Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials

 This article describes the almost sure infinite volume asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials. The random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered Gibbs measure with a superstable pair interaction. There is no restriction on the strength of the pair interaction: it may be taken, e.g., at a critical point. The potential is scaled with the box size in a critical way, i.e. the scale is determined by the typical size of large deviations in the Gibbsian cloud. The almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities. The qualitative behaviour of the ground state energy asymptotics is analysed: Depending on the dimension and on the Hölder exponents of the free energy density, it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy.