Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics.

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension d_{s}≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension d_{a}<d and are characterized by the absorption rate w_{a} which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (w_{a}≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1-(d-d_{a})/d_{w}, where d_{w} is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f(t)∼t^{-α} with the same exponent α as for the stretched exponential regime. For strong absorption w_{a}≳w, including the limit of perfect traps w_{a}→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.