Universal percolation-threshold limits in the continuum

Abstract

The existing criteria for the onset of percolation in the continuum are limited to systems of parallel-aligned, equal-size, penetrable, convex objects. We present here criteria for the much more general case of macroscopically isotropic or anisotropic systems in which the objects may also be of variable sizes and of random orientations. It is found that the critical fractional occupied area (or the equivalent average critical total excluded area 〈$A_{\mathrm{ex}\mathrm{〉}}$) as well as the critical fractional occupied volume (or the equivalent 〈$V_{\mathrm{ex}\mathrm{〉}}$) are confined within two limits. The upper limit is that of the ‘‘parallel objects’’ systems and the lower limit is that of the (newly introduced) ‘‘horizontal-vertical’’ system. These limits are 3.2≤〈$A_{\mathrm{ex}\mathrm{〉}\mathrm{\le }4.5}$ and 0.7≤〈$V_{\mathrm{ex}\mathrm{〉}\mathrm{\le }2.8}$. .AE