RUI: Extremal Paths in Complex Systems

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0083204Halpin-HealyThis grant for Research at an Undergraduate Institution (RUI) provides support for theoretical research on the statistical mechanics of directed polymers in random media (DPRM) and domain-walls in impurity-stricken magnets. Because of common technical aspects, the DPRM has been christened a baby-version of the spin-glass problem. As such, the T=0 DPRM is a matter of global optimization, exhibiting an ultrametric tree structure reminiscent of those found in river delta basins, capillary blood vessel networks, and neuronal arrays in the brain. A Hopf-Cole transformation maps the DPRM to the far-from-equilibrium dynamics of Kardar-Parisi-Zhang (KPZ) stochastic growth models. A final payoff follows from the blood relation of KPZ and noisy Burgers equations, the latter relevant to nonequilibrium kinetics of driven lattice gases. An important goal at hand is further exploration of this rich KPZ triumvirate: DPRM, kinetic roughening, and driven lattice gases. With the DPRM becoming a classic problem of ill-condensed matter, however, future research will focus on diverse, general issues concerning geometric complexity, including scaling of extremal trajectories and networks in geomorphology, evolutionary biology, and other multidisciplinary statistical mechanics contexts.%%%This grant for Research at an Undergraduate Institution (RUI) provides support for theoretical research on the statistical mechanics of directed polymers in random media (DPRM) and domain-walls in impurity-stricken magnets. These statistical models have far-reaching applications in addition to their intrinsic fundamental interest. The research involves extensive undergraduate participation.***