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 Statistical Mechanics Theory and Experiment

Critical exponents testing of a random number generator with the Wolff cluster algorithm

Dongjie Zhu, Yun Lin, Guoya Sun, Fuming Wang

Published Jun 17, 2024• 0 citations
Publisher

Abstract

Abstract Finite-size scaling (FSS) of critical exponents including γ, β and α of 2D Ising models of sizes up to 327682 are studied using the Wolff clustering algorithm and are used to assess the quality of pseudorandom number generators (PRNGs). Critical exponents of PRNGs with quality issues are found to diverge from their theoretical values at large lattice sizes, similar to previous reports that used the Metropolis algorithm to simulate the Ising lattice. Four high-quality PRNGs, including Mersenne Twister, an additive lagged Fibonacci generator, Xorshift and Xorwow are tested and assessed with their FSS behaviors. Dynamic exponent z is also used to assess the quality of the four tested PRNGs and corroborating results are obtained.

DOI
10.1088/1742-5468/ad4e28

Related papers

Critical dynamics of cluster algorithms in the random-bond Ising model

Ulvi Kanbur, Zeynep Demir Vatansever

In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.

Physical review. E•Feb 28, 2024•2 citations

Numerical simulation study of the dynamical behavior of the Niedermayer algorithm

Daniel Girardi, N. S. Branco

We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter E 0 . For E 0 = − 1 we regain the Metropolis algorithm and for E 0 = 1 we regain the Wolff algorithm. For − 1 < E 0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L , but eventually saturates at a given lattice size , which depends on E 0 . For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent. For E 0 > 1, the autocorrelation time is always greater than for E 0 = 1 (Wolff) and, more important, it also grows faster than a power of L . Therefore, we show that the best choice of cluster algorithm is the Wolff one, when comparing against the Niedermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusively, we propose a scaling law for the dependence of the autocorrelation time on L .

arXiv:1003.3655•Journal of Statistical Mechanics Theory and Experiment•Apr 16, 2010•2 citations

A New Measurement of Dynamic Critical exponent of Wolff Algorithm by Dynamic Finite Size Scaling

M. Dilaver, Semra Gündüç, Meral Aydın +1 more

In this work we have calculated the dynamic critical exponent $z$ for 2-, 3- and 4-dimensional Ising models using the Wolff's algorithm through dynamic finite size scaling. We have studied time evolution of the average cluster size, the magnetization and higher moments of the magnetization. It is observed that dynamic scaling is independent of the algorithm. In this sense, universality is established for a wide range of algorithms with their own dynamic critical exponents. For scaling, we have used the literature values of critical exponents to observe the dynamic finite size scaling and to obtain the value of $z$. From the simulation data a very good scaling is observed leading to vanishingly small $z$ values for all three dimensions.

arXiv:cond-mat/0409696•Sep 27, 2004•0 citations

Empirical relations between static and dynamic exponents for Ising model cluster algorithms

Paul David Coddington, CLIVE F. BAILLIE

We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{W}}$=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{SW}}$=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.

Physical Review Letters•Feb 17, 1992•45 citations

The critical region of the random-bond Ising model

A. L. Talapov, Щур Лев Николаевич

We describe results of the cluster algorithm special purpose processor simulations of the 2D Ising model with impurity bonds. We use large lattices with up to 10 6 spins to define critical temperature regions where both finite-size corrections and corrections to scaling are small. High-accuracy data unambiguously show an increase in the magnetization and magnetic susceptibility effective exponents beta and gamma , caused by impurities. The Mand chi -singularities become sharper, while the specific heat singularity is smoothed.

arXiv:hep-lat/9404011•Journal of Physics Condensed Matter•Oct 3, 1994•25 citations

CLUSTER SIMULATIONS OF THE ISING MODEL ON DYNAMICALLY TRIANGULATED LATTICE

Radel Ben-Av, J. Kinar, S. Solomon

We have performed high precision finite size scaling measurements of the critical behavior of the Ising model on dynamically triangulated surfaces of genus 0. The results are in perfect agreement with the matrix model predictions.1,2 Values of the critical exponents equal to the regular lattice 2D Ising model are excluded by our error bars. In order to accumulate sufficient statistics we used cluster methods which eliminated critical slowing down to a great extent.

International Journal of Modern Physics C•Apr 1, 1992•10 citations
Abstract Finite-size scaling (FSS) of critical exponents including γ, β and α of 2D Ising models of sizes up to 327682 are studied using the Wolff clustering algorithm and are used to assess the quality of pseudorandom number generators (PRNGs). Critical exponents of PRNGs with quality issues are found to diverge from their theoretical values at large lattice sizes, similar to previous reports that used the Metropolis algorithm to simulate the Ising lattice. Four high-quality PRNGs, including Mersenne Twister, an additive lagged Fibonacci generator, Xorshift and Xorwow are tested and assessed with their FSS behaviors. Dynamic exponent z is also used to assess the quality of the four tested PRNGs and corroborating results are obtained.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter E 0 . For E 0 = − 1 we regain the Metropolis algorithm and for E 0 = 1 we regain the Wolff algorithm. For − 1 < E 0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L , but eventually saturates at a given lattice size , which depends on E 0 . For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent. For E 0 > 1, the autocorrelation time is always greater than for E 0 = 1 (Wolff) and, more important, it also grows faster than a power of L . Therefore, we show that the best choice of cluster algorithm is the Wolff one, when comparing against the Niedermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusively, we propose a scaling law for the dependence of the autocorrelation time on L .
In this work we have calculated the dynamic critical exponent zzz for 2-, 3- and 4-dimensional Ising models using the Wolff's algorithm through dynamic finite size scaling. We have studied time evolution of the average cluster size, the magnetization and higher moments of the magnetization. It is observed that dynamic scaling is independent of the algorithm. In this sense, universality is established for a wide range of algorithms with their own dynamic critical exponents. For scaling, we have used the literature values of critical exponents to observe the dynamic finite size scaling and to obtain the value of zzz. From the simulation data a very good scaling is observed leading to vanishingly small zzz values for all three dimensions.
We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​EW{\mathit{E}}^{\mathrm{W}}EW=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​ESW{\mathit{E}}^{\mathrm{SW}}ESW=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.
We describe results of the cluster algorithm special purpose processor simulations of the 2D Ising model with impurity bonds. We use large lattices with up to 10 6 spins to define critical temperature regions where both finite-size corrections and corrections to scaling are small. High-accuracy data unambiguously show an increase in the magnetization and magnetic susceptibility effective exponents beta and gamma , caused by impurities. The Mand chi -singularities become sharper, while the specific heat singularity is smoothed.