Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ₂, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work Wang et al. , Phys. Rev. E 105, 034118 (2022) 2470-004510. 1103/PhysRevE. 105. 034118 found symmetric properties of χ₂ (n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ₂ behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i. e. , the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ₂ for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ₂ (n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.