Scaling Limits of Coalescent Processes Near Time Zero

TL;DR

This paper investigates the behavior of b5-coalescents near time zero under regular variation, revealing new scaling limits and geometric structures, including tangent cones and constructions via Brownian motion.

Contribution

It introduces new scaling limits for b5-coalescents near zero, including Kingman's and beta coalescents, and describes their geometric tangent cone structures.

Findings

Derived scaling limits for Kingman's and beta coalescents.

Identified tangent cone structures as coalescents with infinite mass.

Constructed limiting spaces using Brownian motion for Kingman's coalescent.

Abstract

In this paper we obtain scaling limits of $\Lambda$-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman's coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman's coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.