Some large deviations in Kingman's coalescent

Abstract

Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. Concerning the structure of the tree-top there are two well-known laws of large numbers: (i) The (shortest) distance, denoted by $T_n$, from the tree-top to the level when there are $n$ lines in the tree satisfies $nT_n \xrightarrow{n\to\infty} 2$ almost surely; (ii) At time $T_n$, the population is naturally partitioned in exactly $n$ families where individuals belong to the same family if they have a common ancestor at time $T_n$ in the past. If $F_{i,n}$ denotes the relative size of the $i$th family, then $n(F_{1,n}^2 + \cdots + F_{n,n}^2) \xrightarrow{n\to \infty}2$ almost surely. For both laws of large numbers we prove corresponding large deviations results. For (i), the rate of the large deviations is $n$ and we can give the rate function explicitly. For (ii), the rate is $n$ for downwards deviations and $\sqrt n$ for upwards deviations. In both cases we give exact rate functions.