The hydrodynamic limit of beta coalescents that come down from infinity

TL;DR

This paper analyzes the hydrodynamic limit of beta coalescents that come down from infinity, providing explicit formulas for the limiting behavior of block counting and size spectrum as the number of blocks grows large.

Contribution

It offers explicit formulas and differential equations describing the deterministic limits of beta coalescents coming down from infinity, including the block size spectrum.

Findings

The rescaled block counting process converges to a deterministic function c(t).

The block size spectrum converges to a limit characterized by differential equations.

Explicit formulas involving Bell polynomials are derived for the limits.

Abstract

We quantify the manner in which the beta coalescent $\Pi=\{ \Pi(t), t\geq 0\},$ with parameters $a\in (0, 1),$ $b>0,$ comes down from infinity. Approximating $\Pi$ by its restriction $\Pi^n$ to $[n]\:= \{1, \ldots, n\},$ the suitably rescaled block counting process $n^{-1}\#\Pi^n(tn^{a-1})$ has a deterministic limit, $c(t)$, as $n\to\infty.$ An explicit formula for $c(t)$ is provided in Theorem 1. The block size spectrum $(\mathfrak{c}_{1}\Pi^n(t), \ldots, \mathfrak{c}_{n}\Pi^n(t)),$ where $\mathfrak{c}_{i}\Pi^n(t)$ counts the number of blocks of size $i$ in $\Pi^n(t),$ captures more refined information about the coalescent tree corresponding to $\Pi$. Using the corresponding rescaling, the block size spectrum also converges to a deterministic limit as $n\to\infty.$ This limit is characterized by a system of ordinary differential equations whose $i$th solution is a complete Bell polynomial, depending only on $c(t)$ and $a,$ that we work out explicitly, see Corollary 1.