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

arXiv:cond-mat/9610088 · Journal of Physics A Mathematical and General

Correlated percolation and the correlated resistor network

Paul J. M. Bastiaansen, H. J. F. Knops

Published Mar 21, 1997• 14 citations
arXivHTMLPDF

Abstract

We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest-neighbour case, which implies that the percolation threshold, , is still equal to the Ising critical temperature . In addition, we present Monte Carlo calculations of the finite size behaviour of the correlated resistor network defined on the Ising model. The thermal exponent, t, of the conductivity that follows from it is found to be . We observe no corrections to scaling in its finite size behaviour.

DOI
10.1088/0305-4470/30/6/007

Related papers

Percolation, clusters, and phase transitions in spin models

Chin‐Kun Hu

The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

Physical review. B, Condensed matter•May 1, 1984•146 citations

Electrical conductivity of finite-size percolation networks

Catalin D Mitescu, Mickaël Allain, E. Guyon +1 more

Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (4*4 to 50*50) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.

Journal of Physics A Mathematical and General•Aug 1, 1982•39 citations

Critical exponent for 3-D percolation conductivity, revisited

Catalin D Mitescu, M. J. Musolf

When we apply finite-size-scaling analysis to Monte-Carlo calculations of the electrical conductivity of simple-cubic, bond- and site-percolation lattices, we obtain a correlation-length critical exponent in good agreement with current values but, for the conductivity exponent, we find t = 2.06 ± 0.16, significantly higher than the value currently cited. This new value appears, however, in essential agreement with a recently proposed theoretical conjecture.

Journal de Physique Lettres•Jan 1, 1983•49 citations

Study of droplets for correlated site-bond percolation in two dimensions

Naeem Jan, Antonio Coniglio A, Dietrich Stauffer

The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.

Journal of Physics A Mathematical and General•Dec 1, 1982•16 citations

Noise exponents of the random resistor network

Youngah Park, A. Brooks Harris, T. C. Lubensky

We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold ${p}_{c}$. We introduce a n\ifmmode\times\else\texttimes\fi{}m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ${]}_{\mathrm{av}}$ and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { ${\mathrm{}}}_{f}$. We evaluate a family of exponents {${\ensuremath{\psi}}_{l}$} (l=2,3,. . .) whose values are 1+O(\ensuremath{\epsilon}) with \ensuremath{\epsilon}=6-d where d is the spatial dimensionality. Each ${\ensuremath{\psi}}_{l}$ governs the critical behavior of the lth cumulant of the resistance between the sites x,x' conditionally averaged subject to the sites being in the same cluster such that ${C}_{R}$\ifmmode\bar\else\textasciimacron\fi{} $^{(l)}$(x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near ${p}_{c}$, where ${\ensuremath{\nu}}_{p}$ is the correlation-length exponent for percolation. Furthermore, ${\ensuremath{\psi}}_{2}$=1+\ensuremath{\epsilon}/105 determines the dependence of the variance of the resistance in a finite network on size L as . .AE

Physical review. B, Condensed matter•Apr 1, 1987•44 citations

The Two-Dimensional Random Bond Ising Model at Criticality—A Monte Carlo Study

Jian‐Sheng Wang, Walter Selke, Vl.S. Dotsenko +1 more

The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.

Europhysics Letters (EPL)•Feb 15, 1990•74 citations
We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest-neighbour case, which implies that the percolation threshold, , is still equal to the Ising critical temperature . In addition, we present Monte Carlo calculations of the finite size behaviour of the correlated resistor network defined on the Ising model. The thermal exponent, t, of the conductivity that follows from it is found to be . We observe no corrections to scaling in its finite size behaviour.
The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability ppp depending on the NN coupling constant JJJ and the temperature TTT, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.
Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (4*4 to 50*50) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.
The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.
We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold pc{p}_{c}pc​. We introduce a n\ifmmode\times\else\texttimes\fi{}m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ]av{]}_{\mathrm{av}}]av​ and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { ${\mathrm{}}}_{f}$. We evaluate a family of exponents {${\ensuremath{\psi}}_{l}$} (l=2,3,. . .) whose values are 1+O(\ensuremath{\epsilon}) with \ensuremath{\epsilon}=6-d where d is the spatial dimensionality. Each ${\ensuremath{\psi}}_{l}$ governs the critical behavior of the lth cumulant of the resistance between the sites x,x' conditionally averaged subject to the sites being in the same cluster such that CR{C}_{R}CR​\ifmmode\bar\else\textasciimacron\fi{} (l)^{(l)}(l)(x,x')\ensuremath{\sim}\ensuremath{\Vert}x- for p near pc{p}_{c}pc​, where ${\ensuremath{\nu}}_{p}$ is the correlation-length exponent for percolation. Furthermore, ${\ensuremath{\psi}}_{2}$=1+\ensuremath{\epsilon}/105 determines the dependence of the variance of the resistance in a finite network on size L as . .AE
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.