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

Abstract

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 $d_s\le 2$, 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 ($w_a\ll w$), we find that $f(t)$ can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent $\alpha=1-(d-d_a)/d_w$, 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 $f(t)\sim t^{-\alpha}$ with the same exponent $\alpha$ as for the stretched exponential regime. For strong absorption $w_a>w$, including the limit of perfect traps $w_a\to \infty$, the stretched exponential regime is absent and the decay of $f(t)$ follows, after a short transient, the aforementioned power law for all times.