Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models - art. no. 026110

We study via renormalization group (RG), numerics, exact bounds, and qualit ative arguments the equilibrium Gibbs measure of a particle in a d-dimensio nal Gaussian random potential with translationally invariant logarithmic sp atial correlations. We show that for any d greater than or equal to1 it exh ibits a transition at T=T-c>0. The low-temperature glass phase has a nontri vial structure, being dominated by a Sew distant states (with replica symme try breaking phenomenology). In finite dimension this transition exists onl y in this "marginal glass'' case (energy fluctuation exponent theta = 0) an d disappears if correlations grow faster (single ground-state dominance the ta >0) or slower thigh-temperature phase). The associated extremal statisti cs problem for correlated energy landscapes exhibits universal features whi ch we describe using a nonlinear Kolmogorov (KPP) RG equation. These includ e the tails of the distribution of the minimal energy (or free energy) and the finite-size corrections, which are universal, The glass transition is c losely related to Derrida's random energy models. In d=2, the connection be tween this problem and Liouville and sinh-Gordon models is discussed. The g lass transition of the particle exhibits interesting similarities with the weak- to strong-coupling transition in Liouville (c=1 barrier) and with a t ransition that we conjecture for the sinh-Gordon model, with correspondence in some exact results and RG analysis. Glassy freezing of the particle is associated with the generation under RG of new local operators and of nonsm ooth configurations in Liouville. Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (c orresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to reco ver the multifractal spectrum in the delocalized regime.