## Electronic Journal of Probability

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

### Rigid representations of the multiplicative coalescent with linear deletion

James B. Martin and Balázs Ráth

#### Abstract

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

**Source**

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

**Dates**

Received: 14 October 2016

Accepted: 31 August 2017

First available in Project Euclid: 14 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1507946758

**Digital Object Identifier**

doi:10.1214/17-EJP100

**Mathematical Reviews number (MathSciNet)**

MR3718711

**Zentralblatt MATH identifier**

1375.60137

**Subjects**

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]

**Keywords**

multiplicative coalescent Erdős-Rényi random graph frozen percolation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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