Some aspects of additive coalescents

On the equivalence of some eternal additive coalescents

In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

耦合 Bertoin 与 Aldous-Pitman 的加性凝聚表示

Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent

本文构造了加性凝聚的 Aldous-Pitman 与 Bertoin 两种表示的耦合，利用布朗 CRT 切割树的探索算法，统一了两种看似不同的构造。

Probabilistic aspects of $Λ$-coalescents in equilibrium and in evolution

We present approximation methods which lead to law of large numbers and fluctuation results for functionals of $Λ$-coalescents, both in the dust-free case and in the case with a dust component. Our focus is on the tree length (or total branch length) and the total external branch length, as well as the time to the most recent common ancestor and the size of the last merger. In the second part we discuss evolving coalescents. For certain Beta-coalescents we analyse fluctuations of a class of functionals in appropriate time scales. Finally we review results of Gufler on the representation of evo…

On a ternary coalescent process

We present a coalescent process where three particles merge at each coagulation step. Using a random walk representation, we prove duality with a fragmentation process, whose fragmentation law we specify explicitly. Furthermore, we give a second construction of the coalescence in terms of random binary forests and study asymptotic properties. Starting from N particles of unit mass, we obtain under an appropriate rescaling when N tends to infinity a well-known binary coalescence, the so-called standard additive coalescent.

A new encoding of coalescent processes. Applications to the additive and multiplicative cases

We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the…

Asymptotics in Knuth's parking problem for caravans

We consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved.