On the quality of random number generators with taps

Recent exact analytical results developed for the random number generators with taps are reported. These results are applicable to a wide class of algorithms, including random walks, cluster algorithms, Ising models. Practical considerations on the improvement of the quality of random numbers are discussed as well.Comment: Conference on Computational Physics, Granada, Spain, 199

Critical amplitude ratios of the Baxter-Wu model

A Monte Carlo simulation study of the critical and off-critical behavior of the Baxter-Wu model, which belongs to the universality class of the 4-state Potts model, was performed. We estimate the critical temperature window using known analytical results for the specific heat and magnetization. This helps us to extract reliable values of universal combinations of critical amplitudes with reasonable accuracy. Comparisons with approximate analytical predictions and other numerical results are discussed.Comment: 13 pages, 13 figure

Test of multiscaling in DLA model using an off-lattice killing-free algorithm

We test the multiscaling issue of DLA clusters using a modified algorithm. This algorithm eliminates killing the particles at the death circle. Instead, we return them to the birth circle at a random relative angle taken from the evaluated distribution. In addition, we use a two-level hierarchical memory model that allows using large steps in conjunction with an off-lattice realization of the model. Our algorithm still seems to stay in the framework of the original DLA model. We present an accurate estimate of the fractal dimensions based on the data for a hundred clusters with 50 million particles each. We find that multiscaling cannot be ruled out. We also find that the fractal dimension is a weak self-averaging quantity. In addition, the fractal dimension, if calculated using the harmonic measure, is a nonmonotonic function of the cluster radius. We argue that the controversies in the data interpretation can be due to the weak self-averaging and the influence of intrinsic noise.Comment: 8 pages, 9 figure

Numerical revision of the universal amplitude ratios for the two-dimensional 4-state Potts model

Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios are tested. A direct calculation of the effective ratio of the energy amplitudes using duality relations explicitly demonstrates this cancelation of logarithms, thus supporting the predictions of RG. We emphasize the role of corrections of background terms on the determination of the amplitudes. The ratios of the critical amplitudes of the susceptibilities obtained in our analysis differ significantly from those predicted theoretically and supported by earlier SE and MC analysis. This disagreement might signal that the two-kink approximation used in the analytical estimates is not sufficient to describe with fair accuracy the amplitudes of the 4-state model.Comment: 32 pages, 9 figures, 11 table

Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold $p_c=1/2$. It is found that the probability of k and more Incipient Spanning Clusters (ISC) has the values $P(k>1) \approx 0.00658(3)$ and $P(k>2) \approx 0.00000148(21)$ provided that the limit of these probabilities for infinite lattices exists. The probability $P(k>3)$ of more than three ISC could be estimated to be of the order of 10^{-11} and is beyond the possibility to compute a such value by nowdays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when $k>>1$. We have detected a single sample with 4 ISC in a total number of about 10^{10} samples investigated. The probability of single event is 1/10 for that number of samples.Comment: 7 pages, 1 table, 5 figures (1PS+4*Latex),uses epsf.sty Int.J.Mod.Phys. C (submitted to