Distance traveled by random walkers before absorption in a random medium

Abstract

We consider the penetration length $l$ of random walkers diffusing in a medium of perfect or imperfect absorbers of number density $\rho$. We solve this problem on a lattice and in the continuum in all dimensions $D$, by means of a mean-field renormalization group. For a homogeneous system in $D>2$, we find that $l\sim \max (\xi ,{\rho }^{-1/2})$, where $\xi$ is the absorber density correlation length. The cases of D=1 and D=2 are also treated. In the presence of long-range correlations, we estimate the temporal decay of the density of random walkers not yet absorbed. These results are illustrated by exactly solvable toy models, and extensive numerical simulations on directed percolation, where the absorbers are the active sites. Finally, we discuss the implications of our results for diffusion limited aggregation (DLA), and we propose a more effective method to measure $l$ in DLA clusters.

Final version: also considers the case of imperfect absorbers

References in corpus (6)

• Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States
• Nontrivial Exponent for Simple Diffusion
• Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes
• Off-lattice noise reduction and the ultimate scaling of DLA in two dimensions
• Perturbation theory for the effective diffusion constant in a medium of random scatterer
• Localization of Growth Sites in DLA Clusters: Multifractality and Multiscaling

Cited by in corpus (2)

• Sinai model in presence of dilute absorbers
• Breaking supersymmetry in a one-dimensional random Hamiltonian