Site Percolation Problem Research Articles

Percolation transitions in two dimensions

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

Percolation on patchwise heterogeneous surfaces under equilibrium conditions

The site percolation problem on square lattices whose sites are grouped in two types of energetically different patches is studied. Several lattices formed by collections of either randomly or orderly localized and no overlapped patches of different sizes are generated. The system is characterized by two parameters, namely, the size of each patch, l , and the energy difference between the two kind of sites, Δ E . Particles are adsorbed at equilibrium on the lattice. The critical coverage is determined by means of Monte Carlo simulations and finite-size scaling analysis. The percolative behavior of the system as a function of the parameters characterizing the heterogeneity of the energetic surface topography is presented and discussed.

Dynamical aspects of two-dimensional quantum percolation

The existence of a quantum-percolation threshold ${p}_{q}<1$ in the two-dimensional (2D) quantum site-percolation problem has been a controversial issue for a long time. By means of a highly efficient Chebyshev expansion technique we investigate numerically the time evolution of particle states on finite disordered square lattices with system sizes not reachable up to now. After a careful finite-size scaling, our results for the particle's recurrence probability and the distribution function of the local particle density give evidence that indeed extended states exist in the 2D percolation model for $p<1$.

Statistical physics on cellular neural network computers

The computational paradigm represented by Cellular Neural/nonlinear Networks (CNN) and the CNN Universal Machine (CNN-UM) as a Cellular Wave Computer, gives new perspectives also for computational statistical physics. Thousands of locally interconnected cells working in parallel, analog signals giving the possibility of generating truly random numbers, continuity in time and the optical sensors included on the chip are just a few important advantages of such computers. Although CNN computers are mainly used and designed for image processing, here we argue that they are also suitable for solving complex problems in computational statistical physics. This study presents two examples of stochastic simulations on CNN: the site-percolation problem and the two-dimensional Ising model. Promising results are obtained using an ACE16K chip with 128×128 cells. In the second part of the work we discuss the possibility of using the CNN architecture in studying problems related to spin-glasses. A CNN with locally variant parameters is used for developing an optimization algorithm on spin-glass type models. Speed of the algorithms and further trends in developing the CNN chips are discussed.

Limiting factors of normal-state conductivity in superconductingMgB2: an application of mean-field theory for a site percolation problem

Normal-state conductivity in polycrystallineMgB2 bulk samples having a systematically varied packing factor wasstudied. The packing factor dependence of phonon term resistivityΔρ(T) = ρ(T)−ρ0 was found to be well explained by the three-dimensional site percolation model.The low packing density of the samples and the wet impurity phases at grainboundaries are suggested to be the main causes of poor electrical connectivity inMgB2. Our model enables quantitative evaluations of the intrinsic resistivityinside the grains, the fraction of the active grains that can carry currentand the anisotropy of the grains in polycrystalline samples. The modelpredicts that the anomaly suppressed connectivity in rather weak-link-freeMgB2 can be understood under a scenario of a percolation problem.

Cluster size distribution in percolation theory and fractal Cantor dust

Results of numerical simulation of cluster size distribution in the site percolation problem are presented. These results disagree with the theoretical data obtained on the basis of the standard drop model of finite cluster structure, in particular, they give a different value of exponent zeta (lnn{s} approximately -s{zeta}). Therefore, a more precise fractal model for describing the structure of clusters in a percolation system is proposed. The consideration is based on the solution of a kinetic equation for the number of finite clusters. In the framework of the proposed approach (fractal model together with kinetic equation), a correct value of exponent zeta is obtained and an explanation is given to the dependence of this exponent on the fraction of occupied sites p, which was revealed by numerical simulations. Additionally, a relation is established between the characteristics of cluster size distribution and fractal dimension of the Cantor dust constructed on the percolation cluster.

PERSPECTIVES FOR MONTE CARLO SIMULATIONS ON THE CNN UNIVERSAL MACHINE

Possibilities for performing stochastic simulations on the analog and fully parallelized Cellular Neural Network UniversalMachine (CNN-UM) are investigated. By using a chaotic cellular automaton perturbed with the natural noise of the CNN-UM chip, a realistic binary random number generator is built. As a specific example for Monte Carlo type simulations, we use this random number generator and a CNN template to study the classical site-percolation problem on the ACE16K chip. The study reveals that the analog and parallel architecture of the CNN-UM is very appropriate for stochastic simulations on lattice models. The natural trend for increasing the number of cells and local memories on the CNN-UM chip will definitely favor in the near future the CNN-UM architecture for such problems.

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝd, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.

Monte Carlo study of the site-percolation model in two and three dimensions

We investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size L(d-1) x infinity, where L specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at p(c) =0.592 746 5 (4) , and determine two universal quantities as Q(gbc) =0.930 34 (1) and Q(gsc) =0.793 72 (3) , which are associated with bulk and surface correlations, respectively. These values agree well with the exact values 2(-5/48) and 2(-1/3) , respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at p(c) =0.311 607 7 (4) . We further determine the bulk thermal and magnetic exponents as y(t) =1.1437 (6) and y(h) =2.5219 (2) , respectively, and the surface magnetic exponent at the ordinary phase transition as y (o)(hs) =1.0248 (3) .

Localization effects in quantum percolation

We present a detailed study of the quantum site percolation problem on simple cubic lattices, thereby focusing on the statistics of the local density of states and the spatial structure of the single particle wave functions. Using the kernel polynomial method we refine previous studies of the metal-insulator transition and demonstrate the nonmonotonic energy dependence of the quantum percolation threshold. Remarkably, the data indicates a ``fragmentation'' of the spectrum into extended and localized states. In addition, the observation of a chequerboardlike structure of the wave functions at the band center can be interpreted as anomalous localization.

Variation of cluster properties in lattice percolation problem: A prototype of phase transition

Properties of clusters appearing in the site percolation problem on square and cubic lattices are expressed in a way that emphasizes the thermodynamic analogy. It is shown that the analog of the specific heat exhibits expected critical behaviour as a function of the analog of the temperature. The results support the notion that the partition of the specific heat of Ising systems (Borstnik and Lukman, Phys. Rev. E 60, 2595 (1999)) into the structural and populational component is a meaningful one. Another cluster property which is taken under the scrutiny is the fractal dimensionality of clusters which also indicates the presence of phase transition.

A probabilistic approach to the site percolation problems using fractal tree

A simple method based on the use of conditional probabilities, for deriving the particle cluster distribution is discussed for the site-percolation problem. The method is applied to a fractal tree (Inhomogeneous Bethe lattice) in the case of finite clusters. The extrema of the corresponding distribution are analytically analyzed and the known results for the homogeneous case are obtained.

From interactions to structures and thermodynamic properties

The standard paradigm of statistical thermodynamics is explored for five model systems. Three systems belong to the class of lattice models and two to the class of models with hard-core interaction potentials. The methodology is based on the concept of clusters. It is shown that the study of clusters in the lattice site percolation problem can give us valuable information about the Ising model and also about another Ising-type spin model which is introduced in the study. In the field of hard-core systems the concept of clusters is realized in the form of virial expansion and applied to the problem of stability of hard-sphere mixtures and also to the problem of the existence of a thermodynamic limit for the model of adhesive spheres. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 416–424, 2000

Percolation in simple patchwise lattices

We study the site percolation problem on square lattices with two kinds of sites, which are assembled in such a way that the resulting structures have patchwise topographies. Lattices formed by collections of either randomly or orderly localized patches of different sizes are generated. The composition of this system is specified by two independient variables, p and q, which are the occupation probabilities of each type of patch. Interesting phase diagrams in $(p,q)$ composition space for the percolative transition are obtained and explained.

Site percolation and random walks ond-dimensional Kagomé lattices

The site percolation problem is studied on d-dimensional generalisations of the Kagome' lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d. The site percolation thresholds are calculated numerically for d= 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagome' lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply p_c ~ 1/(q-1).

Percolation and cluster structure parameters: The enhanced Hoshen-Kopelman algorithm

An enhanced Hoshen-Kopelman (EHK) algorithm for calculating cluster structure parameters is presented for the site percolation problem. The EHK algorithm enables efficient calculation of cluster spatial moments, perimeters, and bounding boxes for very large Monte Carlo simulated lattices. The algorithm is used to compute the squared radius of gyration and internal perimeters of clusters in $3000\ifmmode\times\else\texttimes\fi{}3000$ square simulated lattices. The squared radius of gyration was used to calculate the value of the correlation length exponent.

Quantum percolation in electronic transport of metal-insulator systems: numerical studies of conductance

The quantum site-percolation problem defined by a tight-binding one-electron Hamiltonian on regular simple cubic lattice with binary probability distribution of site energies P( ϵ n) = pσ( ϵ n) + (1 − p) σ( ϵ n − ∞) is studied using the Landauer-Büttiker formalism and Green's function method. The dimensionless conductance g according to Landauer-Büttiker formula is calculated for a finite system of size L × L × L. The arithmetic and geometric ( e 〈ln g〉 ) averages of g over many realizations of the disordered system are calculated. Plotting g for different L as a function of concentration p has enabled to find a critical p = p q such that g decreases (exponentially) with L for p < p q and it increases (linearly) with L when p > p q . Thus, we have demonstrated the Anderson metal-insulator transition at critical concentration p q from the behaviour of the conductance itself. We have also estimated the critical conductance, g c as g c = g( p q ). By estimating the critical point for different values of electron Fermi energy E we have estimated the mobility-edge trajectory and it has been found to be consistent with the corresponding line in the p- E plane obtained by Soukoulis et al. (1987; 1992).

Thresholds and Universality of the Site Percolation on the Sierpinski Carpets11The project supported by National Basic Research Project “Nonlinear Science” and National Natural Science Foundation of China

Following the methods proposed by Yonezawa, Sakamoto and Hori, we have calculated the percolation thresholds , their error bars , and the correlation length exponents of a family of the Sierpinski carpets for the site percolation problems by making use of MonteCarlo simulations and finite size scaling. We have found the dependence of and on the fractal dimensionality and the lacunarity. We ascertain that the site percolation problems on a family of Sierpinski carpets with central cutouts and different belong to different universal classes, and those on Sierpinski carpets with same but of different lacunarities belong to different universal classes.

Renormalization group approach to the multi-component site percolation on Sierpinski carpets

The multi-component site percolation problems on a family of Sierpinski carpets (SCs) are solved by the real-space renormalization group (RG) method. Three sorts of sites on an SC lattice are distinguished and three independent kinds of occupation probabilities are assigned to them. We develope the usual choice of a cell on the translationally invariant lattices and choose suitable cells to construct the RG transformation scheme based on the connectivity. The non-trivial percolation threshold values and correlation length exponents on such fractals are obtained.

On the surface properties of two-dimensional percolation clusters

The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation-length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, $\eta_s = 0.6664 \pm 0.0008$, consistent with the analytical value $\eta_s = 2/3$. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value $y_s = -1$ for the irrelevant exponent.