  Motivated by recent experimental proposals to search for extragalactic cosmic
rays (including anti-matter from distant galaxies), we study particle
propagation through the intergalactic medium (IGM). We first use estimates of
the magnetic field strength between galaxies to constrain the mean free path
for diffusion of particles through the IGM. We then develop a simple analytic
model to describe the diffusion of cosmic rays. Given the current age of
galaxies, our results indicate that, in reasonable models, a completely
negligible number of particles can enter our Galaxy from distances greater than
$\sim 100$ Mpc for relatively low energies ($E$ $< 10^6$ GeV/n). We also find
that particle destruction in galaxies along the diffusion path produces an
exponential suppression of the possible flux of extragalactic cosmic rays.
Finally, we use gamma ray constraints to argue that the distance to any
hypothetical domains of anti-matter must be roughly comparable to the horizon
scale.
