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

Abstract : 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 classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous [Ann. Probab., vol. 25, pp. 812--854, 1997]: we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $\lambda$, converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $B_t+\lambda t−t^2/2,t≥0)$, but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi, Budhiraja and Wang [Probab. Theory Rel., to appear] using an additional Poisson point process.
Type de document :
Article dans une revue
Probability Theory and Related Fields,, 2016, 166 (1), pp.515--552. 〈http://dx.doi.org/10.1007/s00440-015-0665-1〉. 〈10.1007/s00440-015-0665-1〉
Liste complète des métadonnées

https://hal.inria.fr/hal-01092562
Contributeur : Nicolas Broutin <>
Soumis le : mardi 9 décembre 2014 - 07:03:29
Dernière modification le : vendredi 25 mai 2018 - 12:02:03

Lien texte intégral

Identifiants

Collections

Citation

Nicolas Broutin, Jean-Francois Marckert. A new encoding of coalescent processes. Applications to the additive and multiplicative cases. Probability Theory and Related Fields,, 2016, 166 (1), pp.515--552. 〈http://dx.doi.org/10.1007/s00440-015-0665-1〉. 〈10.1007/s00440-015-0665-1〉. 〈hal-01092562〉

Partager

Métriques

Consultations de la notice

257