Target annihilation by diffusing particles in inhomogeneous geometries

The survival probability of immobile targets annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks, and gels, is analytically investigated. It is shown that, while it cannot in general be related to the number of distinct visited points as in the case of homogeneous lattices, in the case of bounded coordination numbers its asymptotic behavior at large times can still be expressed in terms of the spectral dimension $\tilde{d}$ and its exact analytical expression is given. The results show that the asymptotic survival probability is site-independent of recurrent structures $\left(\tilde{d}\le 2\right)$, while on transient structures $\left(\tilde{d}>2\right)$ it can strongly depend on the target position, and such dependence is explicitly calculated.