Stratified continuum percolation: Scaling geometry of hierarchical cascades

Stratified percolation patterns result from hierarchical cascades with continuous overlap. The pattern construction is a hybrid process, combining continuum percolation with random curdling. Stratified percolation is a correlated percolation with fractal dimensions that can be continuously tuned. The percolation threshold is found to vary with changing correlations. The density of percolating sites is approximately log-normal and can be described by a multifractal. Trends in the f(α) curves are studied as fractal dimensions are changed. The finite-size-scaling properties are investigated using Monte Carlo real-space renormalization. Because the patterns have features at all length scales, stratified percolation has intrinsically small-cell renormalization properties. However, the small-cell properties can be removed by transforming the occupation fraction into an equivalent occupancy for standard uncorrelated continuum percolation. The transformed threshold is approximately invariant with respect to changing correlations and is close to the standard value of ${\mathit{a}}_{\mathit{c}}$=0.7 for isotropic continuum systems. Stratified percolation has a correlation length exponent of ν=1.33±0.05.