Distance traveled by random walkers before absorption in a random medium

TL;DR

This paper analyzes how far random walkers travel before absorption in various media, using mean-field renormalization group techniques, and explores implications for diffusion-limited aggregation and related models.

Contribution

It provides a comprehensive solution for the penetration length in different dimensions and correlation scenarios, including new methods for measuring this length in DLA clusters.

Findings

Penetration length scales as max(ξ, ρ^{-1/2}) in homogeneous systems for D>2.

Derived decay estimates for walker density in correlated media.

Proposed improved measurement method for penetration length in DLA.

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.