Conductivity Exponent and Backbone Dimension in 2-d Percolation

TL;DR

This paper reports high-precision simulations of 2D percolation clusters, analyzing their backbone structure and electrical conductivity, revealing complex correction behaviors and challenging existing theoretical conjectures.

Contribution

It provides refined measurements of backbone fractal dimension and conductivity exponent, highlighting significant correction effects not captured by simple power laws.

Findings

Backbone fractal dimension D_b = 1.6432(8) for both site and bond percolation.

Conductivity exponent t'/ν = 0.9826(8), differing from conjectured values.

Strong non-monotonic corrections to scaling observed in conductivity measurements.

Abstract

We present high statistics simulations for 2-d percolation clusters in the "bus bar" geometry at the critical point, for site and for bond percolation. We measured their backbone sizes and electrical conductivities. For all sets of measurements we find large corrections to scaling, most of which do not seem to be described by single powers. Using single power terms for the corrections to scaling of the backbone masses, we would obtain fractal dimensions which are different for site and bond percolation, while the correct result is $D_b = 1.6432(8)$ for both. For the conductivity, the corrections to scaling are strongly non-monotonic for bond percolation. The exponent $t' = t/\nu$ is measured as 0.9826(8), in disagreement with the Alexander-Orbach and other conjectures.