Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on $L_{ 1}\times L_{ 2}$ planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratio $L_{ 1}/L_{ 2}$. We calculate the probability for the appearance of $n$ percolating clusters, $W_{ n},$ the percolating probabilities, $P$, the average fraction of lattice bonds (sites) in the percolating clusters, $<c^{ b}>_{ n}$ ($<c^{ s}>_{ n}$), and the probability distribution function for the fraction $c$ of lattice bonds (sites), in percolating clusters of subgraphs with $n$ percolating clusters, $f_{ n}(c^{ b})$ ($f_{ n}(c^{ s})$). Using a small number of nonuniversal metric factors, we find that $W_{ n}$, $P$, $<c^{ b}>_{ n}$ ($<c^{ s}>_{ n}$), and $f_{ n}(c^{ b})$ ($f_{ n}(c^{ s})$) for random lattices, duals of random lattices, and square lattices have the same universal finite-size scaling functions. We also find that nonuniversal metric factors are independent of boundary conditions and aspect ratios.
Categories: cond-mat.stat-mech