Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models.

Abstract

We study via renormalization group (RG), numerics, exact bounds, and qualitative arguments the equilibrium Gibbs measure of a particle in a d-dimensional Gaussian random potential with translationally invariant logarithmic spatial correlations. We show that for any d>/=1 it exhibits a transition at T=T(c)>0. The low-temperature glass phase has a nontrivial structure, being dominated by a few distant states (with replica symmetry breaking phenomenology). In finite dimension this transition exists only in this "marginal glass" case (energy fluctuation exponent straight theta=0) and disappears if correlations grow faster (single ground-state dominance straight theta>0) or slower (high-temperature phase). The associated extremal statistics problem for correlated energy landscapes exhibits universal features which we describe using a nonlinear Kolmogorov (KPP) RG equation. These include the tails of the distribution of the minimal energy (or free energy) and the finite-size corrections, which are universal. The glass transition is closely related to Derrida's random energy models. In d=2, the connection between this problem and Liouville and sinh-Gordon models is discussed. The glass transition of the particle exhibits interesting similarities with the weak- to strong-coupling transition in Liouville (c=1 barrier) and with a transition 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 nonsmooth configurations in Liouville. Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (corresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to recover the multifractal spectrum in the delocalized regime.