Narrow Results

Monte Carlo simulations alone could not clarify the corrections to scaling for the size-dependent p_c(L) above the upper critical dimension. Including the previous series estimate for the bulk threshold p_c (∞) gives preference for the complicated corrections predicted by renormalization group and against the simple 1/L extrapolation. Additional Monte-Carlo simulations using the Leath method corroborate the series result for p_c.

We study two-dimensional triangulated surfaces of sphere topology by the canonical Monte Carlo simulation. The coordination number of surfaces is made as uniform as possible. The triangulation is fixed in MC so that only the positions X of vertices may be considered as the dynamical variable. The well-known Helfrich energy function S = S₁ + bS₂ is used for the definition of the model where S₁ and S₂ are the area and bending energy functions respectively and b is the bending rigidity. The discretizations of S₁ and S₂ are identical with that of our previous MC study for a model of fluid membranes. We find that the specific heats have peaks at finite bending rigidities and obtain the critical exponents of the phase transition by the finite-size scaling technique. It is found that our model of crystalline membranes undergoes an expected second order phase transition.

The two-dimensional ferromagnetic Blume–Capel model is simulated on a cellular automaton, which based on the Creutz cellular automaton for square lattice. The values of the critical temperature and the static critical exponents are estimated within the framework of the finite-size scaling theory for 0 ≤ D/J ≤ 1.5. The results are compatible with the universal Ising critical behavior.

We have simulated (3+1)-dimensional finite temperature Z₂ gauge theory by using Metropolis algorithm. We aimed to observe the deconfinement phase transitions by using geometric methods. In order to do so we have proposed two different methods which can be applied to three-dimensional effective spin model consisting of Polyakov loop variables. The first method is based on the studies of cluster structures of each configuration. For each temperature, configurations are obtained from a set of bond probability (P) values. At a certain probability, percolating clusters start to emerge. Unless the probability value coincides with the Coniglio–Klein probability value, the fluctuations are less than the actual fluctuations at the critical point. In this method the task is to identify the probability value which yields the highest peak in the diverging quantities on finite lattices. The second method uses the scaling function based on the surface renormalization, which is of geometric origin. Since this function is a scaling function, the measurements done on different-size lattices yield the same value at the critical point, apart from the correction to scaling terms. The linearization of the scaling function around the critical point yields the critical point and the critical exponents.

We introduced strong avalanches in the two-dimensional sand-pile and investigated the distributions F(·, L) of their size corresponding to the different system lengths L. Being appropriately scaled, these distributions coincide. The inverse map F^-1 as the function on L sets up the correspondence between the avalanches associated with different L. As to the strong avalanches, with any fixed first argument δ, the map F^-1(δ, L) is proportional to a power function, where the exponent varies from 2 to 3 depending on δ, while F^-1(δ, L) is proportional to L² for the other avalanches.

Transfer matrix calculations of the critical two-point correlation function in 2D Ising model on a finite-size lattice with periodic boundaries along 〈11〉 direction are extended to L = 21. A refined analysis of the correlation function in 〈10〉 crystallographic direction at the distance r = L indicates the existence of a nontrivial finite-size correction of a very small amplitude with correction-to-scaling exponent ω < 2 in agreement with our foregoing study for L ≤ 20. Here we provide an additional evidence and show that amplitude a of the multiplicative correction term 1 + aL^-ω is about -3.5·10^-8 if ω = 1/4 (the expected value). We calculate also the susceptibility for L ≤ 18 in order to compare our numerical estimates for the constant background contribution with the known very precise value and to look for possible nontrivial corrections to scaling. The numerical analysis reveals a perfect agreement for the background term, as well as shows that the nontrivial correction term, detected by our analysis in the correlation function, likely cancels in the susceptibility.

We introduce a minesweeper percolation model, in which the system configuration is obtained via an automatic minesweeper process. For a variety of candidate networks with different lattice configurations, our process gives rise to a second-order phase transition. Using Monte Carlo simulation, we identify the critical points implied by giant components. A set of critical exponents are extracted to characterize the nature of the minesweeper percolation transition. The determined universality class shows a clear difference from the traditional percolation transition. A proper mine density of the minesweeper game should be set around the critical density.

In this paper, we present a Monte Carlo study of four explosive bond percolation models on the square lattice: (i) product rule which suppresses intrabonds (PR-SI), (ii) sum rule which suppresses intrabonds (SR-SI), (iii) product rule which enhances intrabonds (PR-EI) and (iv) sum rule which enhances intrabonds (SR-EI). By performing extensive simulations and finite-size scaling analysis of the wrapping probability $R^{(x)}$ and a ratio $Q$ for PR-SI, SR-SI, PR-EI, and the composite quantities $Z_R$ and $Z_Q$ for SR-EI (defined by $R^{(x)}$ and $Q$ corresponding to two different $p$-values, respectively), we determine the thresholds $p_c$ of all models with best precision. We also estimate the critical exponents $\beta ∕\nu$ and $\gamma ∕\nu$ for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster $C_1$ and the second moment $M_2$ of sizes of all clusters. For SR-EI, from $C_1$ and $M_2$, we only obtain pseudo-critical exponents, which are nonphysical. Precisely at $p_c$, we study the critical cluster-size distribution $n(s,L)$ (number density of the clusters of size $s$) for all models and find that it can be described by $n(s,L)\sim s^{-{\tau }^{\prime }}ñ(s∕L^{d^{\prime }})$, where ${\tau }^{\prime }=1+d∕d^{\prime }$ with $d^{\prime }=d_{\mathrm{\text{F}}}$ (fractal dimension) for PR-SI, SR-SI, PR-EI, $d^{\prime }=d$ (spatial dimension) for SR-EI, and $ñ(x)$ with $x=s∕L^{d^{\prime }}$ is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of $\beta ∕\nu$ (and $\gamma ∕\nu$ with the help of a scaling relation) for SR-EI. It is found that the exponents for PR-SI and SR-SI are consistent with each other, but the ones for PR-EI and SR-EI are different. Our results disclose two facts: (1) all models investigated here undergo continuous phase transitions, since their behaviors can be described by typical scaling formulas for continuous phase transitions; (2) PR-SI and SR-SI belong to a same universality class, however, PR-EI and SR-EI belong to different universality classes, and all of them differ from which random percolation belongs to. This work provides a testing ground for future theoretical studies.

We report some selected recent developments in the finite-size scaling theory of critical phenomena occurring in systems with strong spatial anisotropies. Such systems are characterized by correlation lengths divergent with different exponents (ν_⊥, ν_||) along different directions. Attention is focused on the driven diffusive lattice gas that exhibits a second order nonequilibrium phase transition. We present in detail the phenomenology and its comparison with computer simulation. Novel features of finite-size effects in anisotropic nonequilibrium systems are emphasized.

In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.

Describing asymmetric first-order phase transitions by an order parameter distribution, there is a controversy in the literature concerning the relative normalization of the two occurring peaks. There are two rules for normalizing the distribution, called the “equal weight”-rule and the “equal height”-rule, which lead to different predictions of the finite-size scaling. We tested these predictions for an asymmetric model with two co-existing phases by means of a Monte Carlo simulation. We find overwhelming numerical evidence in favour of the “equal weight”-rule, showing at the same time that the shift of the susceptibility maximum with respect to the infinite volume transition point h_t is proportional to L^−2d, and not to L^−d. In addition we tested a new method to determine the transition point h_t which was recently proposed by Borgs and Janke.

We report tests of finite-size scaling ansatzes in the low temperature phase of the two-dimensional Ising model. For moments of the magnetisation density, we find good agreement with the new ansatz of Borgs and Kotecký, and clear evidence of violations of the double Gaussian ansatz. We note that certain consequences of the convexity of the free energy are not adequately treated in either of these approaches.

A Hopfield-type neural network has content addressable memory which emerges from its collective properties. I reinvestigate the controversial question of its critical storage capacity at zero temperature. To locate the discontinuous transition from good retrieval to bad retrieval in infinite systems the decreasing average quality of retrieved information is traced until it falls below a threshold. The cutoff points found for different system sizes are extrapolated towards infinity and yield α_c=0.143±0.002.

We demonstrate that the standard O(n) symmetric φ⁴ field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d>4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n →∞ and n=1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-dimensional Ising model are not conclusive.

Simulations with up to 112⁵ spins confirm the prediction of Chen and Dohm of three different finite-size exponents for the spontaneous magnetization near the Curie point.

We study the standard three-dimensional driven diffusive system on a simple cubic lattice where particle jumping along a given lattice direction are biased by an infinitely strong field, while those along other directions follow the usual Kawasaki dynamics. Our goal is to determine which of the several existing theories for critical behavior is valid. We analyze finite-size scaling properties using a range of system shapes and sizes far exceeding previous studies. Four different analytic predictions are tested against the numerical data. Binder and Wang's prediction does not fit the data well. Among the two slightly different versions of Leung, the one including the effects of a dangerous irrelevant variable appears to be better. Recently proposed isotropic finite-size scaling is inconsistent with our data from cubic systems, where systematic deviations are found, especially in scaling at the critical temperature.

The four-dimensional Ising model is simulated on the Creutz cellular automaton with increased precision. The data are analyzed according to the finite-size scaling relations available. The precision of the critical values related to magnetic susceptibility is improved by one digit, but in order to reach to the same precision for those related to the specific heat more simulation runs at the critical temperatures of the finite-size lattices are required.

A recently introduced method for determining the critical indices of the deconfinement transition in gauge theories, already tested for the case of 3D SU(3) pure gauge theory, is applied here to 4D SU(2) pure gauge theory. The method is inspired by universality and based on the finite size scaling behavior of the expectation value of simple lattice operators, such as the plaquette. We obtain an accurate determination of the critical index ν, in agreement with the prediction of the Svetitsky–Yaffe conjecture.

The finite-size scaling functions of thermodynamic functions in anisotropic systems have been shown to be dependent on the spatial anisotropy [X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004)]. Here we extend this study to the correlation length ξ_‖ of the anisotropic O(n) symmetric φ⁴ model in an L^d−1 × ∞ cylindric geometry with periodic boundary conditions. We calculate the exact finite-size scaling function of correlation length ξ_‖ for T ≥ T_c in 2 < d < 4 dimensions and in the limit n → ∞. The finite-size scaling function of ξ_‖ is dependent on a normalized symmetric (d − 1) × (d − 1) matrix defined by the anisotropy matrix of anisotropic systems.

In this paper, we used a self-propelled particle model to study the transition between phases of collective behavior observed in animal aggregates. In these systems, transitions occur when individuals shift from one collective state to another. We investigated transitions induced by both the speed and the noise. Statistical quantities that characterize the phase transition driven by noise, such as order parameter, the Binder cumulant and the susceptibility were analyzed, and we used the finite-size scaling theory to estimate the critical exponent ratios $\beta /\nu$ and $\gamma /\nu$.