  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 $\theta=0$) and disappears
if correlations grow faster (single ground state dominance $\theta>0$) 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.
