Rigid representations of the multiplicative coalescent with linear deletion

TL;DR

This paper introduces the multiplicative coalescent with linear deletion, a Markov process modeling block merging and deletion, with novel representations connecting to random functions and applications in self-organized criticality models.

Contribution

It extends the classical coalescent model by incorporating linear deletion and provides new rigid representations involving tilt and shift mechanisms.

Findings

Generalizes previous coalescent models without deletion.

Introduces a novel tilt-and-shift representation for the process.

Connects the process to models like forest-fire and frozen-percolation.

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.