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

LONGITUDINAL CORRELATION LENGTH IN DIRECTED PERCOLATION AND RELATED MODELS: A POSSIBLE NEW SCALING MECHANISM

Malte Henkel, Vladimir Privman A

Published Apr 10, 1991• 2 citations
Publisher

Abstract

The finite-size scaling of the correlation length for directed and undirected systems is reviewed with emphasis on the asymptotic eigenvalue degeneracy associated with long-range order or long-range connectivity. The standard scaling mechanism for matching asymptotic behaviors on approach to criticality applies for Ising and other models. However, numerical evidence suggests that for directed percolation in 2D a modified formulation is appropriate, incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling.

DOI
10.1142/s0217984991000666

Related papers

New mechanism for mass gap scaling and transfer-matrix study for (1+1)D directed percolation

Malte Henkel, Vladimir Privman A

The standard scaling mechanism for matching asymptotic behaviors on the approach to criticality fails for directed percolation, as indicated by transfer-matrix calculations for sizes up to N=15. A formulation is proposed incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling. Numerical data support these theoretical predictions and, among other results, yield a new estimate, \ensuremath{\lambda}=0.37\ifmmode\pm\else\textpm\fi{}0.04, for the leading irrelevant-variable exponent.

Physical Review Letters•Oct 1, 1990•14 citations

Finite-size scaling of directed percolation above the upper critical dimension

S. Lübeck, H. K. Janssen

We consider analytically as well as numerically the finite-size scaling behavior in the stationary state near the nonequilibrium phase transition of directed percolation within the mean field regime, i.e., above the upper critical dimension. Analogous to equilibrium, usual finite-size scaling is valid below the upper critical dimension, whereas it fails above. Performing a momentum analysis of associated path integrals we derive modified finite-size scaling forms of the order parameter and its higher moments. The results are confirmed by numerical simulations of corresponding high-dimensional lattice models.

arXiv:cond-mat/0505212•Physical Review E•Jul 19, 2005•14 citations

FINITE-SIZE SCALING FOR THE SPHERICAL MODEL WITH LONG-RANGE INTERACTION

K. Nojima

The finite-size scaling property of the correlation length for the spherical model with long-range interaction is examined above the critical temperature. The analysis is performed below the upper critical dimension.

Modern Physics Letters B•Aug 10, 1995•0 citations

Finite-size scaling for directed bond percolation with and without cycles on a triangular lattice

Anna Chame, S. L. A. de Queiroz, Raimundo R. dos Santos

The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.

Journal of Physics A Mathematical and General•Jun 21, 1986•0 citations

PERCOLATION AND FINITE SIZE SCALING IN SEVEN DIMENSIONS

Lotfi Zekri

Numerical investigation of critical exponents on a hypercubic lattice with Ld random sites with L up to 33 and d up to 7 showed that above the critical dimension the phase transitions in Ising model and percolation are not alike.

•Jan 1, 2004•1 citation

PERCOLATION AND FINITE SIZE SCALING IN SEVEN DIMENSIONS

LOTFI ZEKRI

Numerical investigation of critical exponents on a hypercubic lattice with Ld random sites with L up to 33 and d up to 7 showed that above the critical dimension the phase transitions in Ising model and percolation are not alike.

arXiv:cond-mat/0406322•International Journal of Modern Physics C•Nov 1, 2004•0 citations
The finite-size scaling of the correlation length for directed and undirected systems is reviewed with emphasis on the asymptotic eigenvalue degeneracy associated with long-range order or long-range connectivity. The standard scaling mechanism for matching asymptotic behaviors on approach to criticality applies for Ising and other models. However, numerical evidence suggests that for directed percolation in 2D a modified formulation is appropriate, incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling.
The standard scaling mechanism for matching asymptotic behaviors on the approach to criticality fails for directed percolation, as indicated by transfer-matrix calculations for sizes up to N=15. A formulation is proposed incorporating irrelevant-variable corrections in a new pattern of asymptotic scaling. Numerical data support these theoretical predictions and, among other results, yield a new estimate, \ensuremath{\lambda}=0.37\ifmmode\pm\else\textpm\fi{}0.04, for the leading irrelevant-variable exponent.
We consider analytically as well as numerically the finite-size scaling behavior in the stationary state near the nonequilibrium phase transition of directed percolation within the mean field regime, i.e., above the upper critical dimension. Analogous to equilibrium, usual finite-size scaling is valid below the upper critical dimension, whereas it fails above. Performing a momentum analysis of associated path integrals we derive modified finite-size scaling forms of the order parameter and its higher moments. The results are confirmed by numerical simulations of corresponding high-dimensional lattice models.
The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.