Target annihilation by diffusing particles in inhomogeneous geometries

TL;DR

This paper analytically investigates the survival probability of targets annihilated by random walkers in inhomogeneous structures, revealing dependence on spectral dimension and target position, with explicit formulas for large-time behavior.

Contribution

It provides the first analytical expressions for survival probabilities in inhomogeneous geometries, linking asymptotic behavior to spectral dimension and target location.

Findings

Survival probability depends on spectral dimension for large times.

On recurrent structures, survival probability is site-independent.

On transient structures, survival probability strongly depends on target position.

Abstract

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 behaviour at large times can still be expressed in terms of the spectral dimension $\widetilde {d}$, and its exact analytical expression is given. The results show that the asymptotic survival probability is site independent on recurrent structures ($\widetilde{d}\leq2$), while on transient structures ($\widetilde{d}>2$) it can strongly depend on the target position, and such a dependence is explicitly calculated.