The talk deals with the spectral analysis of Jacobi matrices superimposed
with random perturbations that decay in a certain sense.

We shall focus our attention on two problems: The first is the analysis of
spectral stability. We show that the absolutely continuous spectrum
associated with bounded generalized eigenfunctions, for Jacobi matrices with
a mild growth restriction on the off-diagonal terms, is stable under random
Hilbert-Schmidt perturbations. We also give some results for singular
spectral types. This is joint work with Yoram Last.

The second problem is the spectral analysis of Jacobi matrices arising in
the study of Gaussian \beta ensembles of Random Matrix Theory. These
matrices may be viewed as simple Jacobi matrices (with growing off-diagonal
terms) with a random perturbation that decays in a certain sense. With the
help of the appropriately modified methods, we analyze the behavior of the
generalized eigenfunctions and the Hausdorff dimension of the spectral
measure. Some of this work is joint with Peter Forrester and Uzy Smilansky.