  We study the dynamical exponents $d_{w}$ and $d_{s}$ for a particle diffusing
in a disordered medium (modeled by a percolation cluster), from the regime of
extreme disorder (i.e., when the percolation cluster is a fractal at $p=p_{c}$)
to the Lorentz gas regime when the cluster has weak disorder at $p>p_{c}$ and
the leading behavior is standard diffusion. A new technique of relating the
velocity autocorrelation function and the return to the starting point
probability to the asymptotic spectral properties of the hopping transition
probability matrix of the diffusing particle is used, and the latter is
numerically analyzed using the Arnoldi-Saad algorithm. We also present evidence
for a new scaling relation for the second largest eigenvalue in terms of the
size of the cluster, $|\ln{\lambda}_{max}|\sim S^{-d_w/d_f}$, which provides a
very efficient and accurate method of extracting the spectral dimension $d_s$
where $d_s=2d_f/d_w$.
