Electronic Journal of Probability

Rigid representations of the multiplicative coalescent with linear deletion

James B. Martin and Balázs Ráth

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We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is deleted with rate $\lambda x$ (where $\lambda \geq 0$ is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case $\lambda =0$ (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a “tilt” of a random function, which increases with time; for $\lambda >0$ we find a novel representation in which this tilt is complemented by a “shift” mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are “rigid”, in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 83, 47 pp.

Received: 14 October 2016
Accepted: 31 August 2017
First available in Project Euclid: 14 October 2017

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Zentralblatt MATH identifier

Primary: 60J99: None of the above, but in this section 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 05C80: Random graphs [See also 60B20]

multiplicative coalescent Erdős-Rényi random graph frozen percolation

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Martin, James B.; Ráth, Balázs. Rigid representations of the multiplicative coalescent with linear deletion. Electron. J. Probab. 22 (2017), paper no. 83, 47 pp. doi:10.1214/17-EJP100. https://projecteuclid.org/euclid.ejp/1507946758

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