The hydrodynamic limit of beta coalescents that come down from infinity

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.