Distance traveled by random walkers before absorption in a random medium

Details Distance traveled by random walkers before absorption in a random medium Distance traveled by random walkers before absorption in a random medium

2006-04-19

arxiv.org/abs/cond-mat/0604456v3

Phys. Rev. E 73, 066130 (2006)

Physics

Condensed Matter

Statistical Mechanics

Final version: also considers the case of imperfect absorbers

Scientific paper

10.1103/PhysRevE.73.066130

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.

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