Some large deviations in Kingman’s coalescent

Kingman’s coalescent is a random tree that arises from classical population ge-netic models such as the Moran model. The individuals alive in these models corre-spond to the leaves in the tree and the following two laws of large numbers concern-ing the structure of the tree-top are well-known: (i) The (shortest) distance, denoted by Tn, from the tree-top to the level when there are n lines in the tree satisfies nTn n→∞−−−− → 2 almost surely; (ii) At time Tn, the population is naturally partitioned in exactly n families where individuals belong to the same family if they have a com-mon ancestor at time Tn in the past. If Fi,n denotes the size of the ith family, then n(F 21,n + · · ·+ F 2n,n) n→∞−−−− → 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 n for upwards deviations. For both cases we give the exact rate function.