  We study random Hamiltonians on finite-size cubes and waveguide segments of
increasing diameter. The number of random parameters determining the operator
is proportional to the volume of the cube. In the asymptotic regime where the
cube size, and consequently the number of parameters as well, tends to
infinity, we derive deterministic and probabilistic variational bounds on the
lowest eigenvalue, i.e. the spectral minimum, as well as exponential
off-diagonal decay of the Green function at energies above, but close to the
overall spectral bottom.
