The collision spectrum of $\Lambda$-coalescents

TL;DR

This paper investigates the asymptotic behavior of the collision spectrum in $ ext{Lambda}$-coalescents, revealing how different measures influence the distribution of collision sizes in large systems.

Contribution

It provides a detailed analysis of the asymptotics of the collision spectrum for $ ext{Lambda}$-coalescents, including functional limits and distributional results depending on the measure near zero.

Findings

Asymptotic distributions vary with the parameter $a$ in beta$(a,b)$-coalescents.

Different limit behaviors occur for $0<a extless=1$, $1<a extless 2$, $a=2$, and $a extgreater 2$.

The measure $ ext{Lambda}$ near zero critically influences the collision spectrum asymptotics.

Abstract

$\Lambda$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially $n$ singletons we study the collision spectrum $(X_{n,k}:2\le k\le n)$, where $X_{n,k}$ counts, throughout the history of the process, the number of collisions involving exactly $k$ blocks. Our focus is on the large $n$ asymptotics of the joint distribution of the $X_{n,k}$'s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure $\Lambda$ in the vicinity of $0$. In particular, for beta$(a,b)$-coalescents different types of limit distributions occur depending on whether $0<a\leq 1$, $1<a<2$, $a=2$ or $a>2$.