Journal of Physics A Mathematical and General

Two-dimensional percolation: logarithmic corrections to the critical behaviour from series expansions

Joan Adler, Vladimir Privman A

Published Nov 1, 1981• 29 citations

Abstract

Analyses several extant series for the mean cluster size, the zeroth moment of the pair connectedness and the percolation probability for bond and site percolation on two-dimensional lattices with a view to detecting possible logarithmic corrections. The logarithmic correction exponent z is found to be in the range 0 - <or=z<or=0.15, and our analysis also provides some new information about the critical exponents gamma and beta .

DOI: 10.1088/0305-4470/14/11/008

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Journal of Physics A Mathematical and General

Two-dimensional percolation: logarithmic corrections to the critical behaviour from series expansions

Joan Adler, Vladimir Privman A

Published Nov 1, 1981• 29 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/14/11/008

Related papers

ON LOGARITHMIC CORRECTIONS IN TWO-DIMENSIONAL PERCOLATION

Matteo Marsili A, Giancarlo Jug

Modern Physics Letters BNov 10, 19930 citations

Series study of random percolation in three dimensions

David S. Gaunt, M F Sykes

New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p c , obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p c for both bond and site percolation on all four lattices. Using the bond estimates of p c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p c , obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p c for both bond and site percolation on all four lattices. Using the bond estimates of p c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.

Journal of Physics A Mathematical and GeneralMar 11, 1983130 citations

Measures of Critical Exponents in the Four Dimensional Site Percolation

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Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...

Jan 1, 199632 citations

Universal amplitude ratio Γ−∕Γ+ for two-dimensional percolation

Iwan Jensen, Robert M. Ziff

The amplitude ratio of the susceptibility (or second size moment) for two-dimensional percolation is calculated by two series methods and also by Monte Carlo simulation. The first series method is an approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value Gamma-/Gamma+ = 162.5+/-2 , a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.The amplitude ratio of the susceptibility (or second size moment) for two-dimensional percolation is calculated by two series methods and also by Monte Carlo simulation. The first series method is an approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value Gamma-/Gamma+ = 162.5+/-2 , a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.

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Bond and site percolation in three dimensions

Junfeng Wang, Zongzheng Zhou, Wei Zhang +2 more

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$ , and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$ . 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 ${d}_{\mathrm{min}}=1.3756(3)$ . 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}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$, and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent ${d}_{\mathrm{min}}=1.3756(3)$. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

arXiv:1302.0421Physical Review EMay 7, 2013170 citations

Directed percolation in 3+1 dimensions

Joan Adler, Jorge Berger, J. A. M. S. Duarté +1 more

Directed percolation in four dimensions is of direct physical relevance to the world with three space and one time dimension. We present a comprehensive analysis of recently extended series for the moments of the cluster-size distribution and for the percolation probability in a ``field'' on the hypercubic lattice. We find a critical threshold, ${p}_{c}$ =0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent ${\ensuremath{\Delta}}_{1}$=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.Directed percolation in four dimensions is of direct physical relevance to the world with three space and one time dimension. We present a comprehensive analysis of recently extended series for the moments of the cluster-size distribution and for the percolation probability in a ``field'' on the hypercubic lattice. We find a critical threshold, ${p}_{c}$=0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent ${\ensuremath{\Delta}}_{1}$=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.

Physical review. B, Condensed matterMay 1, 198813 citations

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to

L=512

, and estimate the percolation thresholds to be

{p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)

and

{p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(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

{d}_{\mathrm{min}}=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 ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

{p}_{c}

=0.3025\ifmmode\pm\else\textpm\fi{}0.0010, and dominant critical exponents \ensuremath{\gamma}=1.21\ifmmode\pm\else\textpm\fi{}0.05, for the mean cluster size; \ensuremath{\beta}=0.82\ifmmode\pm\else\textpm\fi{}0.03 and 1/\ensuremath{\delta}=0.45\ifmmode\pm\else\textpm\fi{}0.02 for the percolation probability in the thermal and field directions respectively; and a gap exponent of \ensuremath{\Delta}=2.03\ifmmode\pm\else\textpm\fi{}0.06. We find a thermal-correction exponent ${\ensuremath{\Delta}}_{1}$=0.55\ifmmode\pm\else\textpm\fi{}0.15 and a field correction of \ensuremath{\Omega}=0.3\ifmmode\pm\else\textpm\fi{}0.1. We also calculate some universal critical-amplitude ratios.