Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters

Details Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters Universal scaling functions for bond percolation on planar random and square lattices with multiple percolating clusters

2001-01-09

arxiv.org/abs/cond-mat/0101112v2

Physics

Condensed Matter

Statistical Mechanics

15 pages, 11 figures

Scientific paper

10.1103/PhysRevE.64.016127

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, $_{n}$ ($_{n}$), 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 $W_{n}$, $P$, $_{n}$ ($_{n}$), 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.

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