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

The European Physical Journal B

Precise determination of the conductivity exponent of 3D percolation using exact numerical renormalization

J. P. Clerc, Viktor A. Podolskiy, Andrey Karlovich Sarychev

Published May 1, 2000• 56 citations
Publisher
DOI
10.1007/s100510051153

Related papers

Precise calculation of the dynamical exponent of two-dimensional percolation

J. -M. Normand, Hans Jürgen Herrmann, M. Hajjar

Journal of Statistical Physics•Jul 1, 1988•100 citations

Critical exponents and percolation thresholds in two-dimensional systems with a finite interplane coupling

C. Thomsen

Classical site percolation was used to study numerically the effect of interplane coupling in the range 10(-1)-10(-6) of the in-plane coupling on the static correlation length exponent nu and the critical dimension D. It was found that even for the smallest coupling values the exponents take their three-dimensional (3D) values for sufficiently large system sizes. The percolation threshold p(c), however, varies continuously from the 2D to the 3D value with a power-law exponent kappa=0.41(2), which, to within error, is the same for simple cubic, bcc, and fcc lattices. As predicted by renormalization-group theory this exponent equals the inverse of the susceptibility exponent gamma=43/18.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 28, 2002•5 citations

Critical Exponents of Dynamic Percolation Universality Class: the Renormalization Group Approach

Michal Hnatič, Matej Kecer, T. Lučivjanský +3 more

In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.

Nonlinear Phenomena in Complex Systems•Dec 16, 2025•0 citations

Precision calculation of elasticity for percolation

John G. Zabolitzky, David J. Bergman, Dietrich Stauffer

Journal of Statistical Physics•Jul 1, 1986•94 citations

Potts-model formulation of the random resistor network

Amanda Harris, T. C. Lubensky

The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in \ensuremath{\epsilon}=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the \ensuremath{\epsilon} expansion.

Physical review. B, Condensed matter•May 1, 1987•15 citations

A real-space renormalization group study of correlated percolation

Bikas K. Chakrabarti, Kimmo K. Kaski, János Kertész

Physics Letters A•Mar 1, 1981•6 citations
Classical site percolation was used to study numerically the effect of interplane coupling in the range 10(-1)-10(-6) of the in-plane coupling on the static correlation length exponent nu and the critical dimension D. It was found that even for the smallest coupling values the exponents take their three-dimensional (3D) values for sufficiently large system sizes. The percolation threshold p(c), however, varies continuously from the 2D to the 3D value with a power-law exponent kappa=0.41(2), which, to within error, is the same for simple cubic, bcc, and fcc lattices. As predicted by renormalization-group theory this exponent equals the inverse of the susceptibility exponent gamma=43/18.
In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.
The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in \ensuremath{\epsilon}=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the \ensuremath{\epsilon} expansion.