Physical Review Letters

Universal Scaling Functions in Critical Phenomena

Chin‐Kun Hu, Chai‐Yu Lin, Jau-Ann Chen

Published Jul 10, 1995• 109 citations

Abstract

A histogram Monte Carlo method is used to evaluate the existence probability ${E}_{p}$ and the percolation probability $P$ 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 ${E}_{p}$ and $P$ 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.

DOI: 10.1103/physrevlett.75.193

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Physical Review Letters

Universal Scaling Functions in Critical Phenomena

Chin‐Kun Hu, Chai‐Yu Lin, Jau-Ann Chen

Published Jul 10, 1995• 109 citations

Publisher

Abstract

A histogram Monte Carlo method is used to evaluate the existence probability

{E}_{p}

and the percolation probability

P

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

{E}_{p}

and

P

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.

DOI: 10.1103/physrevlett.75.193

Related papers

Chin‐Kun Hu, Chai‐Yu Lin, Jau-Ann Chen

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arXiv:cond-mat/0303166Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 26, 200338 citations

A histogram Monte Carlo method is used to evaluate the existence probability

{E}_{p}

and the percolation probability

P

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

{E}_{p}

and

P

{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.

We use heat bath dynamics to evaluate the dynamic critical exponent

z

and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that

z

is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research.