Masses of blocks of the $Λ$-coalescent with dust via stochastic flows
Grégoire Véchambre
TL;DR
This work investigates the masses of blocks in the $Λ$-coalescent with dust by embedding the coalescent in the $Λ$-Fleming-Viot flow and its flow of inverses. The authors derive Poisson and stochastic integral representations for block masses via a nested interval-partition constructed from the inverse flow, enabling sharp large- and small-time asymptotics. They establish a cut-off phenomenon driven by dust, with a finite index $N(Λ)$ separating regimes where decay rates grow with the block rank from a universal dust-dominated rate $e^{-tH(Λ)}$, and they provide explicit formulas for the long-time decay rates $λ_k(Λ)$. The results yield tractable analytic and probabilistic bounds that enhance parameter inference for $Λ$-coalescents and illuminate the dust-driven emergence of new blocks, offering new tools for studying evolving genealogies in populations with dust phenomena.
Abstract
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 $Λ$.