65 research outputs found
On the Aizenman exponent in critical percolation
The probabilities of clusters spanning a hypercube of dimensions two to seven
along one axis of a percolation system under criticality were investigated
numerically. We used a modified Hoshen--Kopelman algorithm combined with
Grassberger's "go with the winner" strategy for the site percolation. We
carried out a finite-size analysis of the data and found that the probabilities
confirm Aizenman's proposal of the multiplicity exponent for dimensions three
to five. A crossover to the mean-field behavior around the upper critical
dimension is also discussed.Comment: 5 pages, 4 figures, 4 table
The RANLUX generator: resonances in a random walk test
Using a recently proposed directed random walk test, we systematically
investigate the popular random number generator RANLUX developed by Luescher
and implemented by James. We confirm the good quality of this generator with
the recommended luxury level. At a smaller luxury level (for instance equal to
1) resonances are observed in the random walk test. We also find that the
lagged Fibonacci and Subtract-with-Carry recipes exhibit similar failures in
the random walk test. A revised analysis of the corresponding dynamical systems
leads to the observation of resonances in the eigenvalues of Jacobi matrix.Comment: 18 pages with 14 figures, Essential addings in the Abstract onl
Correction-to-scaling exponent for two-dimensional percolation
We show that the correction-to-scaling exponents in two-dimensional
percolation are bounded by Omega <= 72/91, omega = D Omega <= 3/2, and Delta_1
= nu omega <= 2, based upon Cardy's result for the critical crossing
probability on an annulus. The upper bounds are consistent with many previous
measurements of site percolation on square and triangular lattices, and new
measurements for bond percolation presented here, suggesting this result is
exact. A scaling form evidently applicable to site percolation is also found
Critical Binder cumulant in two-dimensional anisotropic Ising models
The Binder cumulant at the phase transition of Ising models on square
lattices with various ferromagnetic nearest and next-nearest neighbour
couplings is determined using mainly Monte Carlo techniques. We discuss the
possibility to relate the value of the critical cumulant in the isotropic,
nearest neighbour and in the anisotropic cases to each other by means of a
scale transformation in rectangular geometry, to pinpoint universal and
nonuniversal features.Comment: 7 pages, 4 figures, submitted to J. Phys.
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