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Directed polymer model in hydrodynamic dispersion and fracture.

We introduce a generalization of the directed polymer model, where the energies of the bonds building the paths are not random variables, but solutions of Kirchhoff's equations for a network of random resistors. This applies exactly to a hydrodynamic dispersion problem, namely, finding the first arrival time of a tracer convected by a fluid through a porous medium at high Peclet numbers. Our model may also have some relevance to fracture of heterogeneous media. It is shown to be in the same universality class as the directed polymer, but corrections to scaling are generated in two dimensions by the nonlocality of the Laplace field