Site Percolation Threshold Research Articles

Square-lattice site percolation at increasing ranges of neighbor bonds

We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same pc = 0.592... . The fourth-nearest neighbors (5N) give pc = 0.298... . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN), (NN+4N), (NN+NNN+4N), and (NN+5N). The calculated associated thresholds are respectively pc = 0.407..., 0.337..., 0.288..., and 0.234... . The existing Galam-Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.

DiluteKagoméLattice Magnetism withS= 1/2 on Rb2(Pd1-xMx)3S4(M = Co, Mn)

An attempt to substitute Co 2+ , Mn 2+ for Pd 2+ was performed to yield dilute kagome lattice magnetism of S = 1/2 on a ternary transition metal chalocogenide Rb 2 Pd 3 S 4 . This compound takes a crystal structure, Pd sites of which constitute kagome lattice nets. Because the divalent Pd 2+ (4d) 8 's on the kagome lattice are in a low-spin state of spin-less owing to specific characteristics of the crystal field influencing on the Pd sites. Magnetic properties of specimens thus obtained were examined through magnetic susceptibility, ESR and specific heat measurements. Several data were obtained for confirming the syntheses of S = 1/2 systems in spite of dilute magnetic ones with substitutional ratio x's below the site percolation threshold x p = 0.62 of the kagome lattice.

The Optimal Fraction of Hydrophobic Residues Required to Ensure Protein Collapse

The hydrophobic interaction is the main driving force for protein folding. Here, we address the question of what is the optimal fraction, f of hydrophobic (H) residues required to ensure protein collapse. For very small f (say f<0.1), the protein chain is expected to behave as a random coil, where the H residues are "wrapped" locally by polar (P) residues. However, for large enough f this local coverage cannot be achieved and the thermodynamic alternative to avoid contact with water is burying the H residues in the interior of a compact chain structure. The interior also contains P residues that are known to be clustered to optimize their electrostatic interactions. This means that the H residues are clustered as well, i.e. they effectively attract each other like the H-monomers in Dill's HP lattice model. Previously, we asked the question: assuming that the H monomers in the HP model are distributed randomly along the chain, what fraction of them is required to ensure a compact ground state? We claimed there that f approximately p(c), where p(c) is the site percolation threshold of the lattice (in a percolation experiment, each site of an initially empty lattice is visited and a particle is placed there with a probability p. The interest is in the critical (minimal) value, p(c), for which percolation occurs, i.e. a cluster connecting the opposite sides of the lattice is created). Due to the above correspondence between the HP model and real proteins (and assuming that the H residues are distributed at random) we suggest that the experimental f should lead to percolating clusters of H residues over the highly dense protein core, i.e. clusters of the core size. To check this theory, we treat a simplified model consisting of H and P residues represented by their alpha-carbon atoms only. The structure is defined by the C(alpha)-C(alpha) virtual bond lengths, angles and dihedral angles, and the X-ray structure is best-fitted onto a face-centered cubic lattice. Percolation experiments are carried out for 103 single-chain proteins using six different hydrophobic sets of residues. Indeed, on average, percolating clusters are generated, which supports our theory; however, some sets lead to a better core coverage than others. We also calculate the largest actual hydrophobic cluster of each protein and show that, on average, these clusters span the core, again in accord with our theory. We discuss the effect of protein size, deviations from the average picture, and implications of this study for defining reliable simplified models of proteins.

Effect of particle size ratio on the conducting percolation threshold of granular conductive–insulating composites

In this paper, we apply Monte Carlo simulation to investigate the conductive percolation threshold of granular composite of conductive and insulating powders with amorphous structure. We focus on the effect of insulating to conductive particle size ratio λ = di/dc on the conducting percolation threshold pc (the volume fraction of the conductive powder). Simulation results show that, for λ = 1, the percolation threshold pc lies between simple cubic and body centred cubic site percolation thresholds, and that as λ increases the percolation threshold decreases. We also use the structural information obtained by the simulation to study the nonlinear current–voltage characteristics of composite with solid volume fraction of conductive powder below pc in terms of electron tunnelling for nanoscale powders, dielectric breakdown for microscale or larger powders, and pressing induced conduction for non-rigid insulating powders.

Orientational ordering in monolayers of ortho–para hydrogen

We discuss orientational ordering in monolayers of solid hydrogen in view of recent experimental findings in NMR studies of (ortho)c–(para)1−c-hydrogen mixtures on a boron nitride substrate. Analysis of the temperature–concentration behavior for the observed NMR frequency splitting is given on the basis of a two-dimensional (J=1)c–(J=0)1−c-rotor model with the quadrupolar coupling constant Γ0=(0.50±0.03) K and the crystalline field amplitude V0=(0.70±0.10) K derived from experiment. The two distinct pararotational short-range ordered structures are described in terms of the local alignment and orientation of the polar principal axis and are shown to be due to the interplay between the positive and negative crystalline fields. It is shown that the local structures observed below the 2D site-percolation threshold cp=0.72 are rather different from the ferromagnetic-type para–rotational ordering suggested earlier by Harris and Berlinsky.

New types of short-range orientational ordering in ortho- para hydrogen monolayers

Orientational ordering in monolayers of solid hydrogen is discussed in view of recent experimental findings in NMR studies of ( ortho) c -( para) 1− c -hydrogen mixtures on boron nitride substrate reported for ortho-H 2 concentrations 0.35≤ c≤0.92 and temperatures 0.14≤ T≤1.80 K. Analysis of the temperature-concentration behavior for the observed NMR frequency splitting is given on the basis of a two-dimension ( J=1) c -( J=0) 1− c -rotor model for which the fundamental parameters, quadrupolar coupling constant Γ 0=0.50±0.03 K and the crystal-field amplitude V 0=0.70±0.10 K, are derived from experiment. The two distinct para-rotational (PR) short-range ordered structures driven by positive and negative crystalline fields are described in terms of the local alignment and orientation of the polar principal axis. It is shown that they are rather different from the ferromagnetic-type ordering suggested earlier by Harris and Berlinsky as a unique PR structure. The common quadrupolar glass and a new hindered rotor phases found below the 2D site percolation threshold c p=0.72 are also discussed.

Critical percolation in high dimensions.

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4-13 dimensions. For d<6 they are preliminary, for d>or =6 they are between 20 and 10(4) times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 Mbytes memory; (ii) a histogram method that allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to approximately 30 and more. Based on these data we propose a scaling law for finite cluster size corrections.

Quantum versus geometric disorder in a two-dimensional Heisenberg antiferromagnet.

We present a numerical study of the spin-1/2 bilayer Heisenberg antiferromagnet with random interlayer dimer dilution. From the temperature dependence of the uniform susceptibility and a scaling analysis of the spin correlation length we deduce the ground state phase diagram as a function of nonmagnetic impurity concentration p and bilayer coupling g. At the site percolation threshold, there exists a multicritical point at small but nonzero bilayer coupling g(m)=0.15(3). The magnetic properties of the single-layer material La(2)Cu(1-p)(Zn,Mg)(p)O4 near the percolation threshold appear to be controlled by the proximity to this new quantum critical point.

Accuracy of universal formulas for percolation thresholds based on dimension and coordination number.

Recent mathematical results regarding percolation thresholds are relevant to efforts to find universal formulas for the percolation threshold. This Brief Report uses exact solutions and recent rigorous bounds for site and bond percolation thresholds to demonstrate that any universal formula based on only the dimension and the coordination number must provide estimates differing substantially from the true threshold value for some lattices.

Two-dimensional percolation and cluster structure of the random packing of binary disks.

In this paper we study the short-range correlated percolation and the cluster structure of two-dimensional (2D) random packing of binary disks with size ratio lambda in the range of 1-5. A Monte Carlo simulation model is used to generate the configuration of random packing first. Then a from-neighbor-to-neighbor propagation method is used to identify the number and sizes of the clusters. Results show that for lambda=1 the percolation threshold p(c) lies between the square and triangular site percolation thresholds. As lambda increases the percolation threshold p(c) (the area fraction of small disks) decreases. To characterize the cluster structure at the percolation threshold, we scale the cluster size s(c) with the cluster radius R as s(c) proportional, variant R(D). The fractal dimension D obtained lies between 1.86 and 1.88 and is independent of the size ratio lambda. This value is in good agreement with the 2D theoretical fractal dimension which is equal to 91/48.

Site percolation for a class of constrained honeycomb lattices

We consider site percolation in a class of correlated random media derived from randomly bi-colored triangular lattices. Media are constructed due to specific state-dependent local constraints. Constraints destroy stochastic independence of elementary events and have specific impact on site percolation thresholds. Further, a connection to critical concentrations of spontaneous magnetization in random 2 D± J Ising models, at groundstates, is discussed.

Polymer collapse, protein folding, and the percolation threshold.

We study the transition of polymers in the dilute regime from a swollen shape at high temperatures to their low-temperature structures. The polymers are modeled by a single self-avoiding walk (SAW) on a lattice for which l of the monomers (the H monomers) are self-attracting, i.e., if two nonbonded H monomers become nearest neighbors on the lattice they gain energy of interaction (epsilon = -/epsilon/); the second type of monomers, denoted P, are neutral. This HP model was suggested by Lau and Dill (Macromolecules 1989, 22, 3986-3997) to study protein folding, where H and P are the hydrophobic and polar amino acid residues, respectively. The model is simulated on the square and simple cubic (SC) lattices using the scanning method. We show that the ground state and the sharpness of the transition depend on the lattice, the fraction g of the H monomers, as well as on their arrangement along the chain. In particular, if the H monomers are distributed at random and g is larger than the site percolation threshold of the lattice, a collapsed transition is very likely to occur. This conclusion, drawn for the lattice models, is also applicable to proteins where an effective lattice with coordination number between that of the SC lattice and the body centered cubic lattice is defined. Thus, the average fraction of hydrophobic amino acid residues in globular proteins is found to be close to the percolation threshold of the effective lattice.

Estimation of the percolation thresholds in dextromethorphan hydrobromide matrices

Percolation theory is a multidisciplinary theory that studies chaotic systems. It has been applied in the pharmaceutical field since 1987. Knowledge of the percolation threshold — one of the most important concepts in percolation theory — results in a clear improvement of the solid dosage form design. The percolation threshold is the concentration showing the maximum probability to obtain, for the first time, a percolating cluster of a substance. In this work, the percolation thresholds of dextromethorphan·HBr/Eudragit ® RS-PM inert matrices were estimated. The drug percolation threshold was estimated as 0.3691±0.0541 ( P=0.05) of the total porosity (ranging between 23 and 36% w/w of drug). The SEM micrographs of the matrices are consistent with the estimated percolation range. In agreement with previous reports, different percolation thresholds were found for the matrix forming excipient Eudragit ® RS-PM. The site percolation threshold (based on the release properties) ranged between 10 and 20% v/v of the excipient, the site-bond percolation threshold (estimated from the mechanical properties) between 29.5 and 34% v/v of the excipient and the swelling percolation threshold between 34.3 and 46.9% v/v of the excipient. These percolation ranges are in agreement with those found previously for Eudragit ® RS-PM matrices containing naltrexone·HCl and morphine·HCl.

PERCOLATION-LIKE PHASE TRANSITION IN A NONEQUILIBRIUM STEADY STATE

We study the Gierer–Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value pc, the steady state consists of stripe-like patterns with long-range connectivity. For p &lt; pc, the connectivity is lost. The value of pc is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of pc, the cluster-related quantities exhibit power-law scaling behavior. The method of finite-size scaling is used to determine the values of the fractal dimension df, the ratio, γ/ν, of the average cluster size exponent γ and the correlation length exponent ν. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.

Phase transitions in a two-component site-bond percolation model

A method to treat a N-component percolation model as effective one component model is presented by introducing a scaled control variable $p_{+}$. In Monte Carlo simulations on $16^{3}$, $32^{3}$, $64^{3}$ and $128^{3}$ simple cubic lattices the percolation threshold in terms of $p_{+}$ is determined for N=2. Phase transitions are reported in two limits for the bond existence probabilities $p_{=}$ and $p_{\neq}$. In the same limits, empirical formulas for the percolation threshold $p_{+}^{c}$ as function of one component-concentration, $f_{b}$, are proposed. In the limit $p_{=} = 0$ a new site percolation threshold, $f_{b}^{c} \simeq 0.145$, is reported.

NONGAUSSIAN DISTRIBUTION OF PERCOLATION THRESHOLDS IN FINITE SIZE LATTICES

The distribution of site percolation thresholds for finite size lattices is a nonGaussian distribution. In this paper, we try to find out the real form of it.

MARKET VOLATILITY AND THE DISTRIBUTION OF MEAN CLUSTER SIZES FOR PERCOLATION

The tail of the distribution for the "mean cluster size" (susceptibility) at the site percolation threshold is found by Monte Carlo simulations to be exponential. Similar distributions might be expected for the market volatility, if stock price fluctuations are described by Cont–Bouchaud percolation.

Epidemics and percolation in small-world networks.

We study some simple models of disease transmission on small-world networks, in which either the probability of infection by a disease or the probability of its transmission is varied, or both. The resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network, and we give exact solutions for the position of this threshold in a variety of cases. We confirm our analytic results by numerical simulation.

An analytical method to compute an approximate value of the site percolation threshold

An analytical method to compute the site percolation threshold is introduced. This method yields an approximate value of Pc larger or equal to the real value. As examples, the computation of Pc is presented for 4 lattices in 2 dimensions: square, triangular, honeycomb and kagome. The results obtained are 0.592 871 6, 0.5, 0.765 069, 0.654 653 7, to be compared with the real values 0.592 746 0, 0.5, 0.697 043, 0.652 703 6. The method is not limited to 2 dimensions.

Similarity of percolation thresholds on the hcp and fcc lattices

Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (p hcp c, S =0.199 255 5±0.000 001 0) and bond (p hcp c, B =0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of p c are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.