  We consider operators with random potentials on graphs, such as the lattice
version of the random Schroedinger operator. The main result is a general bound
on the probabilities of simultaneous occurrence of eigenvalues in specified
distinct intervals, with the corresponding eigenfunctions being separately
localized within prescribed regions. The bound generalizes the Wegner estimate
on the density of states. The analysis proceeds through a new multiparameter
spectral averaging principle.
