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This paper extends earlier work by Cox and Durrett, who studied thecoalescence times for two lineages in the stepping stone model on thetwo-dimensional torus. We show that the genealogy of a sample of size n isgiven by a time change of Kingman's coalescent. With DNA sequence data in mind,we investigate mutation patterns under the infinite sites model, which assumesthat each mutation occurs at a new site. Our results suggest that the spatialstructure of the human population contributes to the haplotype structure and aslower than expected decay of genetic correlation with distance revealed byrecent studies of the human genome.

The stepping stone model. II: Genealogies and the infinite sites model