Abstract. Consider a one dimensional stepping stone model with colonies of size N and per generation migration rate ν, or a voter model on Z with dispersal of order K. Sample one lineage at the origin and one at L. We show that if Nν/L and L/K2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one dimensional parabolic differential equation with an interesting boundary condition at 0.
Preprint (pdf file) Revised version - Feb 7, 2007