The existence of phase V in the Mandelbrot percolation process

Dekking and Meester defined six phases for a subclass of random Cantor sets consisting of those generated by Bernoulli random substitutions. They proved that the random Sierpinski carpet passed through all these phases asp tended from 0 to 1, but the were not able to prove the existencne of phase V in the Mandelbrot percolation process. In this paper, we accomplish the proof by improving their methods.Research supported by the Chinese Natural Science Foundation.     

Erdös Rényi随机网络上爆炸渗流模型相变性质的数值模拟研究

基于改进的Newman和Ziff算法以及有限尺寸标度理论, 通过对表征渗流相变特征物理量的序参量、平均集团尺寸、二阶矩、标准偏差及尺寸不均匀性的数值模拟, 分析研究了Erdös Rényi随机网络上Achlioptas爆炸渗流模型的相变性质.研究表明: 尽管序参量表现出了不连续相变的特征, 但序参量以及其他特征物理量仍具有连续相变的幂律标度行为.因此严格地说, Erdös Rényi随机网络中的爆炸渗流相变是一种奇异相变, 它既不是标准的不连续相变, 又与常规随机渗流表现出的连续相变处于不同的普适类. 关键词： Erdös Rényi随机网络 爆炸渗流模型 相变 幂律标度行为     

Covering by random intervals and one-dimensional continuum percolation

A brief historical introduction is given to the problem of covering a line by random overlapping intervals. The problem for equal intervals was first solved by Whitworth in the 1890s. A brief resume is given of his solution. The advantages of the present author's approach, which uses a Poisson process, are outlined, and a solution is derived by Laplace transforms. The method of Hammersley for dealing with a stochastic distribution of intervals is described, and a solution can still be derived by Laplace transforms. The asymptotic behavior as the line becomes long is calculated and is related to the one-dimensional continuum percolation problem. It is shown that as long as the mean interval size is finite, the probability of complete coverage decays exponentially, so that the critical percolation probabilityp _c =1. However, as soon as the mean interval size becomes infinite, the critical percolation probabilityp _c switches to 0. This is in accord with previous results for a lattice model by Chinese workers, but differs from those of Schulman. A possible reason for the discrepancy is a difference in boundary conditions.On sabbatical leave from Physics Department, Bar Ilan University, Ramat Gan, Israel.     

The Critical Point of k-Clique Percolation in the Erdős–Rényi Graph

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdős–Rényi graph. When the probability p of two nodes being connected is above a certain threshold p _c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some assumptions that are expected to be valid below the threshold, we determine the average size of the k-clique percolation clusters, using a generating function formalism. From the divergence of this average size we then derive an analytic expression for the critical linking probability p _c (k).     

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the q > 1 random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.     

多孔介质通道的分形随机行走描述

多孔介质中的输运过程,如导热、渗流过程,关注的是热量从高温壁面穿过介质到达低温壁面、流体从多孔介质的边界沿孔隙流到另外一端的过程。此类现象可归结为载流子在多孔介质通道(基质或孔隙)中沿外部势差方向的运动过程。多孔介质通道具有分形特征,可以采用分形维数来描述其通道的通透性。本文基于现象的相似性特征,提出并发展了粒子在多孔介质中的方向随机行走模型,用粒子在基质中的方向随机行走过程来模拟真实的热流传输过程;根据分形统计规律得到粒子方向随机行走分形谱维数,并用其描述基质结构的连通性和方向性。研究结果表明,在孔隙率相同情况下,粒子在基质中的方向随机行走分形谱维数与有效导热系数大小有相同的变化趋势。     

Landscape structure and the speed of adaptation

The role of fragmentation in the adaptive process is addressed. We investigate how landscape structure affects the speed of adaptation in a spatially structured population model. As models of fragmented landscapes, here we simulate the percolation maps and the fractal landscapes. In the latter the degree of spatial autocorrelation can be suited. We verified that fragmentation can effectively affect the adaptive process. The examination of the fixation rates and speed of adaptation discloses the dichotomy exhibited by percolation maps and fractal landscapes. In the latter, there is a smooth change in the pace of the adaptation process, as the landscapes become more aggregated higher fixation rates and speed of adaptation are obtained. On the other hand, in random percolation the geometry of the percolating cluster matters. Thus, the scenario depends on whether the system is below or above the percolation threshold.     

随机多孔介质逾渗模型渗透率的临界标度性质

研究了一类非零键渗透率满足均匀分布的随机多孔介质逾渗模型-数值计算了该模型系统渗透率在临界点处的标度指数-结果表明该指数并不能看作是普适常数,而与均匀分布的参数有关-这意味着即使非零键渗透率值的概率密度函数满足负一阶矩存在条件,系统渗透率在逾渗临界点处的标度指数仍然依赖于分布函数的具体参数,并不是常数-这一数值结果与Sahimi对此问题的结论不同- 关键词： 逾渗 随机多孔介质 标度指数 渗透率     

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.     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

Theory of self-avoiding walks on percolation fractals

A phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self-avoiding walks (SAW) is used to derive a new expression for the Flory exponent describing the average radius of gyration of SAWs on fractals. We focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of SAWs. The mean-field version of our new formula corresponds to the Aharony and Harris expression, who used the standard Flory approach for its derivation.On leave from Santipur College, Nadia 741404, India.     

Impurity diffusion in a harmonic potential and nonhomogeneous temperature

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.     

Probability distribution for percolation clusters generated on a cayley tree at criticality

We present analytical and numerical results for the probability distributions of the number of sitesS as a function of the number of shellsl for several ensembles of percolation clusters generated on a Cayley tree at criticality. We find that for the incipient infinite percolation cluster the probability distribution isP(S¦l)~(S/l ⁴)exp(- aS/l ²) for Sl1.     

Pattern Structure of Deterministic Displacement in Random Porous Media with Dispersion Effect

A new model-model of random porous mediz degradation via several fluid displacing,freezing,and thawing cycles is introduced and investigated in this paper.The fluid transport is based on the deterministic method with dispersion effect.The result shows that the topology and the geometry of the porous media have a strong effect on displacement processes.The cluster size of viscous fingering (VF) pattern in percolation cluster increases with the increase of iteration parameter n.When iteration parameter n≥10,VF pattern does not change with n.We find that the displacement fluid forms trapping regions in random porous media with dispersion effect.And the trapping regions will expand with the increasing of the iteration parameter n.When r (throat size)→1 and n≥5,the peak value of the distribution Nmat(r) increases as n increases,where Nmat(r) is the normalized distribution of throat sizes after different displacement-damages but before freezing.The peak value of the distribution Ninv(r) reaches a maximum when n≥10 and r=1,where Ninv(r) is the normalized distribution of the size of invaded throat.This result is different from invasion percolation.It is found that the sweep efficiency E increases along with the increasing of iteration parameter n and decreases with the network size L,and E has a minimum as L increases to the maximum size of lattice.The VF pattern in percolation cluster has one frozen zone and one active zone.     

Lévy-flight spreading of epidemic processes leading to percolating clusters

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an -expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection . Received: 17 July 1998 / Revised: 20 July 1998 / Accepted: 28 July 1998     

Modeling stock price dynamics by continuum percolation system and relevant complex systems analysis

The continuum percolation system is developed to model a random stock price process in this work. Recent empirical research has demonstrated various statistical features of stock price changes, the financial model aiming at understanding price fluctuations needs to define a mechanism for the formation of the price, in an attempt to reproduce and explain this set of empirical facts. The continuum percolation model is usually referred to as a random coverage process or a Boolean model, the local interaction or influence among traders is constructed by the continuum percolation, and a cluster of continuum percolation is applied to define the cluster of traders sharing the same opinion about the market. We investigate and analyze the statistical behaviors of normalized returns of the price model by some analysis methods, including power-law tail distribution analysis, chaotic behavior analysis and Zipf analysis. Moreover, we consider the daily returns of Shanghai Stock Exchange Composite Index from January 1997 to July 2011, and the comparisons of return behaviors between the actual data and the simulation data are exhibited.     

Graph partitioning induced phase transitions

We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree k. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if nonoptimal) that partitions the graph into essentially equal sized connected components (clusters), the system undergoes a percolation phase transition at f = fc = 1-2/k where f is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find S approximately N 0.4 where S is the size of the clusters and l approximately N 0.25 where l is their diameter. Also, we find that S undergoes multiple nonpercolation transitions for f相似文献   

Diffusion on the Scaling Limit of the Critical Percolation Cluster in the Diamond Hierarchical Lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.     

Theta-point exponent for polymer chains on percolation fractals

We derive a new expression for the Flory exponent describing the average radius of gyration of polymer chains at the theta point. For this we make use of the appropriate distribution function for the radius of gyration. We start from Euclidean lattices and extend the results to percolation fractals, by taking into account the basic geometry and the topology of such structures. We show that such basic features have a very prominent effect on the Flory exponent of the chain polymer on fractals at the theta point.     

Effective dielectric function of a metal-dielectric composite with nonrandomly distributed particles

Summary The optical properties of a metal-dielectric binary composite with nonrandomly distributed particles were studied within the framework of the effective medium theory. All the particles, as usual, were assumed to be spherical and the nonrandomness of their distribution was achieved by forcing onto the system the correlation that prevents the metal particles from touching each other or the one that forces them to aggregate in pairs, either oriented at random or all alike. The drawbacks of the effective medium theory due to the assumption that the particles be very small were overcome by describing the scattering properties of the particles through their exact Mie amplitudes. Both kinds of correlation we considered turned out to have substantial effects on the onset of the percolation phenomena. The transition induced by the correlation of exclusion from a behaviour typical of the Bruggeman model to one more appropriate to the Maxwell-Garnett model was followed as a function of the exclusion radius. In the case of aggregation the onset of the percolation phenomena was found to be strongly influenced by the further introduction of an orientational order of the pairs. Based on work supported in part by the U. S. Army European Research Office through Contract DAJA45-86-C-0003 and in part by the Consiglio Nazionale delle Ricerche through the Gruppo Nazionale Struttura della Materia.