Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

TL;DR

This paper studies how shape anisotropy affects percolation thresholds of aligned, overlapping objects on lattices, revealing complex dependencies on shape dimensions and introducing a semi-continuum percolation model.

Contribution

It introduces a lattice-based excluded volume theory for anisotropic shapes and validates it with Monte Carlo simulations, highlighting the impact of shape anisotropy on percolation behavior.

Findings

Percolation threshold decreases with length for stick-like rectangles.

Threshold is independent of length for rectangles of width two.

The semi-continuum limit depends on shape width and diverging dimensions.

Abstract

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape - rectangle - on a square lattice and show that for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This 'continuum' limit with the lattice spacing tending to zero only along a subset of the possible directions in d-dimensions results in a novel 'semi-continuum' percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. Our results show that shape-anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units are made to diverge.