Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Modern Physics Letters B

ON LOGARITHMIC CORRECTIONS IN TWO-DIMENSIONAL PERCOLATION

Matteo Marsili A, Giancarlo Jug

Published Nov 10, 1993• 0 citations
Publisher

Abstract

The possibility of unusual leading logarithmic corrections to the asymptotic behavior of the percolation connectedness length ξ in two dimensions is explored through a finite-size transfer-matrix analysis on strips of widths L≤12. It is found that, for both square-site and triangular-site percolation problems, no such corrections arise and the accepted exact value of the critical exponent ν is recovered.

DOI
10.1142/s0217984993001697

Related papers

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

Joan Adler, Vladimir Privman A

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 .

Journal of Physics A Mathematical and General•Nov 1, 1981•29 citations

Percolation transitions in two dimensions

Xiaomei Feng, Youjin Deng, Henk W. J. Blöte

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

arXiv:0901.1370•Physical Review E•Sep 30, 2008•142 citations

On the surface properties of two-dimensional percolation clusters

S L A de Queiroz

The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, eta s =0.6664+or-0.0008, consistent with the analytical value eta s =2/3. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value y s =-1 for the irrelevant exponent.

arXiv:cond-mat/9506005•Journal of Physics A Mathematical and General•Jul 7, 1995•4 citations

Transfer-matrix approach to percolation and phenomenological renormalization

Bernard Derrida, J. Vannimenus

A transfer-matrix method is used to calculate the correlation length for strips of finite width in the bond and site percolation problem. From the knowledge of these correlation lengths we compute the thresholds and the critical exponent ν by the phenomenological renormalization method.

Journal de Physique Lettres•Jan 1, 1980•68 citations

Transfer matrix and Monte Carlo tests of critical exponents in lattice models

J. Kaupužs

The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.

arXiv:cond-mat/0201221•Jan 14, 2002•0 citations

Finite-size effects in one-dimensional percolation: a verification of scaling theory

Stanley Greenspoon

The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.

Canadian Journal of Physics•Apr 1, 1979•3 citations
The possibility of unusual leading logarithmic corrections to the asymptotic behavior of the percolation connectedness length ξ in two dimensions is explored through a finite-size transfer-matrix analysis on strips of widths L≤12. It is found that, for both square-site and triangular-site percolation problems, no such corrections arise and the accepted exact value of the critical exponent ν is recovered.
We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.
The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, eta s =0.6664+or-0.0008, consistent with the analytical value eta s =2/3. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value y s =-1 for the irrelevant exponent.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.