Rigid representations of the multiplicative coalescent with linear deletion
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 p...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Published: |
Institute of Mathematical Statistics
2017
|
| _version_ | 1797082374320685056 |
|---|---|
| author | Martin, J Rath, B |
| author_facet | Martin, J Rath, B |
| author_sort | Martin, J |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T01:27:14Z |
| format | Journal article |
| id | oxford-uuid:92624616-0ab9-4ff7-91e9-c19a22c0b909 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:27:14Z |
| publishDate | 2017 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:92624616-0ab9-4ff7-91e9-c19a22c0b9092022-03-26T23:25:06ZRigid representations of the multiplicative coalescent with linear deletionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:92624616-0ab9-4ff7-91e9-c19a22c0b909Symplectic Elements at OxfordInstitute of Mathematical Statistics2017Martin, JRath, BWe 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. |
| spellingShingle | Martin, J Rath, B Rigid representations of the multiplicative coalescent with linear deletion |
| title | Rigid representations of the multiplicative coalescent with linear deletion |
| title_full | Rigid representations of the multiplicative coalescent with linear deletion |
| title_fullStr | Rigid representations of the multiplicative coalescent with linear deletion |
| title_full_unstemmed | Rigid representations of the multiplicative coalescent with linear deletion |
| title_short | Rigid representations of the multiplicative coalescent with linear deletion |
| title_sort | rigid representations of the multiplicative coalescent with linear deletion |
| work_keys_str_mv | AT martinj rigidrepresentationsofthemultiplicativecoalescentwithlineardeletion AT rathb rigidrepresentationsofthemultiplicativecoalescentwithlineardeletion |