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

Journal of Physics A Mathematical and General

Test of universality for Ising-correlated site percolation

Naeem Jan, Dietrich Stauffer

Published Dec 1, 1982• 4 citations
Publisher

Abstract

A universal amplitude ratio was determined from cluster numbers in the Ising model at twice the critical temperature. The authors' Monte Carlo data agree with those for random percolation (infinite temperature), in agreement with expectations from renormalisation group arguments, but in disagreement with earlier Monte Carlo simulations of Stoll and Domb (1979) on much smaller systems.

DOI
10.1088/0305-4470/15/12/009

Related papers

Coarse graining, Monte Carlo renormalisation, percolation threshold and critical temperature in the Ising model

Dietrich Stauffer

Finite-size scaling suggests that the super-spins corresponding to large cells in a Monte Carlo renormalisation scheme form at T=T c and infinite network of connected down spins. The finite-size scaling assumption, M b =f((T-T c )b 1 nu /) for the renormalised block spin magnetisation is tested by computer simulation for lattice sizes 128 3 and 256 3 .

Journal of Physics A Mathematical and General•Dec 1, 1984•8 citations

Phenomenological renormalisation of Monte Carlo data for percolation

Muhammad Sahimi

The accuracy of a phenomenological renormalisation which is based on Monte Carlo data is tested by investigating site percolation in a simple cubic lattice. The method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes. The site percolation threshold of the lattice is predicted to be 0.3115+or-0.0005. If nu is the exponent of correlation length, the author finds beta / nu approximately=0.48+or-0.01 for the exponent of percolation probability and beta B / nu approximately=1.1+or-0.03 for the backbone exponent, in agreement with most accurate data currently available. An analysis of the data at the percolation threshold based on the standard finite-size scaling method also supports the results.

Journal of Physics A Mathematical and General•Dec 21, 1985•12 citations

Criticality of the random-site Ising model: Metropolis, Swendsen-Wang and Wolff Monte Carlo algorithms

Dmytro Ivaneyko, Jaroslav Ilnytskyi, Bertrand Berche +1 more

We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio $Γ^+/Γ^-$ for the magnetic susceptibility critical amplitudes. Our estimate $Γ^+/Γ^- = 1.67 \pm 0.15$ is in a good agreement with the recent MC analysis of the random-bond Ising model giving further support that both random-site and random-bond dilutions lead to the same universality class.

arXiv:cond-mat/0501291•Condensed Matter Physics•Jan 1, 2005•18 citations

Critical droplets and phase transitions in two dimensions

Santo Fortunato

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.

arXiv:cond-mat/0205211•Physical review. B, Condensed matter•Jan 15, 2003•16 citations

Single-cluster Monte Carlo study of the Ising model on two-dimensional random lattices

Wolfhard Janke E, Mohammad W. Katoot, Ramon Villanova

We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.

arXiv:hep-lat/9402008•Physical review. B, Condensed matter•Apr 1, 1994•60 citations

Critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model

Щур Лев Николаевич, Oleg A. Vasilyev

We present a different way of probing the universality class of the site-diluted two-dimensional Ising model. We analyze Monte Carlo data for the magnetic susceptibility, introducing a fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude, and the sample-dependent pseudocritical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration q of the empty sites for all investigated values of q < or =0.25. At the same time the average effective exponent gamma(eff) is found to vary with the concentration q, which may be argued to be due to logarithmic corrections to the power law of the pure system. These corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.

arXiv:cond-mat/0108369•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Dec 14, 2001•22 citations
A universal amplitude ratio was determined from cluster numbers in the Ising model at twice the critical temperature. The authors' Monte Carlo data agree with those for random percolation (infinite temperature), in agreement with expectations from renormalisation group arguments, but in disagreement with earlier Monte Carlo simulations of Stoll and Domb (1979) on much smaller systems.
The accuracy of a phenomenological renormalisation which is based on Monte Carlo data is tested by investigating site percolation in a simple cubic lattice. The method appears to be very accurate and can yield precise estimates of the quantities of interest with small to moderate lattice sizes. The site percolation threshold of the lattice is predicted to be 0.3115+or-0.0005. If nu is the exponent of correlation length, the author finds beta / nu approximately=0.48+or-0.01 for the exponent of percolation probability and beta B / nu approximately=1.1+or-0.03 for the backbone exponent, in agreement with most accurate data currently available. An analysis of the data at the percolation threshold based on the standard finite-size scaling method also supports the results.
We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio Γ+/Γ−Γ^+/Γ^-Γ+/Γ− for the magnetic susceptibility critical amplitudes. Our estimate Γ+/Γ−=1.67±0.15Γ^+/Γ^- = 1.67 \pm 0.15Γ+/Γ−=1.67±0.15 is in a good agreement with the recent MC analysis of the random-bond Ising model giving further support that both random-site and random-bond dilutions lead to the same universality class.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point Tc{T}_{c}Tc​ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability pB<1,{p}_{B}<1,pB​<1, the corresponding site-bond clusters keep on percolating at Tc{T}_{c}Tc​ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability pc{p}_{c}pc​ such that, for any pB> pc,{p}_{B}>~{p}_{c},pB​> pc​, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for pc<pB< 1{p}_{c}<{p}_{B}<~1pc​<pB​< 1 but, for pB=pc,{p}_{B}{=p}_{c},pB​=pc​, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.
We present a different way of probing the universality class of the site-diluted two-dimensional Ising model. We analyze Monte Carlo data for the magnetic susceptibility, introducing a fitting procedure in the critical region applicable even for a single sample with quenched disorder. This gives us the possibility to fit simultaneously the critical exponent, the critical amplitude, and the sample-dependent pseudocritical temperature. The critical amplitude ratio of the magnetic susceptibility is seen to be independent of the concentration q of the empty sites for all investigated values of q < or =0.25. At the same time the average effective exponent gamma(eff) is found to vary with the concentration q, which may be argued to be due to logarithmic corrections to the power law of the pure system. These corrections are canceled in the susceptibility amplitude ratio as predicted by theory. The central charge of the corresponding field theory was computed and compared well with the theoretical predictions.