On High-Level Exceedances of Gaussian Fields and the Spectrum of Random Hamiltonians

We study the almost sure asymptotic structure of high-level exceedances by Gaussian random field (x), xV with correlated values, where {V} is a family of -dimensional cubes increasing to Z. The results are applied to the study of the asymptotic behaviour of extreme eigenvalues of random Schrödinger operator in V.