Diffusion in a Medium with a Random Distribution of Static Traps

Particles diffusing in $d$-dimensional space among a random distribution of stationary spherical traps are considered. Given a particle at the origin at time $t=0\mathrm{}$, it is shown that the density of particles at the origin as $t\to \infty$ must decay at least as fast as $\exp [-t^{\frac{d}{\mathrm{}(d+2\mathrm{})\mathrm{}}}]$. The density here is obtained by averaging the diffusive field for a given configuration of traps over all configurations. The present upper bound coincides with the lower bound recently derived by Grassberger and Procaccia.