The hydrodynamic limit of beta coalescents that come down from infinity

We quantify the manner in which the beta coalescent $\Pi={ \Pi(t), t\geq 0},$withparameters$a\in (0, 1),$ $b>0,$ comes down from infinity. Approximating $\mathrm{\Pi }$ by its restriction ${\mathrm{\Pi }}^n$ to $[n]=\{1,\dots ,n\},$ the suitably rescaled block counting process $n^{-1}\mathrm{\#}{\mathrm{\Pi }}^n(t{n}^{a-1})$ has a deterministic limit, $c(t)$, as $n\to \mathrm{\infty }\mathrm{.}$ 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 ${\mathrm{\Pi }}^n(t),$ captures more refined information about the coalescent tree corresponding to $\mathrm{\Pi }$. Using the corresponding rescaling, the block size spectrum also converges to a deterministic limit as $n\to \mathrm{\infty }\mathrm{.}$ 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.