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Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent (1996)
| Citations: | 106 - 8 self |
BibTeX
@MISC{Aldous96brownianexcursions,,
author = {David J. Aldous},
title = {Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent},
year = {1996}
}
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Abstract
Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (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 \Gamma1: the existence of such a process is not obvious. AMS 1991 subject classifications. 60C05, 60J50, Key words and phras...
Keyphrases
multiplicative coalescent critical random graph brownian excursion nonnegative real cluster size standard multiplicative coalescent component size inhomogeneous brownian motion subject classification random graph drift gamma asymptotic joint distribution key word joint distribution small cluster multiplicative coalescent process excursion length time gamma1 feller process random graph limit gamma1 tn gamma4
