Transport on the percolation backbone

Abstract

We investigated numerically the number of sites visited ${\mathit{S}}_{\mathit{N}}$ by random walks on the backbone structure of the percolation cluster at the critical threshold. This quantity can be predicted by the scaling conjecture in terms of the fractal and the random-walk dimensions (${\mathit{d}}_{\mathit{f}}$ and ${\mathit{d}}_{\mathit{w}}$). Our results confirm this scaling with time, similarly to the critical cluster. The scaling exponent (spectral dimension) is numerically calculated, and it is found to be ${\mathit{d}}_{\mathit{s}}^{\mathrm{BB}}$=1.23, while the scaling conjecture predicts a value of 1.19, suggesting that there are uncertainties in the ${\mathit{d}}_{\mathit{f}}^{\mathrm{BB}}$ and ${\mathit{d}}_{\mathit{w}}^{\mathrm{BB}}$ values. This value is also smaller (by about 5%) than ${\mathit{d}}_{\mathit{s}}$, the spectral dimension on the full percolation cluster, suggesting that the walk is less efficient on the backbone. Previous estimates of the ${\mathit{d}}_{\mathit{w}}^{\mathrm{BB}}$ suggested that the walk should be more efficient on the backbone. We investigate this apparent contradiction by calculating and comparing the full distributions of ${\mathit{S}}_{\mathit{N}}$ for the backbone and the full percolating cluster. We investigated a few higher moments of this quantity and we found that they exhibit constant-gap scaling, similar to the percolation cluster. The backbone considerations help our understanding of the diffusion on the percolation cluster, especially the contribution of the dangling ends and the ramified parts of the structure, which are so characteristic of percolation at criticality.