  Some results on the ordered statistics of eigenvalues for one-dimensional
random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric
quantum mechanics with disorder, the existence of low energy delocalized states
induces eigenvalue correlations and makes the ordered statistics problem
nontrivial. The resulting distributions are used to analyze the problem of
classical diffusion in a random force field (Sinai problem) in the presence of
weakly concentrated absorbers. It is shown that the slowly decaying averaged
return probability of the Sinai problem, $\mean{P(x,t|x,0)}\sim \ln^{-2}t$, is
converted into a power law decay, $\mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}}$,
where $g$ is the strength of the random force field and $\rho$ the density of
absorbers.

Disordered Systems and Neural Networks