Percolation Probability Research Articles

Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

We investigate four types of percolation models—node, edge, bootstrap, and diffusion percolation—in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents (ν, γ, and β) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation dν=γ+2β is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension d is replaced by the Hausdorff dimension.

Level-set percolation of Gaussian random fields on complex networks.

We provide an explicit solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. The solution requires an analysis of the heterogeneous microstructure of the percolation problem, i.e., a self-consistent determination of locally varying percolation probabilities. This is achieved using a cavity or message passing approach. It can be evaluated, both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class.

Neuron-astrocyte interaction-inspired percolative networks with metal microdendrites and nanostars for ultrasensitive and transparent electronic skins

Biological systems provide innovative designs for electronic devices, optimizing network configurations for high-performance signal transmission with minimal energy consumption. The brain, as one of the most complex biological structures, demonstrates efficient network design through the multiscale radial networks of neurons and astrocytes. Emulating these brain networks offers a blueprint for the development of ultrasensitive pressure sensors for electronic skin, aiming to provide a more intuitive and sensitive mode of interaction between humans and machines. Herein, we propose a neuromorphic percolative network inspired by neuron-astrocyte interactions for ultrasensitive pressure sensors employing metal microdendrites and nanostars. Electromechanical investigation through representative volume elements simulation reveals that the optimized arrangement of microdendrites and nanostars in the neuromorphic percolative system enhances the percolation threshold and probability. Following these simulation results, we developed a neuromorphic percolative polyurethane (NP-PU) matrix utilizing the metal microdendrite-nanostar networks. The augmented quantum tunneling effect in the NP-PU matrix was investigated through electrochemical impedance spectroscopy and capacitance analysis. The fabricated piezoresistive pressure sensor with the NP-PU matrix shows ultrahigh sensitivity (160.3 kPa−1) at a low pressure range and a low limit of detection resolution (4 Pa), enabled by multi-channel quantum tunneling in the metal particle networks. Furthermore, the sensor maintains excellent mechanical flexibility and high optical transparency (75.4 %), improving its efficacy in applications like electronic skin and force touch panel. Our study highlights the potential of leveraging biological system-inspired network designs for crafting advanced electronic devices.

Effect of Voids on the Percolation Conductivity of Carbon Nanotube Networks

Two-dimensional (2D) networks consisting of one-dimensional (1D) wires, such as carbon nanotubes, metal nanowires, and graphene nanoribbons, are promising candidates for next-generation flexible transparent conducting electrodes in devices such as organic light-emitting diodes (OLEDs), solar cells, touch screens, smart windows, transparent heaters, and liquid crystal displays. Nanotube networks also have broad application potential in flexible electronics, such as thin film transistors, wearables, electronic skin, and internet of things (IoT) sensors.The electrical conductivity of carbon nanotube networks is governed by percolation, which deals with the formation of long-range connectivity in random networks. As a result, Monte Carlo simulations need to be employed in order to compute the electrical properties of these networks [1,2]. Understanding the impact of voids, which could be present due to lack of control in the deposition process or introduced intentionally, on the percolation conductivity of carbon nanotube networks is critical for applications such as transparent conductive electrodes, thin film transistors, sensing, and hardware security.In this work, we generate two-dimensional square carbon nanotube networks with square voids located at the center of the nanotube network. We define the relative void size as the ratio of the length of the side of the void to the length of the side of the nanotube network. We first study the impact of voids on networks consisting of randomly oriented and straight nanotubes. We compute the percolation probability in these networks as a function of nanotube density for different relative void sizes ranging from 0 (no void) to 0.8. Assuming a Gaussian percolation probability density function (PDF), we find that both the mean and standard deviation of the PDF increase with increasing relative void size. We then compute the relative conductivity change as a function of nanotube density for different relative void sizes and find that it increases approximately linearly with relative void size.According to percolation theory, the conductivity of a nanotube network has a power-law dependence on nanotube density. Next, we extract the local power-law critical exponent as a function of nanotube density for different relative void sizes. We find that the critical exponent approaches 2 at high density for all relative void sizes, in agreement with previous observations for junction-resistance dominated networks without voids [1,3].Furthermore, we generate curvy carbon nanotubes using third order Bezier curves characterized by the curviness angle and aligned nanotubes using an orientation characterized by the alignment angle [1,2]. Using the same procedure as randomly oriented and straight nanotubes, we then investigate the impact of voids on the electronic properties of networks consisting of curvy and aligned nanotubes, including percolation probability, mean and standard deviation of the PDF, relative conductivity change, and nanotube density critical exponent.Our results demonstrate the impact voids have on the percolation conductivity of two-dimensional networks consisting of one-dimensional wires such as carbon nanotubes. These results also show that Monte Carlo simulations are an essential predictive tool for providing insights into the electronic properties of nanotube and nanowire networks, which are promising candidates for a wide range of applications such as flexible transparent conductors, thin film transistors, and resistive switching memory.

Sharp Asymptotics for Arm Probabilities in Critical Planar Percolation

Sharp Asymptotics for Arm Probabilities in Critical Planar Percolation

Atomic-molecular perspectives on local high-temperature structure and transport properties of CaCO3-foamed glass

Significant gaps remain in understanding of gas-melt interfaces in foamed glass systems. We attempt to address these gaps by reporting on the transition of local structure, ionic (molecular) diffusion, and melt viscosity of CaCO3-foamed soda-lime-silica glass using molecular dynamics calculations and experimental analyses. Our calculations show that despite the continual increase in network modifying CaO, the melt network experiences an abnormal polymerization process in the initial stage because of Na ion penetration from the melt skeleton to the interface. The percolation probability decays from the surface to the center of melt because of the gradual polymerized network structure, boosting nonmonotonic changes in the [SiO4] species, equilibrium constants, and average residual charge per O atom. Furthermore, the diffusion capacities of melt ions and CO2 molecules peak at 6 wt% CaCO3 addition, attributed to the combined effects of melt depolymerization and interface constraint. The viscosity jump occurring at 1 wt% CaCO3 addition results in the optimal sample with an even cellular glass framework and strong matrix. These findings contribute to the precise control of foamed glass with optimal performance.

A novel numerical reconstruction method of coal based on characteristics of the cleat system

To effectively characterize the characteristics of the pore network of coal and to provide a realistic flow channel for flow simulation, a numerical reconstruction method of coal based on characteristics of the cleat system was proposed. Based on the real coal obtained through high-resolution micrometer CT, the cleat system characteristic parameters are defined and extracted. Then, a series of coals with different pore characteristics at the same scale and the same pore characteristics at different scales were reconstructed. Besides, the applicability of box dimension in calibrating the pore characteristics of coals was quantitatively analyzed. The results show that the coals reconstructed based on the method proposed in this paper can truly restore the pore characteristics of the real coal, especially the non-extension features of cleats; When the cleat system characteristic parameters remain consistent, the reconstructed coals maintain the same pore characteristics, with an average error of 3.37 % in the autocorrelation function, 0.78 % in the variation function and 1.94 % in the average percolation probability function. The box dimension can effectively identify the differences in pore characteristics of coals. At the same scale, as the distance between face cleats increases, the value of the box dimension of the reconstructed coals monotonically decreases, with a fitting slope of −0.22; At different scales, when the cleat system characteristic parameters remain the same, the value of the box dimension of the reconstructed coals is consistent, with an average relative error of 0.34 %. The research results provide a reasonable and effective flow channel for the microscopic/mesoscopic flow simulation of coalbed methane and provide a reference for the quantitative calibration of the pore characteristics of coals.

A computational method for calculating the electrical and thermal conductivity of random composites

In the present work, for the first time, the electrical–thermal conductivity of a composite random material is correlated with the number of conductive paths, the mean path length, and the mean effective path width measured in a representative surface element (RSE) by Monte Carlo sampling. This work is organized in two parts. The theoretical calculation takes place in the first part for the correlation of conductive paths by adopting three approaches (incremental with respect to accuracy and complexity) based on: (i) one-component straight paths; (ii) one-component non-straight paths; (iii) multi-component non-straight paths. In the second part, a novel numerical methodology for the calculation of the conductance of the RSE through the conductive paths, mean path length and the mean effective path is developed for the general model of multi-component non-straight line paths based on Ohm’s law. In addition, a methodology that reduces the calculated conductivity from the microscale to the macroscale is proposed, that takes into account the probability of percolation for finite sizes and the average value of the conductivity from the samples in which percolation occurs. After the necessary consistency steps and verification of the method and its implementation on random binary material systems, it is further applied to solve synthetic continuous percolation problems at the microscale and macroscale. The results and the accuracy of the calculations are in close agreement with existing models. The electrical Direct Current (DC) and thermal conductivity follow the same scaling relation and the proposed methodology can be applied at the same stage without additional cost. In a random binary medium the simulations showed that a scaling law of conductivity is followed by an exponent value close to 1.3. In addition, the number of conductive paths, the average length of the paths, and the effective width follow scaling laws with exponents 1.44±0.01 , −0.25±0.01 and −0.39±0.01, respectively.

Unlocking Electrode Performance of Disordered Rocksalt Oxides Through Structural Defect Engineering and Surface Stabilization with Concentrated Electrolyte

Abstract Li‐excess cation‐disordered rocksalt oxides can boost the energy density of rechargeable batteries by using anionic redox, but the inferior reversibility of anionic redox hinders its use for practical applications. Herein, a binary system of Li3NbO4–LiMnO2 is targeted and the systematic study on factors affecting electrode reversibility, i.e., percolation probability, electronic conductivity, defect concentrations, and electrolyte solutions, is conducted. A Mn‐rich sample, Li1.1Nb0.1Mn0.8O2, delivers a smaller reversible capacity compared with a Li‐rich sample, Li1.3Nb0.3Mn0.4O2, because of the limitation of ionic migration associated with insufficient percolation probability for disordered oxides. Nevertheless, a larger reversible capacity of Li1.3Nb0.3Mn0.4O2 originates from excessive activation of anionic redox, leading to the degradation of electrode reversibility. The superior performance of Li1.1Nb0.1Mn0.8O2, including Li ion migration kinetics and electronic transport properties, is unlocked by the enrichment of structural defects for nanosized oxides. Moreover, electrode reversibility is further improved by using a highly concentrated electrolyte solution with LiN(SO2F)2 through the surface stabilization on high‐voltage exposure. Superior capacity retention, >100 cycles, is achieved for nanosized Li1.1Nb0.1Mn0.8O2. The electrolyte decomposition and surface stabilization mechanisms are also carefully examined, and it is revealed that the use of highly concentrated electrolyte solution can effectively prevent lattice oxygen being further oxidized and transition metal ion dissolution.

Optimal mechanical interactions direct multicellular network formation on elastic substrates

Cells self-organize into functional, ordered structures during tissue morphogenesis, a process that is evocative of colloidal self-assembly into engineered soft materials. Understanding how intercellular mechanical interactions may drive the formation of ordered and functional multicellular structures is important in developmental biology and tissue engineering. Here, by combining an agent-based model for contractile cells on elastic substrates with endothelial cell culture experiments, we show that substrate deformation-mediated mechanical interactions between cells can cluster and align them into branched networks. Motivated by the structure and function of vasculogenic networks, we predict how measures of network connectivity like percolation probability and fractal dimension as well as local morphological features including junctions, branches, and rings depend on cell contractility and density and on substrate elastic properties including stiffness and compressibility. We predict and confirm with experiments that cell network formation is substrate stiffness dependent, being optimal at intermediate stiffness. We also show the agreement between experimental data and predicted cell cluster types by mapping a combined phase diagram in cell density substrate stiffness. Overall, we show that long-range, mechanical interactions provide an optimal and general strategy for multicellular self-organization, leading to more robust and efficient realizations of space-spanning networks than through just local intercellular interactions.

Investigation on the sulfate attack of metakaolin blended recycled concrete based on percolation theory

Recycled aggregate concrete (RAC) is one of the most important ways to achieve sustainable development for concrete industry. However, the research on durability, especially on sulfate transport and erosion mechanism, of RAC needs deep investigation. Pore structure evolution is a vital parameter for sulfate transport, which is difficult to be traced. To investigate the dynamic changes of pore structure in metakaolin blended RAC under sulfate erosion, percolation theory is addressed in this study. Firstly, dry and wet cycle was applied to accelerate the sulfate transport in metakaolin (MK) blended RAC systems. Then the compressive strength of the samples after sulfate attack was measured to evaluate the sulfate erosion degree. Furthermore, backscattered electron imaging (BSE) was used to acquire the percolation parameters, such as wet site ratio, percolation probability and so forth. Finally, sulfate content and compressive strength of the sample were related to the percolation probability. Results present that MK exhibits excellent resistance to sulfate erosion and can be used to enhance the performance of RAC. There is a certain correlation between the degree of sulfate erosion, compressive strength, and percolation parameters. The influence of the number of dry–wet cycles and MK content on sulfate ion transport can be demonstrated using percolation theory, which can be employed for predicting sulfate transport and revealing erosion mechanisms. This enables a clear understanding of the ion transport mechanisms within concrete and holds significant importance on the investigation of durability issues of RAC.

Ising percolation in the hyperbolic plane

We study infinite “+” or “−” clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph G with finite vertex degree. If the critical percolation probability pcsite for the independent identically distributed (IID). Bernoulli site percolation on G is less than 12, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If pcsite>12, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of pcsite. We also obtain an explicit region for the coupling constant when the XOR Ising model (the product of two IID Ising models) does not have a unique infinite contour a.s. and an explicit region for the coupling constant when the XOR Ising model has infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters.

On the Size of Superconducting Islands on the Density-Wave Background in Organic Metals

Most high-Tc superconductors are spatially inhomogeneous. Usually, this heterogeneity originates from the interplay of various types of electronic ordering. It affects various superconducting properties, such as the transition temperature, the magnetic upper critical field, the critical current, etc. In this paper, we analyze the parameters of spatial phase segregation during the first-order transition between superconductivity (SC) and a charge- or spin-density wave state in quasi-one-dimensional metals with imperfect nesting, typical of organic superconductors. An external pressure or another driving parameter increases the transfer integrals in electron dispersion, which only slightly affects SC but violates the Fermi surface nesting and suppresses the density wave (DW). At a critical pressure Pc, the transition from a DW to SC occurs. We estimate the characteristic size of superconducting islands during this phase transition in organic metals in two ways. Using the Ginzburg–Landau expansion, we analytically obtain a lower bound for the size of SC domains. To estimate a more specific interval of the possible size of the superconducting islands in (TMTSF)2PF6 samples, we perform numerical calculations of the percolation probability via SC domains and compare the results with experimental resistivity data. This helps to develop a consistent microscopic description of SC spatial heterogeneity in various organic superconductors.

Dimensionless unified modelling of macroscale electric properties of conductive fibre networks in different regimes

The electric properties of conductive fibre networks, which can be representative of carbon nanotube networks, are investigated by means of numerical simulations and a dimensionless unified macroscale model is developed. Fibres are represented as rigid chains of discrete element method particles and we compress systems of fibres in a representative volume element periodic in three dimensions. In the used electric resistor network method, the particles are used as discretisation points to construct a system of linear equations linking the particle conductivities to their local electric potentials and in turn allowing for predictions on the electric macroscale properties of the fibre system. A carried out dimensional analysis suggests suitable scaling laws to unify different macroscale fibre systems in two different regimes - a percolating and a conductive regime. The dimensionless macroscale conductance is found to depend on the percolation threshold and percolation probability. It is moreover found that the critical solid volume fraction for rigid fibres is not only dependent on the fibre aspect ratio, but can also depend on the compression height of the system. Additionally, correlations between transmittance and solid volume fractions are found allowing for possibly simple solid volume fraction estimations in experiments.

Knowledge percolation threshold and optimization strategies of the combinatorial network for complex innovation in the digital economy

Digital economy expands the source of knowledge for innovation and accelerates the flow and combination of knowledge to form novel knowledge combinations, thereby generating the interdisciplinary knowledge production model. In this context, complex innovation which is characterized by the knowledge production consequence based on the combinations of multiple-field knowledge has become the new way for firms to seize new development opportunities and compete in the digital economy. Given that complex innovation emerged from a gradually forming large, multilayered, combinatorial network consists of collaboration networks in various knowledge fields that are initially separated, the challenge of facillatating the emergence of complex innovation is unveiling the minimum proportion of connected paths in the combinatorial network to trigger effective transmission of multi-fields knowledge and offering applicable optimization strategies to optimize that proportion. This study incorporated Ohm's law into the percolation theoretical framework and calculate the knowledge percolation threshold of the combinatorial network and its subnetworks with patent data of Chinese strategic emerging industries. We further examined the optimization results of six strategies in terms of their optimization effects and time costs. Accordingly, we revealed the probability of knowledge percolation occurring in a combinatorial network and its subnetworks, clarified knowledge transmission characteristics according to knowledge-based cluster dynamics, and determined strategies for optimizing the knowledge percolation threshold. This study is not only highly feasible and exercisable for academics to conduct future studies, but it also has vital implications for the practitioners to utilize and control the knowledge transmission of the combinatorial network to realize the complex innovation.

Percolation in simple directed random graphs with a given degree distribution

AbstractWe study site and bond percolation in simple directed random graphs with a given degree distribution. We derive the percolation threshold for the giant strongly connected component and the fraction of vertices in this component as a function of the percolation probability. The results are obtained for degree sequences in which the maximum degree may depend on the total number of nodes n, being asymptotically bounded by $n^{\frac{1}{9}}$.

Crossing probabilities for planar percolation

We prove a general Russo–Seymour–Welsh (RSW) result valid for any invariant bond percolation measure on Z2 satisfying positive association. This means that the crossing probability of a long rectangle is related by a universal homeomorphism to the crossing probability of a short rectangle.

Sorption-Deformation-Percolation Model for Diffusion in Nanoporous Media.

Diffusion of molecules in porous media is a critical process that is fundamental to numerous chemical, physical, and biological applications. The prevailing theoretical frameworks are challenged when explaining the complex dynamics resulting from the highly tortuous host structure and strong guest-host interactions, especially when the pore size approximates the size of diffusing molecule. This study, using molecular dynamics, formulates a semiempirical model based on theoretical considerations and factorization that offer an alternative view of diffusion and its link with the structure and behavior (sorption and deformation) of material. By analyzing the intermittent dynamics of water, microscopic self-diffusion coefficients are predicted. The apparent tortuosity, defined as the ratio of the bulk to the confined self-diffusion coefficients, is found to depend quantitatively on a limited set of material parameters: heat of adsorption, elastic modulus, and percolation probability, all of which are experimentally accessible. The proposed sorption-deformation-percolation model provides guidance on the understanding and fine-tuning of diffusion.

The design of highly conductive and stretchable polymer conductors with low-load nanoparticles.

Highly conductive and stretchable polymer conductors fabricated from conductive fillers and stretchable polymers are urgently needed in flexible electronics, implants, soft robotics, etc. However, polymer conductors encounter the conductivity-stretchability dilemma, in which high-load fillers needed for high conductivity always result in the stiffness of materials. Herein, we propose a new design of highly conductive and stretchable polymer conductors with low-load nanoparticles (NPs). The design is achieved by the self-assembly of surface-modified NPs to efficiently form robust conductive pathways. We employ computer simulations to elucidate the self-assembly of the NPs in the polymer matrices under equilibrium and tensile states. The conductive pathways retain 100% percolation probability even though the loading of the NPs is lowered to ∼2% volume. When the tensile strain reaches 400%, the percolation probability of the ∼2% NP system is still greater than 25%. The theoretical prediction suggests a way for advancing flexible conductive materials.

1‐independent percolation on ℤ2×Kn

AbstractA random graph model on a host graph is said to be 1‐independent if for every pair of vertex‐disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . For an infinite connected graph , the 1‐independent critical percolation probability is the infimum of the such that every 1‐independent random graph model on in which each edge is present with probability at least almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that tends to a limit in as , and they asked for the value of this limit. We make progress on a related problem by showing that In fact, we show that the equality above remains true if the sequence of complete graphs is replaced by a sequence of weakly pseudorandom graphs on vertices with average degree . We conjecture the answer to Balister and Bollobás's question is also .