Masses of blocks of the $Λ$-coalescent with dust via stochastic flows
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913715397328896 |
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| author | Véchambre, Grégoire |
| author_facet | Véchambre, Grégoire |
| contents | We study the masses of blocks of the $Λ$-coalescent with dust and some aspects of their large and small time behaviors. To do so, we start by associating the $Λ$-coalescent to a nested interval-partition constructed from the flow of inverses, introduced by Bertoin and Le Gall in [Ann. inst. Henri Poincare (B) Probab. Stat. 41(3), 307-333 (2003)], of the $Λ$-Fleming-Viot flow, and prove Poisson representations for the masses of blocks in terms of the flow of inverses. The representations enable us to use the power of stochastic calculus to study the masses of blocks. We apply this method to study the long and small time behaviors. In particular, for all $k>1$, we determine the decay rate of the expectation of the $k$-th largest block as time goes to infinity and find that a cut-off phenomenon, related to the presence of dust, occurs: the decay rate is increasing for small indices $k$ but remains constant after a fixed index depending on the measure $Λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01320 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Masses of blocks of the $Λ$-coalescent with dust via stochastic flows Véchambre, Grégoire Probability 60J25, 60J76, 60J90, 60G57, 60H05, 92D25 We study the masses of blocks of the $Λ$-coalescent with dust and some aspects of their large and small time behaviors. To do so, we start by associating the $Λ$-coalescent to a nested interval-partition constructed from the flow of inverses, introduced by Bertoin and Le Gall in [Ann. inst. Henri Poincare (B) Probab. Stat. 41(3), 307-333 (2003)], of the $Λ$-Fleming-Viot flow, and prove Poisson representations for the masses of blocks in terms of the flow of inverses. The representations enable us to use the power of stochastic calculus to study the masses of blocks. We apply this method to study the long and small time behaviors. In particular, for all $k>1$, we determine the decay rate of the expectation of the $k$-th largest block as time goes to infinity and find that a cut-off phenomenon, related to the presence of dust, occurs: the decay rate is increasing for small indices $k$ but remains constant after a fixed index depending on the measure $Λ$. |
| title | Masses of blocks of the $Λ$-coalescent with dust via stochastic flows |
| topic | Probability 60J25, 60J76, 60J90, 60G57, 60H05, 92D25 |
| url | https://arxiv.org/abs/2503.01320 |