Ráth, Balázs and Martin, James (2016) Rigid representations of the multiplicative coalescent with linear deletion. Electronic Journal of Probability. (In Press)
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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.
| Item Type: | Article |
|---|---|
| Subjects: | Q Science / természettudomány > QA Mathematics / matematika |
| Depositing User: | Dr. Balázs Ráth |
| Date Deposited: | 26 Sep 2017 17:20 |
| Last Modified: | 26 Sep 2017 17:20 |
| URI: | http://real.mtak.hu/id/eprint/63892 |
Available Versions of this Item
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Rigid representations of the multiplicative coalescent with
linear deletion. (deposited 05 Oct 2016 05:22)
- Rigid representations of the multiplicative coalescent with linear deletion. (deposited 26 Sep 2017 17:20) [Currently Displayed]
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