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Journal of Computational Physics

Long range correlations in linear congruential generators

Ora E. Percus, Jerome K. Percus

Published Jul 1, 1988• 10 citations

Abstract

We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen–Wang algorithm for the three-dimensional Ising model at the critical point. For the dynamic critical exponents associated to the integrated autocorrelation times of the “energy-like” observables, we find zint,N=zint,E=zint,E′=0.459±0.005±0.025, where the first error bar represents statistical error (68% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68% subjective confidence interval). For the “susceptibility-like” observables, we find zint,M2=zint,S2=0.443±0.005±0.030. For the dynamic critical exponent associated to the exponential autocorrelation time, we find zexp≈0.481. Our data are consistent with the Coddington–Baillie conjecture zSW=β/ν≈0.5183, especially if it is interpreted as referring to zexp.

DOI: 10.1016/0021-9991(88)90166-0

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We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen–Wang algorithm for the three-dimensional Ising model at the critical point. For the dynamic critical exponents associated to the integrated autocorrelation times of the “energy-like” observables, we find zint,N=zint,E=zint,E′=0.459±0.005±0.025, where the first error bar represents statistical error (68% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68% subjective confidence interval). For the “susceptibility-like” observables, we find zint,M2=zint,S2=0.443±0.005±0.030. For the dynamic critical exponent associated to the exponential autocorrelation time, we find zexp≈0.481. Our data are consistent with the Coddington–Baillie conjecture zSW=β/ν≈0.5183, especially if it is interpreted as referring to zexp.

We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2, 3 by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep. In some cases the dynamic critical exponent for the integrated autocorrelation time is negative. We also show that the dynamic critical exponent zexp is very close (possibly equal) to the rigorous lower bound alpha/nu and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.

We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.

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.