Fluctuating interfaces and paths in disordered and non-equilibrium systems

Directed polymers in random media (DPRM) is a simple lattice model in the Kardar-Parisi-Zhang (KPZ) universality class for stochastic surface growth. The model considers configurations of a directed path traversing a random energy landscape, and can be simulated in polynomial time through a transfer matrix formulation. As a possible application, we examine the expansion of biological populations into new territory using methods from DPRM, and model the stochastic 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 by computing coalescence statistics for pairs of individuals separated at the front, and exploring how environmental heterogeneities can locally suppress fluctuations.