  When a biological population expands into new territory, genetic drift
develops an enormous influence on evolution at the propagating front. In such
range expansion processes, fluctuations in allele frequencies occur through
stochastic \textit{spatial} wandering of both genetic lineages and the
boundaries between genetically segregated sectors. Laboratory experiments on
microbial range expansions have shown that this stochastic wandering,
transverse to the front, is superdiffusive due to the front's growing
roughness, implying much faster loss of genetic diversity than predicted by
simple flat front diffusive models. We study the evolutionary consequences of
this superdiffusive wandering using two complementary numerical models of range
expansions: the stepping stone model, and a new interpretation of the model of
directed paths in random media, in the context of a roughening population
front. Through these approaches we compute statistics for the times since
common ancestry for pairs of individuals with a given spatial separation at the
front, and we explore how environmental heterogeneities can locally suppress
these superdiffusive fluctuations.
