  A method is presented for proving upper bounds on the moments of the position
operator when the dynamics of quantum wavepackets is governed by a random
(possibly correlated) Jacobi matrix. As an application, one obtains sharp upper
bounds on the diffusion exponents for random polymer models, coinciding with
the lower bounds obtained in a prior work. The second application is an
elementary argument (not using multiscale analysis or the Aizenman-Molchanov
method) showing that under the condition of uniformly positive Lyapunov
exponents, the moments of the position operator grow at most logarithmically in
time.
