Scaling inequalities for oriented percolation

We look at seven critical exponents associated with two-dimensional oriented percolation. Scaling theory implies that these quantities satisfy four equalities. We prove five related inequalitites.     

Scaling properties of models of nonequilibrium phenomena

The renormalization group method proposed by 't Hooft is developed for the study of scaling properties of some models of nonequilibrium phenomena. For one of two models studied in detail, the Langevin equation for the random variables contains a bilinear streaming velocity and the stationary probability distribution is Gaussian. The time-dependent Ginzburg-Landau model is chosen as a second example because it illustrates the advantage of the 't Hooft method of not having to specify a particular renormalization point. The scaling exponents for a model of the liquid-gas phase transition are calculated in lowest order to illustrate application of the method to a multifield system.     

Scaling assumption for lattice animals in percolation theory

A scaling assumption for the numberg _ns of different cluster configurations with perimeters and sizen leads to the desired cluster numbers near the percolation threshold. The perimeter distribution function has a mean square width proportional ton for largen. The relation between the average perimeter and the cluster sizen for percolation has three different forms atp _c, belowp _c, and abovep _c and is closely related to the shape of the cluster size distribution.     

The canonical and grand canonical models for nuclear multifragmentation

Many observables seen in intermediate energy heavy-ion collisions can be explained on the basis of statistical equilibrium. Calculations based on statistical equilibrium can be implemented in microcanonical ensemble, canonical ensemble or grand canonical ensemble. This paper deals with calculations with canonical and grand canonical ensembles. A recursive relation developed recently allows calculations with arbitrary precision for many nuclear problems. Calculations are done to study the nature of phase transition in nuclear matter.     

Comparisons of statistical multifragmentation and evaporation models for heavy-ion collisions

The results from ten statistical multifragmentation models have been compared with each other using selected experimental observables. Even though details in any single observable may differ, the general trends among models are similar. Thus, these models and similar ones are very good in providing important physics insights especially for general properties of the primary fragments and the multifragmentation process. Mean values and ratios of observables are also less sensitive to individual differences in the models. In addition to multifragmentation models, we have compared results from five commonly used evaporation codes. The fluctuations in isotope yield ratios are found to be a good indicator to evaluate the sequential decay implementation in the code. The systems and the observables studied here can be used as benchmarks for the development of statistical multifragmentation models and evaporation codes. An erratum to this article is available at .     

Scaling and continuum percolation model for enzyme-catalyzed gel degradation

Enzyme-catalyzed gel degradation is inherently controlled by diffusion of enzymes in the gel. We report kinetics measurements on the gelatin-thermolysin system, varying solvent viscosity as well as gel and enzyme concentrations. Scaling relations and reduced variables are proposed which are shown to account for the experimental results. Finally, we argue that the nontrivial experimental dependence on enzyme concentration for the degradation time demonstrates that enzyme random walk is self-attracting, leading to a continuum percolation model for gel degradation.     

Central limit theorems for percolation models

Letp 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. LetW _x be the cluster of sites connected tox by open paths, and let(n) be any sequence of circuits with interiors . It is shown that for certain sequences of functions {f _n }, converges in distribution to the standard normal law when properly normalized. This result answers a problem posed by Kunz and Souillard, proving that the numberS _n of sites inside(n) which are connected by open paths to(n) is approximately normal for large circuits(n).     

Scaling function for cluster size distribution in two-dimensional site percolation

Exact cluster size distributions of Sykes et al. in the square and triangular lattice for cluster sizes up to 17 are used to extrapolate the scaling function in the site percolation problem. Also the amplitude ratioC ₊/C _- of the second moment is determined.     

Scaling behavior of the directed percolation universality class

In this work we consider five different lattice models which exhibit continuous phase transitions into absorbing states. By measuring certain universal functions, which characterize the steady state as well as the dynamical scaling behavior, we present clear numerical evidence that all models belong to the universality class of directed percolation. Since the considered models are characterized by different interaction details the obtained universal scaling plots are an impressive manifestation of the universality of directed percolation.     

Finite-size effects for some bootstrap percolation models

The consequences of Schonmann's new proof that the critical threshold is unity for certain bootstrap percolation models are explored. It is shown that this proof provides an upper bound for the finite-size scaling in these systems. Comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics. We show that the threshold for a finite system of sizeL scales asO[ln(lnL)] for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled.Related systems have been studied in the context of cellular automata.⁽⁴⁾     

Generalized percolation models for frustrated spin systems

Summary A brief review on how to study frustrated spin models by mapping them into generalized percolation models is given. The percolation models associated to a number of deterministic frustrated models are discussed with particular attention on the properties of critical clusters. Paper presented at the I Internatioanl Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN ₀, the number of distinct infinite clusters, is either 0, 1, or and the caseN ₀= (which might occur in sufficiently high dimension) is analyzed.Alfred P. Sloan Research Fellow, Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.Research supported in part by the U.S.Israel Binational Science Foundation.     

Simultaneous analysis of three-dimensional percolation models

We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, 2013, 87（5）： 052107], it is observed that in comparison with dimensionless ratios based on cluster-size distribution, certain wrapping probabilities exhibit weaker finite-size corrections and are more sensitive to the deviation from percolation threshold Pc, and thus provide a powerful means for determining Pc. We analyze the numerical data of the wrapping probabilities simultaneously such that universal parameters are shared by the aforementioned models, and thus significantly improved estimates of Pc are obtained.     

New critical frontiers for the potts and percolation models

We obtain the critical threshold for a host of Potts and percolation models on lattices having a structure which permits a duality consideration. The consideration generalizes the recently obtained thresholds of Scullard and Ziff for bond and site percolation on the martini and related lattices to the Potts model and to other lattices.     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.