Open Access
November 2019 The eternal multiplicative coalescent encoding via excursions of Lévy-type processes
Vlada Limic
Bernoulli 25(4A): 2479-2507 (November 2019). DOI: 10.3150/18-BEJ1060


The multiplicative coalescent is a mean-field Markov process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3) each extreme eternal version of the multiplicative coalescent was described in three different ways, one of which matched its (marginal) law to that of the ordered excursion lengths above past minima of a certain Lévy-type process.

Using a modification of the breadth-first-walk construction from Aldous (Ann. Probab. 25 (1997) 812–854) and Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3), and some new insight from the thesis by Uribe Bravo (Markovian bridges, Brownian excursions, and stochastic fragmentation and coalescence (2007) UNAM), this work settles an open problem (3) from Aldous (Ann. Probab. 25 (1997) 812–854) in the more general context of Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3). Informally speaking, each eternal version is entirely encoded by its Lévy-type process, and contrary to Aldous’ original intuition, the time for the multiplicative coalescent does correspond to the linear increase in the constant part of the drift of the Lévy-type process. In the “standard multiplicative coalescent” context of Aldous (Ann. Probab. 25 (1997) 812–854), this result was first announced by Armendáriz in 2001, while its first published proof is due to Broutin and Marckert (Probab. Theory Related Fields 166 (2016) 515–552), who simultaneously account for the process of excess (or surplus) edge counts.


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Vlada Limic. "The eternal multiplicative coalescent encoding via excursions of Lévy-type processes." Bernoulli 25 (4A) 2479 - 2507, November 2019.


Received: 1 November 2016; Revised: 1 April 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110102
MathSciNet: MR4003555
Digital Object Identifier: 10.3150/18-BEJ1060

Keywords: entrance law , excursion , Lévy process , multiplicative coalescent , near-critical , random graph , stochastic coalescent

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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