Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models

We study via RG, numerics, exact bounds and qualitative arguments the equilibrium Gibbs measure of a particle in a $d$-dimensional gaussian random potential with {\it translationally invariant logarithmic} spatial correlations. We show that for any $d \ge 1$ it exhibits a transition at $T=T_c>0$. The low temperature glass phase has a non trivial structure, being dominated by {\it a few} distant states (with replica symmetry breaking phenomenology). In finite dimension this transition exists only in this ``marginal glass'' case (energy fluctuation exponent $θ=0$) and disappears if correlations grow faster (single ground state dominance $θ>0$) or slower (high temperature phase). The associated extremal statistics problem for correlated energy landscapes exhibits universal features which we describe using a non linear (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 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. Glassy freezing of the particle is associated with the generation under RG of new local operators and of non-smooth configurations in Liouville. Applications to Dirac fermions in random magnetic fields at criticality reveals a peculiar ``quasi-localized'' regime (the glass phase for the particle) where eigenfunctions are concentrated over {\it a finite number} of distant regions, and allows to recover the multifractal spectrum.

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