Noisy traveling waves: Effect of selection on genealogies

For a family of models of evolving population under selection, which can be described by noisy traveling-wave equations, the coalescence times along the genealogical tree scale like ln^α N, where N is the size of the population, in contrast with neutral models for which they scale like N. An argument relating this time scale to the diffusion constant of the noisy traveling wave leads to a prediction for α which agrees with our simulations. An exactly soluble case gives trees with statistics identical to those predicted for mean-field spin glasses by Parisi's theory.