Connectedness percolation of elongated hard particles in an external field

A theory is presented of how orienting fields and steric interactions conspire against the formation of a percolating network of, in some sense, connected elongated colloidal particles in fluid dispersions. We find that the network that forms above a critical loading breaks up again at higher loadings due to interaction-induced enhancement of the particle alignment. Upon approach of the percolation threshold, the cluster dimensions diverge with the same critical exponent parallel and perpendicular to the field direction, implying that connectedness percolation is not in the universality class of directed percolation.     

Random percolation as a gauge theory

Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to flux tube quantum fluctuations like in ordinary confining gauge theories. Wilson loop correlators define a non-trivial spectrum of physical states of increasing mass and spin, like the glueballs of ordinary gauge theory. The crumbling of the percolating cluster when the length of one periodic direction decreases below a critical threshold accounts for the finite temperature deconfinement, which belongs to 2D percolation universality class.     

Correlated percolation in a film-deposition model

Clustering phenomenon has been studied in a film-deposition model in which a monolayer of particles is deposited onto a substrate. The occupation of a given site is assumed to depend on the occupation states of its nearest neighbouring sites as well as on the temperature of the particles being deposited. It is found that the percolating clusters remain ramified in that the number of their boundary sites are proportional to the number of particles in the cluster. The percolation threshold, however is lowered to (54±2)% as compared to 59% density for the uncorrelated case.     

Simulating percolating behavior of conductive particles in anisotropically conductive composite films

A numerical scheme is developed to simulate the percolating behavior of conductive particles within a non-conductive matrix film with a preferential alignment of particles achieved via externally imposed deterministic driving forces. The sharp transition from non-conducting to conducting of the composite film is successfully revealed with the model. The percolation behavior is studied in terms of four percolation parameters, including the percolation probability, the normalized shortest percolation path, normalized gyration radius and density of the percolation cluster, subject to variation in five important system parameters. These include particle concentration, relative importance of the externally applied force, film thickness, film width and particle size. The threshold particle concentration can be reduced by increasing the strength of the deterministic driving force, decreasing film thickness, increasing film width or using smaller size particles. Our study suggests that using stronger applied force for wider and thinner films containing smaller particles may be a good practice to obtain anisotropically conductive films with a light particle loading that possess good conduction capability in the thickness direction and good insulation in the planar direction. Received: 19 February 2001 / Accepted: 30 May 2001 / Published online: 30 August 2001     

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.     

A Mixed-Transfer-Matrix Method for Simulating Normal Conductor/Perfect Insulator/Perfect Conductor Random Networks

We develop a mixed-transfer-matrix approach for computing the macroscopic conductivity of a three-constituent normal conductor/perfect insulator/perfect conductor random network. This is applied to two-dimensional and three-dimensional samples at a percolation threshold. Such networks are simulated in order to test whether a diluted percolating network of normal conducting bonds remains in the same universality class of critical behavior when a finite fraction of those bonds have been replaced by perfectly conducting bonds. Also tested by such simulations is whether a percolating mixture of normal and perfectly conducting bonds remains in the same universality class of critical behavior when a finite fraction of the normal bonds are replaced by perfectly insulating bonds. These questions are crucial for some recently published exact results which connect the macroscopic electrical and elastic responses of percolating networks.     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

The affection on the nature of percolation by concentration of pentagon–heptagon defects in graphene lattice

In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degree of accuracy. This method is also suitable for locating the percolation critical point on other crystalline structures. Through investigating the evolution of the largest cluster size and the cluster sizes distribution, we find that under various lattice sizes and concentrations of pentagon-heptagon defects there is no apparent change for the percolation properties in graphene lattice.     

Low-energy dynamics of the two-dimensional S=1/2 Heisenberg antiferromagnet on percolating clusters

We investigate the quantum dynamics of site diluted S=1/2 Heisenberg antiferromagnetic clusters at the 2D percolation threshold. We use Lanczos diagonalization to calculate the lowest excitation gap Delta and, to reach larger sizes, use quantum Monte Carlo simulations to study an upper bound for Delta obtained from sum rules involving the staggered structure factor and susceptibility. Scaling the gap distribution with the cluster length L, Delta approximately L(-), we obtain a dynamic exponent z approximately 2D(f), where D(f)=91/48 is the fractal dimensionality of the percolating cluster. This is in contrast with previous expectations of z=D(f). We argue that the low-energy excitations are due to weakly coupled effective moments formed due to local imbalance in sublattice occupation.     

Percolation clusters above criticality form Kardar-Parisi-Zhang surfaces

This study is concerned with the characteristics of regular (isotropic) percolation clusters above the critical threshold p{c}. Analytic arguments for the general dimension case, and numerical results for the two-dimensional case, lead to the conclusion that the characteristics of the shortest paths (defined as the chemical distance l) between given two sites on a percolation cluster are similar to the characteristics of optimal paths in the directed polymer model. A corollary which should be valid for the general dimension case, and verified by numerical results for the two-dimensional case, is that a cluster whose sites are at chemical distance l from a given site forms a Kardar-Parisi-Zhang surface.     

Aging dynamics and the topology of inhomogenous networks

We study phase ordering on networks and we establish a relation between the exponent a(x) of the aging part of the integrated auto-response function and the topology of the underlying structures. We show that a(x) > 0 in full generality on networks which are above the lower critical dimension d(L), i.e., where the corresponding statistical model has a phase transition at finite temperature. For discrete symmetry models on finite ramified structures with T(c) = 0, which are at the lower critical dimension d(L), we show that a(x) is expected to vanish. We provide numerical results for the physically interesting case of the 2 - d percolation cluster at or above the percolation threshold, i.e., at or above d(L), and for other networks, showing that the value of a(x) changes according to our hypothesis. For O(N) models we find that the same picture holds in the large-N limit and that a(x) only depends on the spectral dimension of the network.     

Depinning of a domain wall in the 2d random-field Ising model

We report studies of the behaviour of a single driven domain wall in the 2-dimensional non-equilibrium zero temperature random-field Ising model, closely above the depinning threshold. It is found that even for very weak disorder, the domain wall moves through the system in percolative fashion. At depinning, the fraction of spins that are flipped by the proceeding avalanche vanishes with the same exponent as the infinite percolation cluster in percolation theory. With decreasing disorder strength, however, the size of the critical region decreases. Our numerical simulation data appear to reflect a crossover behaviour to an exponent at zero disorder strength. The conclusions of this paper strongly rely on analytical arguments. A scaling theory in terms of the disorder strength and the magnetic field is presented that gives the values of all critical exponent except for one, the value of which is estimated from scaling arguments. Received: 13 February 1998 / Accepted: 30 March 1998     

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

We consider the scaling limit of the two-dimensional q-state Potts model for q ⩽ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one-and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.     

基于逾渗理论的非晶合金屈服行为研究

从非晶合金的微观结构出发,基于处理强无序和具有随机几何结构系统常用的理论方法——逾渗理论来描述非晶合金剪切屈服时的塑性流变.为了更好地理解非晶合金剪切带萌生时的临界问题,结合已有的"自由体积(free volume)模型"和"剪切转变区(shear transformation zone)模型",建立了非晶合金剪切转变的逾渗模型.以Cu_(25)Zr_(75)二元非晶合金为例,计算了在剪切转变区内易发生塑性流动的原子团簇剪切失稳的逾渗阈值,并粗略估算了这些原子团簇的大小.研究发现,剪切失稳的逾渗阈值与临界约化自由体积浓度(x_c～2.4%)有着相似的特性,不同之处在于其值与自由体积的分散度有着密切联系.研究结果作为非晶合金的韧脆转变问题提供了新思路.     

Static and dynamic heterogeneities in a model for irreversible gelation

We study the structure and the dynamics in the formation of irreversible gels by means of molecular dynamics simulation of a model system where the gelation transition is due to the random percolation of permanent bonds between neighboring particles. We analyze the heterogeneities of the dynamics in terms of the fluctuations of the self-intermediate scattering functions: in the sol phase close to the percolation threshold, we find that this dynamic susceptibility increases with the time until it reaches a plateau. At the gelation threshold this plateau scales as a function of the wave vector k as k(eta-2), with eta being related to the decay of the percolation pair connectedness function. At the lowest wave vector, approaching the gelation threshold it diverges with the same exponent gamma as the mean cluster size. These findings suggest an alternative way of measuring critical exponents in a system undergoing chemical gelation.     

Analytical solutions for black-hole critical behaviour

Dynamical Einstein cluster is a spherical self-gravitating system of counterrotating particles, which may expand, oscillate and collapse. This system exhibits critical behaviour in its collapse at the threshold of black hole formation. It appears when the specific angular momentum of particles is tuned finely to the critical value. We find the unique exact self-similar solution at the threshold. This solution begins with a regular surface, involves timelike naked singularity formation and asymptotically approaches a static self-similar cluster.     

Multicritical point in a diluted bilayer Heisenberg quantum antiferromagnet

The S=1/2 Heisenberg bilayer antiferromagnet with randomly removed interlayer dimers is studied using quantum Monte Carlo simulations. A zero-temperature multicritical point (p(*),g(*)) at the classical percolation density p=p(*) and interlayer coupling g(*) approximately equal 0.16 is demonstrated. The quantum critical exponents of the percolating cluster are determined using finite-size scaling. It is argued that the associated finite-temperature quantum critical regime extends to zero interlayer coupling and could be relevant for antiferromagnetic cuprates doped with nonmagnetic impurities.     

Superscaling of percolation on rectangular domains

For percolation on (RL)xL two-dimensional rectangular domains with a width L and aspect ratio R, we propose that the existence probability of the percolating cluster E(p)(L,epsilon,R) as a function of L, R, and deviation from the critical point epsilon can be expressed as F(epsilonL(y(t))R(a)), where y(t) identical with1/nu is the thermal scaling power, a is a new exponent, and F is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with a=0.14(1) for R>2. We also propose superscaling for other critical quantities.