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.
Highlights
1.1 The multiplicative coalescent and MCLD(λ)The multiplicative coalescent is a continuous-time Markov process describing the evolution of a collection of blocks
The MCLD arises as a scaling limit of certain discrete processes of coalescence and fragmentation or deletion, such as the mean-field forest-fire model introduced by Ráth and Tóth in [36] and studied by Crane, Freeman and Tóth in [21], and the mean-field frozen percolation process introduced by Ráth in [35]
We show that the resulting MCLD(λ) process is the scaling limit of the list of component sizes in the mean field frozen percolation model of [35] started from a near-critical Erdos-Rényi graph, extending the result [2, Corollary 24], which is the λ = 0 case
Summary
The multiplicative coalescent (or briefly MC) is a continuous-time Markov process describing the evolution of a collection of blocks (components). For initial states in ↓2, a graphical construction is given in [34], which gives rise to a well-behaved continuous-time Markov process taking values in ↓2 This process has the Feller property with respect to the distance d(·, ·) – see Theorem 1.2 of [34]. The MCLD arises as a scaling limit of certain discrete processes of coalescence and fragmentation or deletion, such as the mean-field forest-fire model introduced by Ráth and Tóth in [36] and studied by Crane, Freeman and Tóth in [21], and the mean-field frozen percolation process introduced by Ráth in [35] These processes are of particular interest because of their self-organised criticality properties.
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