"Several works by Tom Wolff related to Anderson-Bernoulli models"


Theorem: Let $\mu$ be a probability measure on $PSL(2,\R)$ and
assume that supp$\mu$ is not contained in a left coset of an
elementary  subgroup. Let $\rho$ be a unitary representation such
that for some $\e$, the direct integral decomposition of $\rho$
does not contain the trivial representation nor the
representations ${\cal C}^u, u>1-\e$. Define
$\rho(\mu)=\int\rho(g)d\mu(g)$. Then $\|\rho(\mu)\|<1$. This
theorem was first written as an appendix to {\it Some harmonic
analysis questions suggested by Anderson-Bernoulli models} by C.
Shubin, R. Vakilian, and T.Wolff.

This theorem can be generalized to noncompact semi-simple groups.
Such groups are connected with products of random matrices; e.g.
the Anderson model on the strip with a single site distribution.
The main result asserts that if $\mu$ is a probability measure on
$G$ contained in $ GL(n,R)$ for which the Zariski closure of the
supp$\mu$ is big, then for suitable unitary representations of
$\rho$ of $G$, $\hat \mu (\rho)$ has norm less than 1. The key
point here being that supp$\mu$ is allowed to be small.

We will mention various corollaries and applications of this
theorem to the Anderson-Bernoulli models on the strip such as a
quantitative bound of the largest Lyapunov exponent for the
measure $\nu$  is  $> C^{-1}\lambda ^2/n$ where $n$ is the width of the
strip, $\lambda$ is the disorder parameter, $C$ a nonzero constant. Also
we obtain a refinement of Le Page's theorem on the Holder continuity of
the density of states for the Bernoulli model when $n=1$.

All three papers will appear in the Journal d'Analyse Mathematique
Thomas Wolff Memorial Issue. Many thanks to Editor Larry Zalcman.