In recent years, studies of diffusion in random media have been
extended to include the effects of media in which the defects
fluctuate randomly in time.  Typically, the diffusive motion of
particles in a static medium persists when the medium is allowed to
fluctuate, with the diffusivity (diffusion constant) $D$ depending on
the character of the fluctuations.  In the present work, we study
random walks on lattices in which the bonds connecting vertices open
and close randomly in time, and the walker is not allowed to cross a
closed bond.  Variations of the model studied here have been used to
model the diffusion of CO through myoglobin, the transport of ions in
polymer solutions, and conduction in hydrogenated amorphous silicon.
The major objective in analyzing these systems is to find efficient
methods for computing the diffusivity.  In this dissertation, we focus
mainly on methods of computing the diffusivity in our model. In
addition, we study the critical behavior of the model and present a
demonstration, valid for a restricted range of model parameters, that
the distribution of the displacement converges in time to a Gaussian
with width $D$.

To compute the diffusivity, we use a numerical renormalization group
(RG) method, power series expansions in model parameters, and Monte
Carlo simulations.  We choose a model with two parameters
characterizing the bond fluctuations--- the time scale of fluctuations
$\tau$ and the mean open-bond density $p$.  We calculate a series
expansion of the diffusivity to about $10$th order in the parameter
$\nu=\exp(-1/\tau)$ on the hypercubic lattice $\Zd$ for $d=1,2,3$, as
well as on the Bethe lattice.  We compute the same power series
expansion to $3$rd order in $\nu$ for arbitrary $d$.  We compute
estimates of the diffusivity on the Bethe lattice using the RG methods
and show by comparison to Monte Carlo data that the RG provides
excellent quantitative predictions of $D$ when $\tau$ is not too
large.