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**Rigid representations of the multiplicative coalescent with linear deletion.**
*(English)*
Zbl 1375.60137

Summary: 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.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

82B43 | Percolation |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

05C80 | Random graphs (graph-theoretic aspects) |