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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universal finite-size scaling functions for percolation on three-dimensional lattices

Chai‐Yu Lin, Chin‐Kun Hu

Published Aug 1, 1998• 50 citations
Publisher

Abstract

Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability ${E}_{p},$ the percolation probability $P,$ and the probability ${W}_{n}$ for the appearance of $n$ percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability ${E}_{p}$ and the percolation probability $P$ for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of ${E}_{p}$ and $P$ in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that ${E}_{p}$ and $P$ have universal finite-size scaling functions. This implies that the critical ${E}_{p}$ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that ${W}_{n}$ for site and bond percolation on sc lattices have universal finite-size scaling functions.

DOI
10.1103/physreve.58.1521

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Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep,{E}_{p},Ep​, the percolation probability P,P,P, and the probability Wn{W}_{n}Wn​ for the appearance of nnn percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L.L.L. Using the HMCSM, we calculate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep{E}_{p}Ep​ and PPP in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep{E}_{p}Ep​ and PPP have universal finite-size scaling functions. This implies that the critical Ep{E}_{p}Ep​ is a universal quantity, which is 0.265\ifmmode±\else\textpm\fi0.0050.265\ifmmode\pm\else\textpm\fi{}0.0050.265\ifmmode±\else\textpm\fi0.005 for free boundary conditions and 0.924\ifmmode±\else\textpm\fi0.0050.924\ifmmode\pm\else\textpm\fi{}0.0050.924\ifmmode±\else\textpm\fi0.005 for periodic boundary conditions. We also find that Wn{W}_{n}Wn​ for site and bond percolation on sc lattices have universal finite-size scaling functions.
In this paper, we briefly review our recent results in the calculation of universal scaling functions for site and bond percolation on finite square, plane triangular, and honeycomb lattices.We find that, by choosing an aspect ratio for each lattice and a very small number of nonuniversal metric factors, all scaled data of the existance probability E p and the percolation probability P fallon the same universal scaling functions.We also find that free and periodic boundary conditions share the same nonuniversal metric factors.When the aspect ratio of each lattice is reduced by the same factor, the nonouniversal metric factors remain the same.The probabilities for the "appearance of n, n = 1, 2, 3, ..., percolating clusters for bond and site percolations on various planar lattices also have universal scaling functions.The implications of such results on numerical and experimental studies of critical phenomena are discussed.
A histogram Monte Carlo method is used to evaluate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP of bond and site percolation on finite square, plane triangular, and honeycomb lattices. We find that, by choosing a very small number of nonuniversal metric factors, all scaled data of Ep{E}_{p}Ep​ and PPP may fall on the same universal scaling functions. We also find that free and periodic boundary conditions share the same nonuniversal metric factors. This study may be extended to many critical systems.
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512L=512L=512, and estimate the percolation thresholds to be pc(bond)=0.24881182(10){p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)pc​(bond)=0.24881182(10) and pc(site)=0.3116077(2){p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)pc​(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent dmin=1.3756(3){d}_{\mathrm{min}}=1.3756(3)dmin​=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
We use a histogram Monte Carlo simulation method to calculate the scaling functions of the existence probability E p and the percolation probability P of the site percolation model on square lattices with free and periodic boundary conditions. We find that different boundary conditions give quite different scaling functions near the critical region. However, they give the consistent critical point, critical exponents, and the thermodynamic order parameter from renormalization-group calculations. Similar results are found for other percolation models. The implications of our calculated results for some theoretical problems of current interest are discussed.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.