Two-dimensional Bond Percolation Research Articles

Size distributions of the largest hole in the largest percolation cluster and backbone

At criticality in two dimensions, the size distribution of holes within a large percolation cluster or backbone was recently found to be governed by a power law with the exponent given by a hyperscaling relation. It was also found that, though the boundary of the largest hole is a fractal, the size of the largest hole itself is proportional to Ld, where L represents the linear system size and d=2 is the spatial dimension. In this work, we study the probability distribution of the size of the largest hole in the largest percolation cluster and that in the largest backbone, and compare them with the probability distribution of the size of the largest percolation cluster and that of the largest backbone themselves. We obtain these four size distributions in the critical region by Monte Carlo simulations for the two-dimensional bond percolation model on the square lattice with periodic boundary conditions. It is found that each of these distributions can be described by a different single-variable function P(C,L)dC=P̃(x)dx, with x≡C∕LdC, where C is the size of the structure whose (fractal) dimension is dC. The function P̃(x) is generally not unimodal, and interestingly, for the largest cluster hole, symmetric properties are found in its size distribution due to duality, e.g., P̃(x) is symmetric around the mean value x=1 at the percolation threshold pc. We characterize these size distributions by calculating the variance and different dimensionless cumulant ratios, which exhibit rich critical behaviors at the pseudocritical and critical points. For example, the extremum of the excess kurtosis and the zero point of the skewness locate at the same (pseudocritical) point, whose values are different for the four structures but approach pc in the limit L→∞. We also perform wrapping analyses for each size distribution, showing that different wrapping types contribute to the overall distribution in different forms.

Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

We investigate the influence of the range of interactions in the two-dimensional bond percolation model, by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number $z$ of equivalent neighbors. We also consider the $z \to \infty$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite $z$ are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for $q=3$ and $q=4$ Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as $y_r \approx 2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.

Extreme paths in oriented two-dimensional percolation

Abstract A useful result about leftmost and rightmost paths in two-dimensional bond percolation is proved. This result was introduced without proof in Gray (1991) in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete-time contact process and two-dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.

Monte Carlo renormalization-group analysis of percolation

We describe a Monte Carlo renormalization group approach to the calculation of critical behavior for percolation models. This approach can be utilized to determine the renormalized bond probabilities and the values of the critical exponents. We illustrate the method for two-dimensional bond percolation, but the method is also applicable to other percolation models and other dimensions.

Lattice-oriented percolation system applied to volatility behavior of stock market

In this paper, a discrete time series of stock price process is modeled by the two-dimensional lattice-oriented bond percolation system. Percolation theory, as one of statistical physics systems, has brought new understanding and techniques to a broad range of topics in nature and society. According to this financial model, we studied the statistical behaviors of the stock price from the model and the real stock prices by comparison. We also investigated the probability distributions, the long memory and the long-range correlations of price returns for the actual data and the simulative data. The empirical research exhibits that for proper parameters, the simulative data of the financial model can fit the real markets to a certain extent.

Monte-Carlo study of scaling exponents of rough surfaces and correlated percolation

We calculate the scaling exponents of the two-dimensional correlated percolation cluster's hull and unscreened perimeter. Correlations are introduced through an underlying correlated random potential, which is used to define the state of bonds of a two-dimensional bond percolation model. Monte-Carlo simulations are run and the values of the scaling exponents are determined as functions of the Hurst exponent H in the range -0.75 <= H <= 1. The results confirm the conjectures of earlier studies.

Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdoes-Renyi random graphs with finite connectivity and for two-dimensional bond percolation. Probabilities as small as 10^-180 are accessed using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the Erdoes-Renyi ensemble agree well with previously obtained analytical results. The results for the percolation problem, where no analytical results are available, are qualitatively similar, but the shapes of the distributions are somehow different and the finite-size corrections are sometimes much larger. Furthermore, for both problems, a first-order phase transition at low temperatures T within the artificial ensemble is found in the percolating regime, respectively.

From complex structures to complex processes: Percolation theory applied to the formation of a city

We investigate the morphology of the spatial pattern resulting from the division of land into the parcels that is observed in the centers of the cities, by analyzing the distribution function of the parcel areas. A simple model based on a two-dimensional bond percolation is employed to mimic the process of the formation of the city. The model reproduces the empirical distribution of the parcel areas that is found to exhibit the power law with the exponent tau=2.0. We argue that the city emerges from a collection of separated settlements in a process that can be described as a structural phase transition.

Crystal Structure and Diffusion Path in the Fast Lithium-Ion Conductor La0.62Li0.16TiO3

We report the results of a neutron powder diffraction study of the La(0.62)Li(0.16)TiO3 perovskite that determined the diffusion path of lithium cations at room temperature. At 77 K, the Li cations are located at the 2c site (Wycoff notation of the Cmmm space group) on the (002) La deficient layer, while, at room temperature, they are spread over a wide area and migrate following the 2c-4f-2c or 2c-2d-2c tie line on the (002) layer. The probability density of Li cations has a minimum between the 2c and 4f or between the 2c and 2d positions on the diffusion path in contrast to the previous reports where the bottleneck has been thought to be located at the 2c, 2d, and 4f positions. On the basis of the present structural model, the Li-cation conductivity is discussed in terms of a two-dimensional bond-percolation model for Li-cation diffusion. It was found that the vacancy at the La site is essential for the Li-cation conduction.

Anisotropic thermally activated diffusion in percolation systems.

We present a study of static and frequency-dependent diffusion with anisotropic thermally activated transition rates in a two-dimensional bond percolation system. The approach accounts for temperature effects on diffusion coefficients in disordered anisotropic systems. Static diffusion shows an Arrhenius behavior for low temperatures with an activation energy given by the highest energy barrier of the system. From the frequency-dependent diffusion coefficients, we calculate a characteristic frequency omega(c) approximately 1/t(c), related to the time t(c) needed to overcome a characteristic barrier. We find that omega(c) follows an Arrhenius behavior with different activation energies in each direction.

Growth, morphology, and optical and electrical properties of semicontinuous metallic films

The growth and optical properties of semicontinuous silver films on insulator substrates were studied experimentally and theoretically. In the experimental studies, films were synthesized by the pulsed-laser deposition technique, characterized by electron microscopy and in situ optical and dc electrical resistance measurements and studied using near-field optical microscopy. The percolation threshold of the films was found to be at \ensuremath{\sim}65% metal filling fraction, higher than the 50%--60% range of values predicted for two-dimensional (2D) bond or site percolation films and suggesting the importance of grain coalescence and 3D growth in our system. Local optical properties measured by near-field optical microscopy were compared with theoretical results obtained using the block-elimination method, with good agreement. Local-field distributions were found to depend strongly on the metal concentration and wavelength of illumination. The degree of localization was found to increase at metal concentrations both above and below the percolation threshold. At a very high metal coverage, very strong local fields were observed in submicron voids of a metal dielectric film. These fields are likely due to localized surface plasmon polaritons.

Traveling time and traveling length in critical percolation clusters

We study traveling time and traveling length for tracer dispersion in two-dimensional bond percolation, modeling flow by tracer particles driven by a pressure difference between two points separated by Euclidean distance r. We find that the minimal traveling time t(min) scales as t(min) approximately r(1.33), which is different from the scaling of the most probable traveling time, t* approximately r(1.64). We also calculate the length of the path corresponding to the minimal traveling time and find l(min) approximately r(1.13) and that the most probable traveling length scales as l* approximately r(1.21). We present the relevant distribution functions and scaling relations.

Percolation study on electrical resistivity of SiC/Si3N4 composites with segregated distribution

Experimental and theoretical studies are made to account for the formation of segregated networks in SiC/Si3N4 particulate composite. It was observed in the present experimental work that fine SiC particles are uniformly dispersed along boundaries of Si3N4 matrix grains. The proposed theoretical model takes into account the volume fraction of dispersed particles necessary to form a percolating network as a function of the size ratio of SiC particle to Si3N4 grain. Threshold particle loading for percolation is estimated by applying a modified two-dimensional bond percolation model and then electrical resistivity of the composite by using percolation and general effective media equations. Both predicted values of threshold particle loading and electrical resistivity were found to agree closely with the present experimental values.

Series expansion analysis of the backbone properties of two-dimensional percolation clusters

Low-density series expansions for the backbone properties of two-dimensional bond percolation clusters are derived and analysed. Expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice. Earlier series work was confined to three properties of the square lattice and was to order . The fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest Monte Carlo results. The union, intersection and length of the longest self-avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for . On the other hand, the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection, the triangular and square lattices are found to have sigificantly different dimensions. The fractal dimension of the shortest path also appears to be non-universal and we find for the square lattice and for the triangular lattice. Critical amplitude ratios are considered and found to be in agreement with theoretical inequalities.

Landau Hamiltonians with random potentials: Localization and the density of states

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.

Neutron spectroscopy in rba2cu3Ox (R  Ho, Er; 6 &lt;x⩽7) compounds

Abstract Inelastic neutron scattering has been employed to study the perovskite-type high c superconducting compounds RBa 2 Cu 3 O ϵ (R  Er, Ho; 6 ϵ ⩽ 7). For ErBa 2 Cu 3 O ϵ , we have determined the crystalline electric field (CEF) level scheme as a function of the oxygen content ϵ. The variation of the energies and intensities of the CEF transitions versus ϵ is shown to be predominantly related to a charge transfer process between the chains and the planes. The observed energy spectra are the result of a superposition of two different metallic components and a semiconducting one, i.e. there is clear experimental evidence for phase separation. A two-dimensional bond percolation model explains the appearance of superconductivity as well as the critical oxygen concentrations associated with the two-plateau structure of T c . For HoBa 2 Cu 3 O 7 , we have measured the low-lying magnetic excitations of Ho 3+ ions in a grain-aligned sample. From the dispersive behavior of the spin-wave excitations, we derive the nearest-neighbor magnetic coupling parameters in the Heisenberg approximation. From complimentary experiments on a magnetically diluted Ho 0.1 Y 0.9 Ba 2 Cu 3 O 7 compound, we have been able to observe Ho 3+ dimer excitations which allow a direct determination of the magnetic coupling strength in the ( a , b ) plane. The line shapes of the Ho 3+ excitations turn out to be highly asymmetric which we interpret in terms of an exchange interaction between the Ho 3+ spins and the fluctuating Cu 2+ spins.

Crystal field, cluster formation and percolative superconductivity in ErBa 2Cu 3O x

Inelastic neutron scattering has been employed to study the crystal-field excitations of Er 3+ in ErBa 2Cu 3O x (6<×<7). The lowest-lying crystal-field state exhibits a most unusual behavior versus the oxygen content x, which we interpret in terms of cluster formation in the superconducting CuO 2 planes. Three different cluster types are identified corresponding to local regions of semiconducting and metallic (T c ≈ 60 K, T c ≈ 90 K) character. The superconductivity is shown to result from the formation of a two-dimensional percolative network. A two-dimensional bond percolation model correctly predicts the critical oxygen contents associated with the two-plateau structure of T c.

Neutron spectroscopic evidence for cluster formation and percolative superconductivity in ErBa2Cu3Ox.

Inelastic neutron scattering has been employed to study in detail the low-energy crystal-field excitations of ${\mathrm{Er}}^{3+}$ in ${\mathrm{ErBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{\mathit{x}}$ (6x7). The observed energy spectra are found to be the result of a superposition of three stable states, which we interpret in terms of local regions of semiconducting and metallic character. The superconductivity is shown to result from the formation of a two-dimensional percolative network. A two-dimensional bond percolation model correctly predicts the critical oxygen contents associated with the two-plateau structure of ${\mathit{T}}_{\mathit{c}}$.

Vibrational properties of bond-diluted triangular lattices

Vibrational properties of two-dimensional (2D) triangular elastic bond percolation systems with scalar, central, and bond-bending forces are studied by using the HCPA, the recursion method and the replica trick. The densities of states calculated by means of the HCPA agree qualtatively well with corresponding results of the recursion method. The low-frequency behaviour of the density of states can be interpreted in terms of a phonon-fracton crossover. The dynamic structure factor calculated with the aid of the HCPA for the scalar model exhibits for any wave vector only a single peak, contrary to threedimensional systems.

Moment of inertia of doubly connecting bonds in two-dimensional bond percolation

By applying a theorem from conformal field theory to bond percolation in two dimensions the author obtains an equation for the second moment I of bonds which doubly connect a percolation cluster: (I)=A(5 square root 3/27 pi 2)(p-pc)-2. The equation is exact as the occupancy probability p approaches the critical probability pc from either side. A is a calculable lattice dependent number which for a square lattice equals 1/2.