Ráth, Balázs and Martin, James
(2016)
Rigid representations of the multiplicative coalescent with
linear deletion.
Other.
MTA-BME Stochastics Research Group.
(Submitted)
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.
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Rigid representations of the multiplicative coalescent with
linear deletion. (deposited 05 Oct 2016 05:22)
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