Percolation in two-dimensional, macroscopically anisotropic systems

Measurements of the dependence of conductivity on area fraction has been made on macroscopically anisotropic, two-dimensional systems. The systems consist of squares of anisotropic conducting paper into which circular, square, or elliptical holes are punched. Measurements of both principal conductivities were made on each sample. The percolation threshold for the circles and ellipses is 0.4, that for the squares 0.6. The latter difference appears to be due to a difference in coordination number between the squares and the other systems. The ratio of principal conductivities appears to agree with the predictions of Shklovskii (1978) but, because of large statistical fluctuations, this conclusion is tentative. The data on all samples can be fitted by an expression of the form sigma varies as (p-p c ) t . Because of statistical fluctuation the fit is poor for p near p c , but is good for larger values of p up to p=1. Values of t range from 1.1 to 1.6. For the circles and squares both principal conductivities can be fitted with the same value of t, but for the ellipses the two principal conductivities have different exponents.