Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

On the continuity of the critical value for long range percolation in the exponential case

We show that for a long range percolation model with exponentially decaying connections, the limit of critical values of any sequence of long range percolation models approaching the original model from below is the critical value for the original long range percolation model. As an interesting corollary, this implies that if a long range percolation model with exponential connections is supercritical, then it still percolates even if all long bonds are removed. We also show that the percolation probability is continuous (in a certain sense) in the supercritical regime for long range percolation models with exponential connections.Research supported by a grant from the Swedish National Science Foundation.     

New universality classes in “Percolative” dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.     

Multifractality in elastic percolation

We present numerical results on the distribution of forces in the central-force percolation model at threshold in two dimensions. We conjecture a relation between the multifractal spectrum of scalar and vector percolation that we test for central-foce percolation. This relation is in excellent agreement with our numerical data.     

Ising model in a quasiperiodic transverse field,percolation, and contact processes in quasiperiodic environments

Quantum Ising models in a transverse field are related to continuous-time percolation processes whose oriented percolation versions are contact processes. We study such models in the presence of quasiperiodic disorder and prove localization in the ground state, no percolation, and extinction, respectively, for sufficiently large disorder.     

Tricritical point in explosive percolation

The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.     

Percolation transition in an irreversible-surface reaction model

Monte Carlo simulation studies of percolation transition in a surface reaction model describing the oxidation of carbon mono-oxide on a catalytic surface are presented. The percolation transition for adsorbed oxygen atoms occurs below the poisoning transition where carbon mono-oxide completely covers the surface of the catalyst and takes place for an oxygen coverage of about 0.525 which is close to the percolation transition in an Ising lattice gas with nearest-neighbor attractive interactions. In several respects the oxygen clusters near the percolation threshold resemble those of the Ising lattice gas near its critical point.     

Limited universality at the percolation threshold in 2 to 6 dimensions

The universality of the spanning fraction R(p) of percolation is confirmed by comparing bond percolation with site-bond percolation in four to six dimensions. However, different boundary conditions change the universality class, as shown also for site percolation in two dimensions.     

On the Fractal dimension and correlations in percolation theory

We discuss the fractal dimension of the infinite cluster at the percolation threshold. Using sealing theory and renormalization group we present an explicit expression for the two-point correlation function within percolation clusters. The fractal dimension is given by direct integration of this function.See especially Ref. 1 for a discussion of the general aspects of percolation.     

Metastability Thresholds for Anisotropic Bootstrap Percolation in Three Dimensions

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability thresholds for a fairly general class of models. In our proofs, we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the ‘easiest growth direction’ in the model. In contrast to anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metastability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.     

Numerical studies of gravity destabilized percolation in 2D porous media

Two dimensional simulations of percolation are realized on square networks of pore throats with a random capillary pressure distribution. We analyse the influence of a destabilizing gravity field (g) and of the standard deviation of the distribution of the capillary pressure thresholds (W_t). The fragmentation process is not taken into account in this study. For an increase of g or/and when W_t decreases, two transitions are analyzed with three different regimes displacement patterns: Invasion percolation, invasion percolation in a gradient, and invasion in a pure gradient. The transitions are controlled both by the ratio g/W_t and by the sample size (L). A scaling law between the saturation at the percolation threshold and g/W_t allows delineating the three regimes in agreement with theoretical argument of the percolation in a gradient.     

Percolation transport in random flows with weak dissipation effects

In the present paper, we consider the influence of weak dissipative effects on the passive scalar behavior in the framework of continuum percolation approach. The renormalization method of a small parameter in continuum percolation models is reviewed. It is shown that there is a characteristic velocity scale, which corresponds to the dissipative process. The modification of the renormalization condition of the small percolation parameter is suggested in accordance with additional external influences superimposed on the system. In the framework of mean-field arguments, the balance of correlation scales is considered. This gives the characteristic time that corresponds to the percolation regime. The expression for the effective coefficient of diffusion is obtained.     

Topology invariance in percolation thresholds

An universal invariant for site and bond percolation thresholds ( and respectively) is proposed. The invariant writes where and are positive constants, and d the space dimension. It is independent of the coordination number, thus exhibiting a topology invariance at any d. The formula is checked against a large class of percolation problems, including percolation in non-Bravais lattices and in aperiodic lattices as well as rigid percolation. The invariant is satisfied within a relative error of for all the twenty lattices of our sample at d=2, d=3, plus all hypercubes up to d=6. Received: 7 July 1997 / Accepted: 5 November 1997     

炭黑橡胶复合材料导热逾渗行为

研究了导电炭黑40b2填充天然橡胶复合材料的导热性能和力学性能随炭黑体积分数的变化规律,并采用扫描电子显微镜观察了炭黑橡胶体系内部的炭黑分布状况.结果表明,导热性能随炭黑体积分数的变化规律存在类似于导电逾渗现象的导热逾渗现象,逾渗阈值在8.3%～13.63%之间.在逾渗阈值之后,复合材料的拉伸强度下降.炭黑橡胶复合材料...     

Scale invariance in percolation theory and fractals

The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot transformation, leading to a fractal behavior of the percolation probability in the complex plane. The hierarchical chains of impedances, reducing to a nonlinear mapping of the impedance space onto itself, are studied. An infinite continuation of the procedure leads to a fixed point. It is shown that the number of steps required to reach a neighborhood of this point has a fractal distribution. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 6, 427–432 (25 September 1996)     

Percolation and epidemic thresholds in clustered networks

We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scale-free networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks, thus extending this result to a wide variety of real scale-free networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations.     

Explosive percolation transition is actually continuous

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.     

Evolution of a Magnetoresonance Line Near to a Limit of Percolation in Dilute Magnetics

In paper the results of numerical modeling of a magnetic resonance in dilute magnetics near to a threshold of a percolation are discussed. The classical equation of motion of magnetic moments is used in view of an exchange interaction such as RKKI and imitation of spin-phonon interaction by Monte-Carlo method. It is shown, that cluster structure of a magnetic and threshold of percolation are determined by critical distance, on which there is a change of a sign of an exchange interaction. In an examination of percolation phase transition the jump change of breadth of a line of a magnetic resonance is set, that can form the basis for experimental definition of a threshold percolation and parameters of an exchange interaction by methods of a radiospectroscopy.     

二维孔隙裂隙双重介质逾渗规律研究

在孔隙介质逾渗理论的基础上，将另外一个非常重要的渗透通道——裂隙引入到介质的逾渗研究中，提出了更为普遍的孔隙裂隙双重介质的逾渗研究方法.通过对二维平面孔隙裂隙双重介质的数值计算，得到了孔隙裂隙双重介质三个重要参数：孔隙率，裂隙分形维数、裂隙数量分布初值与逾渗概率的关系，给出了孔隙裂隙双重介质逾渗阈值的数学表达式，揭示了孔隙裂隙双重介质的逾渗规律. 关键词： 孔隙 裂隙 双重介质 逾渗 逾渗阈值