Journal of Statistical Physics

Histogram Monte Carlo position-space renormalization group: Applications to the site percolation

Chin‐Kun Hu, Chi-Ning Chen, Fa Yueh Wu

Published Feb 1, 1996• 13 citations

Publisher

DOI: 10.1007/bf02179808

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Journal of Statistical Physics

Histogram Monte Carlo position-space renormalization group: Applications to the site percolation

Chin‐Kun Hu, Chi-Ning Chen, Fa Yueh Wu

Published Feb 1, 1996• 13 citations

Publisher

DOI: 10.1007/bf02179808

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We present a histogram Monte Carlo method to calculate the existence probability ${\mathit{E}}_{\mathit{p}}$ , the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use ${\mathit{E}}_{\mathit{p}}$ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.We present a histogram Monte Carlo method to calculate the existence probability ${\mathit{E}}_{\mathit{p}}$, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use ${\mathit{E}}_{\mathit{p}}$ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.

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Chai‐Yu Lin, Chin‐Kun Hu

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.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.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 1, 199850 citations

We present a histogram Monte Carlo method to calculate the existence probability

{\mathit{E}}_{\mathit{p}}

, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use

{\mathit{E}}_{\mathit{p}}

and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.

{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.