### Brownian excursions, critical random graphs and the multiplicative coalescent

David Aldous
Source: Ann. Probab. Volume 25, Number 2 (1997), 812-854.

#### Abstract

Let $(B^t (s), 0 \leq s < \infty)$ be reflecting inhomogeneous Brownian motion with drift $t - s$ at time $s$, started with $B^t (0) = 0$. Consider the random graph $\mathscr{G}(n, n^{-1} + tn^{-4/3})$, 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 $B^t$ (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 $\mathsf{x}$ 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$. The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time$-\infty\$; the existence of such a process is not obvious.

First Page:
Primary Subjects: 60C05, 60J50
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1024404421
Mathematical Reviews number (MathSciNet): MR1434128
Digital Object Identifier: doi:10.1214/aop/1024404421
Zentralblatt MATH identifier: 0877.60010

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