Fluctuating interfaces and paths in disordered and non-equilibrium systems

Abstract

In this thesis, we study the statistics of fluctuating paths and interfaces in the presence of disorder. Specifically, we consider systems in the Kardar-Parisi-Zhang universality class for stochastic interface growth, from the perspectives of both fundamental statistical mechanics and applications to real world problems. We show numerically that the probability distribution associated with directed polymers in random media, a lattice model in this universality class, interpolates between Tracy-Widom and Gaussian distributions when spatial correlations are added to the random energy landscape. As a possible application, we examine the statistics of optimal paths on actual road networks as given by GPS routing, exploring connections and distinctions to directed polymers. We investigate also the effects of roughness in the growth front of a bacterial range expansion. There, we find that such roughness can account for the experimentally observed super-diffusivity, and leads to a rapid loss of genetic diversity. Finally, we explore the complete eigenvalue spectrum of products of random transfer matrices, as relevant to a finite density of non-intersecting directed polymers. We identify a correspondence in distribution to eigenvalues of Gaussian random matrices, and show that the density of states near the edge of the spectrum is altered by the presence of disorder.