Percolation in semicontinuum geometries

We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting semicontinuum problem describes the percolation of overlapping shapes in parallel layers or lanes with positional constraints for the placement of the objects along the discrete directions. Several semicontinuum percolation systems are analyzed like hypercuboids with a particular focus on 2D and 3D cases, disks, and parallelograms. Adapting the excluded volume arguments to the semicontinuum setting, we show that for the semicontinuum problem of hypercuboids, for fixed side-lengths of the hypercuboids along the directions in which a lattice structure is maintained, the percolation threshold is always independent of the side-lengths along the continuum directions. The result holds even when there is a distribution for the side-lengths along the continuum directions. Trends in the variation of the thresholds, as we vary the linear measure of the shapes along the continuum directions, are obtained for other semicontinuum models like disks and parallelograms in 2D. The results are compared with those of corresponding continuum and lattice models. For the 2D and 3D models considered, using Monte Carlo simulations, we verify the excluded volume predictions for the trends and numerical values of the percolation thresholds. Very good agreement is seen between the predicted numerical values and the simulation results. The semicontinuum setting also allows us to establish a connection between the percolation problem of overlapping shapes in 2D continuum and triangular lattice. We also verify that the isotropy of the threshold for anisotropic shapes and standard percolation universality class is maintained in the semicontinuum setting.