DisorderedXYModels and Coulomb Gases: Renormalization via Traveling Waves

We present a novel renormalization approach to 2D random XY models using direct and replicated Coulomb gas (CG) methods. By including fusion of environments (charge fusion in the replicated CG), the distribution of local disorder is found to obey a Kolmogorov nonlinear equation (KPP) with traveling wave solutions. At low T and weak disorder it yields a glassy XY phase with broad distributions and precise connections to random energy models. Finding marginal operators at the disorder-induced transition is related to the front velocity selection problem in KPP equations that yield new critical behavior. The method is applied to critical random Dirac problems.