Seminar Talk: Probability and Biomath

“Large deviations for the absorption time of Beta-coalescents”

Abstract:
The Beta-coalescent is a stochastic process used to model the random genealogy of large populations, for instance in marine species or other organisms with highly skewed reproduction. Starting from $n$ ancestral lineages, it describes how these lineages progressively merge until reaching a single common ancestor --- the time required for this to happen is called the absorption time $\tau_n$.  In this talk, we present two sets of results depending on the parameter regime. When $a > 1$, we establish a large deviation principle for $\tau_n / \log n$, which quantifies the probability of rare events of the form $\tau_n \approx x \log n$. The rate function is constructed from ratios of Gamma functions via a Legendre transform and is numerically tractable for any parameter values. When $a \in (0,1)$, the coalescent comes down from infinity and $\tau_n$ converges in distribution to a finite limit; in this regime, we obtain quantitative bounds on the total variation distance between the law of $\tau_n$ and its limit distribution, as well as sharp estimates for record probabilities.  Both sets of results rely on a novel analysis of Laplace transforms of integral functionals of the coalescent, for which we develop a strategy inspired by statistical mechanics. Based on arXiv:2512.17418.