a process is not obvious.