Site Percolation Model Research Articles

Iterative site percolation on triangular lattice

The site percolation on the triangular lattice stands out as one of the few exactly solved statistical systems. By initially configuring critical percolation clusters of this model and randomly reassigning the color of each percolation cluster, we obtain coarse-grained configurations by merging adjacent clusters that share the same color. It is shown that the process can be infinitely iterated in the infinite-lattice limit, leading to an iterative site percolation model. We conjecture from the self-matching argument that percolation clusters remain fractal for any finite generation, which can even take any positive real number by a generalized process. Extensive simulations are performed, and, from the generation-dependent fractal dimension, a continuous family of previously unknown universalities is revealed. Published by the American Physical Society 2024

The percolating cluster is invisible to image recognition with deep learning

We study the two-dimensional site-percolation model on a square lattice. In this paradigmatic model, sites are randomly occupied with probability p; a second-order phase transition from a non-percolating to a fully percolating phase appears at occupation density pc , called percolation threshold. Through supervised deep learning approaches like classification and regression, we show that standard convolutional neural networks (CNNs), known to work well in similar image recognition tasks, can identify pc and indeed classify the states of a percolation lattice according to their p content or predict their p value via regression. When using instead of p the spatial cluster correlation length ξ as labels, the recognition is beginning to falter. Finally, we show that the same network struggles to detect the presence of a spanning cluster. Rather, predictive power seems lost and the absence or presence of a global spanning cluster is not noticed by a CNN with a local convolutional kernel. Since the existence of such a spanning cluster is at the heart of the percolation problem, our results suggest that CNNs require careful application when used in physics, particularly when encountering less-explored situations.

The Blume–Emery–Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume–Emery–Griffiths (BEG) model on the lattice Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^d$$\\end{document} on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} and a vanishing value in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. Second, using a careful coupling with the Bernoulli site percolation model in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we prove rigorously that under imposing +\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+$$\\end{document} boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line we restrict ourselves to d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^2$$\\end{document}, we get a condition for the vanishing of the infinite-volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

New bounds for the site percolation threshold of the hexagonal lattice

The site percolation threshold of the hexagonal lattice satisfies 0.656 246 < p c < 0.739 695. For comparison, the largest previous lower bound of 0.652 703… was established in 1981, and the smallest previous upper bound of 0.743 359 was derived in 2007. The bound is obtained by using the substitution method to compare the hexagonal lattice site model to an exactly-solved two-parameter site percolation model on the martini lattice. Computational reductions involving graph-welding, symmetry, non-crossing partitions, and network flow computations overcome challenges to establishing stochastic ordering between the models.

The Influence of Pore Distribution of Coal Char in the Char Fragmentation and Included Minerals Partitioning: A Percolation Modeling

The processes of char fragmentation, including mineral partitioning and particulate matter (PM) formation during dense and porous char combustion, were observed by a site percolation model. This model simulated the diffusion-controlled regime of char combustion, and the size distributions of included minerals in typical bituminous coal were determined by the computer-controlled scanning electron microscope (CCSEM), and the data were put into the char matrix randomly. The model presents the influence of initial pore distribution on char oxidation and fragmentation, the impact of the char conversion process on the extent of fragmentation, the change of ash distributions with the char conversion, and the particulate matters (PM) size distribution, which is derived from the consequence of the competition between char fragmentation and included minerals partitioning and coalescence. The results indicate that with increasing initial char porosity (φ), the number of large size pores increases but the number of pores decreases, which leads to open pores increasing, close pores decreasing, and the surface reaction area increasing. While φ ≥ 0.4, char fragmentation obviously occurs during the stage in which the rates of char conversion are 0.4–0.6, and it looks as though the maximum value of fragmentation will transfer to an earlier conversion stage if it has a larger φ. The enhanced φ shows a positive effect on the increase in the number and concentration of PM &lt; 10 μm (nominally aerodynamic diameter), this is attributed to char fragments more drastically, and the probability of mineral coalescence reduces a lot.

Impact of pore distribution characteristics on percolation threshold based on site percolation theory

In order to better simulate the porous media with different pore distribution characteristics, this paper introduced a pore distribution factor m (m represents the randomly generated pore volume in the site percolation model, which is expressed by the number of sites)into the classic site percolation model. On this basis, the influence of pore distribution characteristics of two-dimensional (2D) and three-dimensional (3D) porous media on the percolation threshold is studied using numerical simulations. The study found that, when the porosity is the same, larger the value of m, smaller the number of pores in both the simulated 2D and 3D porous media. The decrease in the number of pores leads to the changes in the pore volume distribution characteristics. The shape of each pore, generated using the simulation in this paper, is random, due to which, larger the value of m is, easier it is for pores to merge to form larger pore clusters, which result in a reduction in the percolation threshold. The percolation threshold of 2D (lattice type: square) and 3D (lattice type: simple cubic) site percolation models decreases with the increase in the value of m. When the value of m is unity, the percolation thresholds of 2D and 3D porous media are found to be 59.27% and 31.16%, respectively, which are in agreement with the existing research results. When the value of m increases to 40, the percolation thresholds of 2D and 3D porous media decrease to 38.38% and 8.65%, respectively. Therefore, the pore connectivity of porous media is determined by the porosity and the pore distribution characteristics, whereas the pore connectivity of porous media can be misjudged if only porosity is taken into account.

Quantitative analysis of meandering and dimensional crossover of conduction path in 3D disordered media by percolation modeling

Conduction in disordered media has been the subject of interest in applications and basic research. Percolation theory is used to study conduction, and mean-field theory has been used to describe macroscopic conduction. However, such an approximation theory cannot provide specific insights into the local limiting factors of conduction. In this study, a finite element simulation based on the site percolation model is used to investigate the effect of microstructural defects in materials on transport current. Statistical analysis of the local current obtained by the simulation experiment shows that the meandering of the transport current because of the disturbance of the current path increases as the concentration of defects increases, thus observing a dimensional crossover of the transport current from 1D to 3D. Λ, which represents the degree of current meandering, has a value of 5.3 in the vicinity of the singularity of the percolation threshold (Pc). The governing mechanisms of the transport current differ depending on the concentration and correlation of defects. The main limiting factors of the transport current are the following: ‘shadow effect’ derived from isolated defects in the quasi-1D region where the defects are dilute; the meandering of the current path including the reverse current in the 3D region where the defects are highly concentrated and begin to correlate. These results can be used to understand the current limiting mechanisms in 3D polycrystalline superconducting materials and are expected to provide guidelines for controlling complex microstructures in iron-based superconductors, MgB2, and BSCCO.

Constrained percolation, Ising model, and XOR Ising model on planar lattices

We study site percolation models on planar lattices including the [m,4,n,4] lattice and the square tilings on the Euclidean plane () or the hyperbolic plane (), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of or . In particular, we obtain a description of the numbers of infinite “+” and “−” clusters of the ferromagnetic Ising model on a vertex‐transitive triangular tiling of for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite “+” and “−” clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.

Convergence of percolation probability functions to cumulative distribution functions on square lattices with [formula omitted]-neighborhood

We consider a percolation model on square lattices with sites weighted by beta-distributed random variables S∼Beta(a,b) with a positive real parameters a>0 and b>0. Using the Monte Carlo method, we estimate the percolation probability P∞ as a relative frequency P∞∗ averaged over the target subset of sites on a square lattice. As a result of the comparative analysis, we formulate two empirical hypotheses: the first on the correspondence of percolation thresholds pc to p0-quantiles (where level p0=0.592746… coincides with the percolation threshold for the site percolation model on a square lattice) of random variables Si weighing sites of the square lattice, and the second on the convergence of statistical estimates of percolation probability functions P∞∗(p) to cumulative distribution functions FSi(p) of these variables Si for the supercritical values of the occupation probability p≥pc.

Tortuosity and Percolation Probability on 3 Dimensional Rock Models with Different Model Sizes

Tortuosity, percolation probability and porosity are important rock parameters because all those things affect the fluid flow in rocks. In this study, 3-dimensional cube-shaped rock models were randomly generated with N × N × N size containing matrices and pores. Rock models were made with porosities ranging from 0.1 to 1 with increasing of value 0.1. The selected N is N = 5, 6, 7, 8, 9, 10, 25, 125 and for every N there are 10,000 random configurations. Then, tortuosity was calculated using a nearest-neighbor sites method with 6 neighboring sides. In addition, with the site percolation model, percolation probability was calculated by determining the comparison between the number of percolate configurations and the total number of configurations created. The results show that in large-size rock models, the average tortuosity will increase. Further, the percolation probability is zero at a small porosity and rises to 1 as the porosity increases. Moreover, the slope of the percolation probability curve will increase for the larger model. In other words, the larger the size of the model, the percolation threshold will become larger or more difficult to percolate for small porosity.

Pathogenic–dynamic epidemic agent model with an epidemic threshold

An agent model of epidemic spreading is proposed. When a susceptible agent is exposed to pathogen levels above its specified infection-tolerance threshold F, its state changes to infected, where a pathogen is a virus, microparasite, or any other disease-causing organism or material. The spreading sources of pathogens are the agents that are in the infected state. The growth and decay of a pathogen in an infected host obeys a given function, ft. This function increases linearly in the early period of infection t1, and then decreases linearly to zero in the latter period t2. The simulation results show that as the agent density increases, the model undergoes a phase transition from a local epidemic phase to a pandemic phase. For immobile agents, transition density ρC equals transition density ρP of the corresponding site percolation model. For random-walking agents, the transition density decreases as ρCU≈ρP×U−0.3, where U denotes the average path length of the walking agent during the period t1+t2. This model provides a reliable alternative to the standard SIR model, which is a simple compartmental model of susceptibility, infection, and pathogen removal. Moreover, it can predict epidemic phenomena using fact-based parameters.

Spreading Dynamics of Corrosion on Material Surface Using Site Percolation Model

Spreading phenomenon is usually characterized by existence of two competing processes. On the spreading of corrosion, competition occurs between the power of solution to corrode with the resistance level of a metal surface against corrosion. In the present study, we simulate a corrosion process using site percolation model to determine the spreading pattern of corrosion. The simulation results showed different spreading patterns and dynamics of corrosion when the value of corrode molecules were varied. The simulation results also indicate the emergence of two regimes that happens, smooth regime and random regime. Smooth regime characterized by the shape of its corrosion face is almost the same as before. While random regime characterized by the appearance of random spreading pattern on the shape of corrosion face.

A review on structural characteristics, lithium ion diffusion behavior and temperature dependence of conductivity in perovskite-type solid electrolyte Li3xLa2∕3−xTiO3

Due to various and indefinite Li-ion distributions within the cuboctahedron surrounded by eight TiO6and local subtle distortions, perovskite-type solid electrolyte Li[Formula: see text]La[Formula: see text]TiO3(LLTO) is suitable to be used as a model system for studying the structure–conductivity relationship. This review is focused on structural characteristics, Li-ion diffusion behavior and conductivity in LLTO. Li-ion concentration, cooling rate of heat treatment and heating temperature are shown to be three main factors influencing the space group structure of LLTO, involving the distributions of Li, La ions and vacancies as well as the distortion of TiO6octahedron. In rhombohedral and some orthorhombic phases, Li ions partially occupy O4windows of cuboctahedrons, whose diffusion could be described by the bond percolation model, whereas in other phases with Li ions inside the cuboctahedrons, the site percolation model is applicable to Li-ion conduction. Li-ion conductivity versus temperature curve exhibits non-Arrhenius behavior, which is divided into three sections with different activation energies, although its first-derivative is continuous. The activation energy is correlated closely with the tilting angle of TiO6octahedron, both of which decrease with the increase of temperature. Experimental studies and theoretical results are discussed in parallel to provide insight into the detailed structure-conductivity relationship.

Percolation on the stationary distributions of the voter model

The voter model on ZdZd is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When d≥3d≥3, the set of (extremal) stationary distributions is a family of measures μαμα, for αα between 0 and 1. A configuration sampled from μαμα is a strongly correlated field of 0’s and 1’s on ZdZd in which the density of 1’s is αα. We consider such a configuration as a site percolation model on ZdZd. We prove that if d≥5d≥5, the probability of existence of an infinite percolation cluster of 1’s exhibits a phase transition in αα. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for d≥3d≥3.

The dilute Temperley–Lieb O(n = 1) loop model on a semi infinite strip: the sum rule

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley–Lieb loop model with the loop weight n = 1 on a semi infinite strip of width L (Garbali and Nienhuis 2017 J. Stat. Mech. 043108). We focus here on the computation of the normalization (otherwise called the sum rule) ZL of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization ZL is a symmetric polynomial in the inhomogeneities of the lattice . This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi–Trudi like determinants. Thus we provide a few determinant expressions for the normalization ZL.

The incipient infinite cluster of the uniform infinite half-planar triangulation

We introduce the Incipient Infinite Cluster ($\mathsf{IIC} $) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation ($\mathsf{UIHPT} $), which is the local limit of large random triangulations with a boundary. The $\mathsf{IIC} $ is defined from the $\mathsf{UIHPT} $ by conditioning the open percolation cluster of the origin to be infinite. We prove that the $\mathsf{IIC} $ can be obtained by adding within the $\mathsf{UIHPT} $ an infinite triangulation with a boundary whose distribution is explicit.

Breaking of the site-bond percolation universality in networks.

The stochastic addition of either vertices or connections in a network leads to the observation of the percolation transition, a structural change with the appearance of a connected component encompassing a finite fraction of the system. Percolation has always been regarded as a substrate-dependent but model-independent process, in the sense that the critical exponents of the transition are determined by the geometry of the system, but they are identical for the bond and site percolation models. Here, we report a violation of such assumption. We provide analytical and numerical evidence of a difference in the values of the critical exponents between the bond and site percolation models in networks with null percolation thresholds, such as scale-free graphs with diverging second moment of the degree distribution. We discuss possible implications of our results in real networks, and provide additional insights on the anomalous nature of the percolation transition with null threshold.

A lower bound on the two-arms exponent for critical percolation on the lattice

We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$. We improve slightly this lower bound in any dimension $d\geq2$. Next, starting only with the hypothesis that $\theta(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.

Reversible first-order transition in Pauli percolation.

Percolation plays an important role in fields and phenomena as diverse as the study of social networks, the dynamics of epidemics, the robustness of electricity grids, conduction in disordered media, and geometric properties in statistical physics. We analyze a new percolation problem in which the first-order nature of an equilibrium percolation transition can be established analytically and verified numerically. The rules for this site percolation model are physical and very simple, requiring only the introduction of a weight W(n)=n+1 for a cluster of size n. This establishes that a discontinuous percolation transition can occur with qualitatively more local interactions than in all currently considered examples of explosive percolation; and that, unlike these, it can be reversible. This greatly extends both the applicability of such percolation models in principle and their reach in practice.

Patterns in the English language: phonological networks, percolation and assembly models

In this paper we provide a quantitative framework for the study of phonological networks (PNs) for the English language by carrying out principled comparisons to null models, either based on site percolation, randomization techniques, or network growth models. In contrast to previous work, we mainly focus on null models that reproduce lower order characteristics of the empirical data. We find that artificial networks matching connectivity properties of the English PN are exceedingly rare: this leads to the hypothesis that the word repertoire might have been assembled over time by preferentially introducing new words which are small modifications of old words. Our null models are able to explain the ‘power-law-like’ part of the degree distributions and generally retrieve qualitative features of the PN such as high clustering, high assortativity coefficient and small-world characteristics. However, the detailed comparison to expectations from null models also points out significant differences, suggesting the presence of additional constraints in word assembly. Key constraints we identify are the avoidance of large degrees, the avoidance of triadic closure and the avoidance of large non-percolating clusters.