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.     

No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p _k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN ¹⁻ k is partitioned intoN ² congruent subsquares, and each of these is retained with probabilityp _k. independent of all others. We show that if Π_k p _k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).     

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.     

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.     

Anomalous diffusion and continuum percolation

Anomalous diffusion for continuum percolation is simulated by considering systems of randomly distributed circles and spheres. Universal behavior is obtained for the case of equal local conductances and nonuniversal behavior for diverging distributions of the local conductances. Diffusion in the continuum has a behavior consistent with that of other transport properties in the continuum. In addition, the results suggest that different algorithms for diffusion, which differ only in the random walker sitting times, are equivalent.     

Self-affine fractal clusters: Conceptual questions and numerical results for directed percolation

In this paper we address the question of the existence of a well defined, non-trivial fractal dimensionD of self-affine clusters. In spite of the obvious relevance of such clusters to a wide range of phenomena, this problem is still open since thedifferent published predictions forD have not been tested yet. An interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self-affine functions. As a much studied example of self-affine structures, we investigate the infinite directed percolation cluster at the threshold. We measuredD ind=2 dimensions by the box counting method. Using a correction to scaling analysis, we obtainedD=1.765(10). This result does not agree with any of the proposed relations, but it favorsD=1+(1- )/ , where and are the correlation length exponents and is a Fisher exponent in the cluster scaling.     

Critical behavior of a dynamical percolation model

The critical behavior of the dynamical percolation model, which realizes the molecular-aggregation conception and describes the crossover between the hadronic phase and the partonic phase, is studied in detail. The critical percolation distance for this model is obtained by using the probability P∞ of the appearance of an infinite cluster. Utilizing the finite-size scaling method the critical exponents γ/v and T are extracted from the distribution of the average cluster size and cluster number density. The influences of two model related factors, I.e. The maximum bond number and the definition of the infinite cluster, on the critical behavior are found to be small.     

Growth, percolation, and correlations in disordered fiber networks

This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameterp which controls the degree of clustering. Forp=1 the deposited network is uniformly random, while forp=0 only a single connected cluster can grow. Forp=0 we first derive the growth law for the average size of the cluster as well as a formula for its mass density profile. Forp>0 we carry out extensive simulations on fibers, and also needles and disks, to study the dependence of the percolation threshold onp. We also derive a mean-field theory for the threshold nearp=0 andp=1 and find good qualitative agreement with the simulations. The fiber networks produced by the model display nontrivial density correlations forp<1. We study these by deriving an approximate expression for the pair distribution function of the model that reduces to the exactly known case of a uniformly random network. We also show that the two-point mass density correlation function of the model has a nontrivial form, and discuss our results in view of recent experimental data on mss density correlations in paper sheets.     

The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

3d Monte Carlo simulation of site-bond continuum percolation of spheres

We present off-lattice Monte Carlo simulations of site-bond percolation of semi-penetrable spheres or, equivalently, of hard spheres with a finite bond range. We will show that the crucial parameter is the effective volume fraction ( φ_e), i.e. the volume that is occupied or within the bond range of at least one particle. For the equivalent system of semi-penetrable spheres 1 - φ_e is the porosity. The bond percolation threshold (p _b) can be described in terms of φ_e by a simple analytical expression: log(φ_e)/log(φ_ec) + log(p _b)/log(p _bc) = 1, with p _bc = 0.12 independent of the bond range and φ_ec a constant that decreases with increasing bond range. Received: 10 March 2003 / Accepted: 23 April 2003 / Published online: 21 May 2003 RID="a" ID="a"e-mail: jean-christophe.gimel@univ-lemans.fr     

A percolation approach to the Kauffman model

A study is made of the spreading of damage in the random but deterministic Kauffman model on the square lattice with the spreading from one edge of the lattice. The critical value of the parameterp _c above which the system becomes chaotic is found to bep _c0.298. The possibility of suppression of the chaotic phase by noise is also studied. It is found that forpp _c, an extremely large noise levelg>0.99 is required, if possible at all.On leave from Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China.     

Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

We establish, using mathematically rigorous methods, that the critical covered volume fraction (CVF) for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls. In addition, we show that the critical CVF is a continuous function of the distribution of the radius random variable, in the sense that if a sequence of random variables converges weakly to some random variable, then the critical CVF based on these random variables converges to the critical CVF of the limiting random variable.     

Bond percolation on a finite lattice: The one-state Potts model reconsidered

Bond percolation on a finite lattice is studied by looking at the Kac mean field model. The investigation utilizes the one-state Potts model connection established by Kasteleyn and Fortuin. To deal with special problems associated with the finite extent of the system we re-cast the partition function, which allows us to investigate the percolation transition in detail. This fundamental new formulation clears up certain ambiguities present in previous treatments and indicates a possible new direction in the study of other replica-type models.     

Stresses and strains in a deformable fractal medium and in its fractal continuum model

The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed.     

Bootstrap percolation in a polluted environment

Let a low densityp of sites on the lattice Z² be occupied, remove a proportionq of them, and call the remaining sites empty. Then update this configuration in discrete time by iteration of the following synchronous rule: an empty site becomes occupied by contact with at least two occupied nearest neighbors, while occupied and removed sites nerver change their states. Ifq/p ² is large most sites remain unoccupied forever, while ifq/p ² is small, this dynamics eventually makes most sites occupied. This demonstrates how sensitive the usual bootstrap percolation rule (theq=0 case) is to the pollution of space.     

Quantum to classical crossover under dephasing effects in a two-dimensional percolation model

Scaling theory predicts complete localization in d = 2 in quantum systems belonging to the orthogonal class(i.e., with timereversal symmetry and spin-rotation symmetry). The conductance g behaves as g^exp(-L/l) with system size L and localization length l in the strong disorder limit. However, classical systems can always have metallic states in which Ohm’s law shows a constant g in d=2. We study a two-dimensional quantum percolation model by controlling dephasing effects. The numerical investigation of g aims at simulating a quantum-to-classical percolation evolution. An unexpected metallic phase, where g increases with L, generates immense interest before the system becomes completely classical. Furthermore, the analysis of the scaling plot of g indicates a metal-insulator crossover.     

Comparison of wurtzite atomistic and piezoelectric continuum strain models: Implications for the electronic band structure

We compare continuum and atomistic models for the electromechanical fields in wurtzite GaN/AlN quantum dots and their relative impact on the electronic band structure. Qualitative agreement between atomistic strain calculations and continuum elastic models for a wurtzite hexagonal quantum-dot structure is demonstrated; however, significant quantitative discrepancies of up to 100 meV are observed. A smaller difference of approximately 15 meV is found between fully coupled and semi-coupled continuum models.     

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers (A + A A) on fractal spaces bear out a recent conjecture: ( ^–1 – ₀ ^–1) t ^f , where is the instantaneous walker density and ₀ the initial one, andf=d _s /2, whered _s is the spectral dimension. For the Sierpinski gaskets:d=2, 2f=1.38 (d _s =1.365);d=3, 2f=1.56 (d _s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation:d= 2,p=0.60, 2f= 1.35 (d _s =1.35); d=3,p=0.32, 2f=1.37 (d _s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2f (from reacting walkers) andd _s (from single walkers) are in good agreement, in both two and three dimensions.Supported by NSF Grant No. DMR 8303919.     

Real space renormalization group theory of the percolation model

A cluster expansion renormalization group method in real space is-developed to determine the critical properties of the percolation model. In contrast to previous renormalization group approaches, this method considers the cluster size distribution (free energy) rather than the site or bond probability distribution (coupling constants) and satisfies the basic renormalization group requirement of free energy conservation. In the construction of the renormalization group transformation, new couplings are generated which alter the topological structure of the clusters and which must be introduced in the original system. Predicted values of the critical exponents appear to converge to presumed exact values as higher orders in the expansion are considered. The method can in principle be extended to different lattice structures, as well as to different dimensions of space.This paper is dedicated to Prof. Philippe Choquard.