Site Percolation Problem Research Articles

Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality.

We define the Q factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability p is increased, the Q factor for the system size L grows systematically to its maximum value Q_{max}(L) at a specific value p_{max}(L) and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of p_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at p_{max} the value of Q_{max}(L) diverges as L^{d} (d denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the Q(ρ,L) factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with ρ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

Determination of the non-Euclidean lower critical dimension for the site percolation problem

Determination of the non-Euclidean lower critical dimension for the site percolation problem

Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone

The site percolation problem is one of the core topics in statistical physics. Evaluation of the percolation threshold, which separates two phases (sometimes described as conducting and insulating), is useful for a range of problems from core condensed matter to interdisciplinary application of statistical physics in epidemiology or other transportation or connectivity problems. In this paper with Newman–Ziff fast Monte Carlo algorithm and finite-size scaling theory the random site percolation thresholds pc for a square lattice with complex neighborhoods containing sites from the sixth coordination zone are computed. Complex neighborhoods are those that contain sites from various coordination zones (which are not necessarily compact). We also present the source codes of the appropriate procedures (written in C) to be replaced in original Newman–Ziff code. Similar to results previously found for the honeycomb lattice, the percolation thresholds for complex neighborhoods on a square lattice follow the power law pc(ζ)∝ζ−γ2 with γ2=0.5454(60), where ζ=∑iziri is the weighted distance of sites in complex neighborhoods (ri and zi are the distance from the central site and the number of sites in the coordination zone i, respectively).

Dependence of the conductivity of two-dimensional site percolation network on the length-ratio of conducting paths to all bonds: the viewpoint of effective path theory

Previous studies have found that the network conductivity of 2-dimensional disordered nanowire networks (DNNs) scaled linearly with the length-ratio of conducting-paths to all nanowires. To show the universality of this rule, the conducting behavior of a 2-dimensional site percolation problem is studied in this article with the assistance of a Monte Carlo based numerical simulation. It is observed that, as the existence probability of site increases in the 2-dimensional site percolated network, more conducting-paths are formed, and the network becomes more conductive. After correlating the site-percolated lattice to DNNs, the normalized network conductivity is observed to scale linearly with the length-ratio of conducting-paths to all bonds, which could be well described by the linear formula using a slope of 2 and an incept of 0.5. As a result, the length-ratio of conducting-paths could again serve as a basic topological parameter in describing the conducting behavior of 2-dimensional site percolation networks. Such universality enables the definition of an ‘effective path theory’, in which the normalized network conductivity scales linearly with the length-ratio of conducting-paths to all bonds.

Random site percolation on honeycomb lattices with complex neighborhoods.

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds p follow the dependency p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i z r-where z and r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.

Momentum signatures of site percolation in disordered two-dimensional ferromagnets

In this work, we consider a two-dimensional square lattice of pinned magnetic spins with nearest-neighbour interactions and we randomly replace a fixed proportion of spins with nonmagnetic defects carrying no spin. We focus on the linear spin-wave regime and address the propagation of a spin-wave excitation with initial momentum $k_0$. We compute the disorder-averaged momentum distribution obtained at time $t$ and show that the system exhibits two regimes. At low defect density, typical disorder configurations only involve a single percolating magnetic cluster interspersed with single defects essentially and the physics is driven by Anderson localization. In this case, the momentum distribution features the emergence of two known emblematic signatures of coherent transport, namely the coherent backscattering (CBS) peak located at $-k_0$ and the coherent forward scattering (CFS) peak located at $k_0$. At long times, the momentum distribution becomes stationary. However, when increasing the defect density, site percolation starts to set in and typical disorder configurations display more and more disconnected clusters of different sizes and shapes. At the same time, the CFS peak starts to oscillate in time with well defined frequencies. These oscillation frequencies represent eigenenergy differences in the regular, disorder-immune, part of the Hamiltonian spectrum. This regular spectrum originates from the small-size magnetic clusters and its weight grows as the system undergoes site percolation and small clusters proliferate. Our system offers a unique spectroscopic signature of cluster formation in site percolation problems.

Mixed site-bond percolation in Archimedean (3,12[formula omitted]) lattices

The site-bond percolation problem in two-dimensional (3,122) lattices has been studied by means of numerical simulation and analytical calculations. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as S∩B and S∪B) were considered. In S∩B (S∪B), two points are connected if a sequence of occupied sites AND (OR) bonds joins them. The analytical method is based on the approximation introduced by Tsallis (2004), which allows to calculate the S∩B and S∪B percolation functions from the percolation functions corresponding to pure site and pure bond percolation problems. Theoretical and numerical data (supplemented by analysis using finite-size scaling theory) were used to determine, for the first time, the complete phase diagram of the system (phase boundary between the percolating and nonpercolating regions). Comparisons between results obtained using the Tsallis’s scheme and simulation data were performed in order to test the reaches and limitations of the approach developed here.

Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices.

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of D_{n} root lattices in n dimensions as well as E_{8}-related lattices. Here, we consider the percolation problem on D_{n} for n=3 to 13 and on E_{8} relatives for n=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_{n} lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as n increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.

Flat-band ferromagnetism of SU(N) Hubbard model on Tasaki lattices

We investigate the para-ferro magnetic transition of the repulsive SU(N) Hubbard model on a type of one- and two-dimensional decorated cubic lattices, referred as Tasaki lattices, which feature massive single-particle ground state degeneracy. Under certain restrictions for constructing localized many-particle ground states of flat-band ferromagnetism, the quantum model of strongly correlated electrons is mapped to a classical statistical geometric site-percolation problem, where the nontrivial weights of different configurations must be considered. We prove rigorously the existence of para-ferro transition for the SU(N) Hubbard model on one-dimensional Tasaki lattice and determine the critical density by the transfer-matrix method. In two dimensions, we numerically investigate the phase transition of SU(3), SU(4) and SU(10) Hubbard models by Metropolis Monte Carlo simulation. We find that the critical density exceeds that of standard percolation, and increases with spin degrees of freedom, implying that the effective repulsive interaction becomes stronger for larger N. We further rigorously prove the existence of flat-band ferromagnetism of the SU(N) Hubbard model when the number of particles equals to the degeneracy of the lowest band in the single-particle energy spectrum.

Dimer site-bond percolation on a triangular lattice

A generalization of the site-percolation problem, in which pairs of neighbor sites (site dimers) and bonds are independently and randomly occupied on a triangular lattice, has been studied by means of numerical simulations. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as and ) have been considered. In (), two points are said to be connected if a sequence of occupied sites and (or) bonds joins them. Numerical data, supplemented by analysis using finite-size scaling theory, were used to determine (i) the complete phase diagram of the system (phase boundary between the percolating and nonpercolating regions), and (ii) the values of the critical exponents (and universality) characterizing the phase transition occurring in the system.

Site-bond percolation on simple cubic lattices: numerical simulation and analytical approach

The site-percolation problem on simple cubic lattices has been studied by means of numerical simulation and analytical calculations based on exact counting of configurations on finite cells. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as and ) have been considered. In (), two points are said to be connected if a sequence of occupied sites and (or) bonds joins them. Theoretical and numerical results, supplemented by analysis using finite-size scaling theory, were used to calculate the complete phase diagram of the system in the () space. Our study allowed us also to determine the critical exponents (and universality) characterizing the phase transition occurring in the system.

A conditionally sure ergodic theorem with an application to percolation

We prove an apparently new type of ergodic theorem, and apply it to the site percolation problem on sparse random sublattices of Zd (d≥2), called “lattices with large holes”. We show that for every such lattice the critical probability lies strictly between zero and one, and the number of the infinite clusters is at most two with probability one. Moreover for almost every such lattice, the infinite cluster, if it exists, is unique with probability one.

Site–bond percolation on triangular lattices: Monte Carlo simulation and analytical approach

A generalization of the pure site and pure bond percolation problems called site–bond percolation on a triangular lattice is studied. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as S∩B and S∪B) for site–bond percolation are used. In S∩B (S∪B), two points are said to be connected if a sequence of occupied sites and (or) bonds joins them. By using finite-size scaling theory, data from S∩B and S∪B are analyzed in order to determine (i) the phase boundary between the percolating and non-percolating regions and (ii) the numerical values of the critical exponents of the phase transition occurring in the system. A theoretical approach, based on exact calculations of configurations on finite triangular cells, is applied to study the site–bond percolation on triangular lattices. The percolation processes have been monitored by following the percolation function, defined as the ratio between the number of percolating configurations and the total number of available configurations for a given cell size and concentration of occupied elements. A comparison of the results obtained by these two methods has been performed and discussed.

Percolation of polyatomic species on a simple cubic lattice

In the present paper, the site-percolation problem corresponding to linear k-mers (containing k identical units, each one occupying a lattice site) on a simple cubic lattice has been studied. The k-mers were irreversibly and isotropically deposited into the lattice. Then, the percolation threshold and critical exponents were obtained by numerical simulations and finite-size scaling theory. The results, obtained for k ranging from 1 to 100, revealed that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the k-mer size; and (ii) the phase transition occurring in the system belongs to the standard 3D percolation universality class regardless of the value of k considered.

Flat-Band Ferromagnetism as a Pauli-Correlated Percolation Problem

We investigate the location and nature of the para-ferro transition of interacting electrons in dispersionless bands using the example of the Hubbard model on the Tasaki lattice. This case can be analyzed as a geometric site-percolation problem where different configurations appear with nontrivial weights. We provide a complete exact solution for the one-dimensional case and develop a numerical algorithm for the two-dimensional case. In two dimensions the paramagnetic phase persists beyond the uncorrelated percolation point, and the grand-canonical transition is via a first-order jump to an unsaturated ferromagnetic phase.

Transmission of packets on a hierarchical network: Statistics and explosive percolation

We analyze an idealized model for the transmission or flow of particles, or discrete packets of information, in a weight bearing branching hierarchical two dimensional network and its variants. The capacities add hierarchically down the clusters. Each node can accommodate a limited number of packets, depending on its capacity, and the packets hop from node to node, following the links between the nodes. The statistical properties of this system are given by the Maxwell-Boltzmann distribution. We obtain analytical expressions for the mean occupation numbers as functions of capacity, for different network topologies. The analytical results are shown to be in agreement with the numerical simulations. The traffic flow in these models can be represented by the site percolation problem. It is seen that the percolation transitions in the 2D model and in its variant lattices are continuous transitions, whereas the transition is found to be explosive (discontinuous) for the V lattice, the critical case of the 2D lattice. The scaling behavior of the second-order percolation case is studied in detail. We discuss the implications of our analysis.

On Equilibrium Distribution of a Reversible Growth Model

We study a probabilistic model of interacting spins indexed by elements of a finite subset of the d-dimensional integer lattice, d≥1. Conditions of time reversibility are examined. It is shown that the model equilibrium distribution converges to a limit distribution as the indexing set expands to the whole lattice. The occupied site percolation problem is solved for the limit distribution. Two models with similar dynamics are also discussed.

Connectedness percolation in monodisperse rod systems: clustering effects

A model is presented that examines the impact of local clustering upon the percolation behaviour of interpenetrable rod-like particles. The percolation threshold, as well as percolation and backbone probabilities, are evaluated as functions of the particle aspect ratio and degree of clustering by way of an analogy to a lattice site percolation problem. The formation of local, physically connected cliques of particles is shown to raise the percolation threshold whilst reducing the percolation and backbone fractions for a fixed volume fraction of particles.

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions

We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is rigorous and based on an established result without the need of a uniqueness assumption, thus firmly establishing all derived results. For kagome-type lattices the exact critical frontier is not known. We derive a closed-form expression for the Potts critical frontier by making use of a homogeneity assumption. The closed-form expression is unique, and we apply it to a host of problems including site, bond, and mixed site-bond percolations on various lattices. It yields exact thresholds for site percolation on kagome, martini, and other lattices and is highly accurate numerically in other applications when compared to numerical determination.

SITE AND BOND PERCOLATION PROBLEM FOR CONSTRUCTION OF MACROSCOPIC SURFACE IN A CUBIC LATTICE

When we distribute particles and bonds onto a cubic lattice, we assume that a plaquette is formed, if four corner sites of a square are occupied or four bonds of a square are occupied. When these plaquettes are produced in a cubic lattice and occupy a certain number of square surfaces in the cubic lattice, then a macroscopic surface is formed following a chain construction. This is the second phase transition added to usual site and bond percolation problems where a spreading chain is formed.