Random Percolation Research Articles

Critical Permeability from Resummation

Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed to characterize the dependencies on porosities, critical points, and indices. Various approximation techniques are applied without involving popular lubrication approximation in any sense. In particular, the Borel summation technique is applied to the effective polynomial approximations with or without optimization. Minimal difference and minimal derivative optimal conditions are adapted to calculations of critical indices and amplitudes for the effective permeability of thin wavy channels. Critical indices, amplitudes, and thresholds are obtained for the effective permeability and wetted perimeter of the two-dimensional percolating random media. Closed-form expressions for all porosities, critical points, and indices are calculated from the polynomial approximations for the first time.

Generating Galton–Watson trees using random walks and percolation for the Gaussian free field

Generating Galton–Watson trees using random walks and percolation for the Gaussian free field

Short-range correlation of stress chains near solid-to-liquid transition in active monolayers.

Using a three-dimensional model of cell monolayers, we study the spatial organization of active stress chains as the monolayer transitions from a solid to a liquid state. The critical exponents that characterize this transition map the isotropic stress percolation onto the two-dimensional random percolation universality class, suggesting short-range stress correlations near this transition. This mapping is achieved via two distinct, independent pathways: (i) cell-cell adhesion and (ii) active traction forces. We unify our findings by linking the nature of this transition to high-stress fluctuations, distinctly linked to each pathway. The results elevate the importance of the transmission of mechanical information in dense active matter and provide a new context for understanding the non-equilibrium statistical physics of phase transition in active systems.

Cascadic failure and preferential decay via pruning mediated percolation on interdependent networks: implications for schizophrenia

IntroductionDuring adolescence the brain is dynamically changing. Destabilization and acceleration of the normal adolescent synaptic pruning process is likely a contributing factor to the neuropathology of schizophrenia. Details on whether normal pruning effects weaker synapses more or uniformly all synapses with different strengths, needs to be further evaluated. Widespread impairment in structural connectivity in schizophrenic patients involving several cortical and subcortical areas, has been previously described. In this computational study, we investigated a stochastic percolation process in interdependent networks, motivated by pathological synaptic pruning. We examined preferential decay in the connectivity decremental process, as well as differential pruning in interconnected subnetworks. Finally, the speed of the percolation process, as well as the potential for pharmacological interventions of percolation in random networks was explored. Statistical structural properties of decaying networks pinpointed several network attributes which the disintegration and phase transitions qualitatively depended on.ObjectivesThe following objectives were explored: 1.) Apart from a random percolation process, we investigated preferential decay of the connections. We introduced different percolation rules for various connection types. 2.) Based on previous experimental results, we assumed that different interconnected neural subpopulations prune differently, therefore we explored differential pruning process in the subnetworks. 3.) The speed of the percolation was studied and the pharmacological synaptic connectivity change was also analyzed.MethodsWe considered two inter-connected randomly connected networks, where the connections were removed during the percolation process. Simulations were partially performed using Octave on a Lenovo Thinkpad running the Linux operating system and partially performed on a supercomputer at UPPMAX (NAISS Small Compute 2023 Dnr: NAISS 2023/22-102).ResultsWe found that the coupled network system shows rich percolation behaviors with phase transitions for various coupling strength and coupling patterns. The phase transitions of both layers are altered qualitatively between discontinuous, mixed and continuous. Recursively developing percolation in interdependent networks can cause complete fragmentation of these networks, resulting in cascadic failure which might be related to schizophrenia symptoms.Conclusions This computational study analyzes the pruning-mediated percolation in interdependent neural networks. Consequences of the pathological overpruning were related to the attributes of the interdependent network properties. Implications for schizophrenia development and predictions for compensatorial iatrogenic percolation was also pinpointed and discussed.Disclosure of InterestNone Declared

Critical Percolation in the Ordering Kinetics of Twisted Nematic Phases.

I report on the experimental confirmation that critical percolation statistics underlie the ordering kinetics of twisted nematic phases in the Allen-Cahn universality class. Soon after the ordering starts from a homogeneous disordered phase and proceeds toward a broken Z_{2}-symmetry phase, the system seems to be attracted to the random percolation fixed point at a special timescale t_{p}. At this time, exact formulas for crossing probabilities in percolation theory agree with the corresponding probabilities in the experimental data. The ensuing evolution for the number density of hull-enclosed areas is described by an exact expression derived from a percolation model endowed with curvature-driven interface motion. Scaling relation for hull-enclosed areas versus perimeters reveals that the fractal percolation geometry is progressively morphed into a regular geometry up to the order of the classical coarsening length. In view of its universality and experimental possibilities, the study opens a path for exploring percolation keystones in the realm of nonequilibrium, phase-ordering systems.

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).

Cluster scaling and critical points: A cautionary tale.

Many systems in nature are conjectured to exist at a critical point, including the brain and earthquake faults. The primary reason for this conjecture is that the distribution of clusters (avalanches of firing neurons in the brain or regions of slip in earthquake faults) can be described by a power law. Because there are other mechanisms such as 1/f noise that can produce power laws, other criteria that the cluster critical exponents must satisfy can be used to conclude whether or not the observed power-law behavior indicates an underlying critical point rather than an alternate mechanism. We show how a possible misinterpretation of the cluster scaling data can lead one to incorrectly conclude that the measured critical exponents do not satisfy these criteria. Examples of the possible misinterpretation of the data for one-dimensional random site percolation and the one-dimensional Ising model are presented. We stress that the interpretation of a power-law cluster distribution indicating the presence of a critical point is subtle and its misinterpretation might lead to the abandonment of a promising area of research.

A technique for the estimation of percolation thresholds in lattice systems: Application to a problem of granular flow through an orifice

A simple method to estimate percolation thresholds pc in lattice models is presented. The technique is based on the calculation of the probabilities RLX(p) for finding a percolating cluster of type X [X could be horizontal (H), vertical (V), average (A), intersection (I), or union (U)] on a finite lattice of side length L at concentration p of occupied sites. The functions RLX(p) are obtained by numerical simulations. Then, (1) effective percolation thresholds pcX,r(L) are estimated from the condition RLX(p)=r (where r is a parameter ranging between 0 and 1); and (2) the percolation threshold in the thermodynamic limit is determined by extrapolating the effective percolation thresholds to infinite L. The methodology presented herein contains and extends the classical scheme developed by Yonezawa, Sakamoto and Hori, offering a more complete and versatile theoretical/simulation framework to calculate percolation thresholds. The approach is validated by successfully comparing with well-known results obtained for the problem of random site percolation on square lattices. Taking advantage of the behavior of the curves of pcX,r(L) for different values of the parameter r, more general scaling relations are proposed for the effective percolation thresholds. Finally, the technique is applied to model experimental data of clogging transitions in a two-dimensional silo with a vibrated base.

Possible new phase transition in the 3D Ising model associated with boundary percolation

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.

Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. This area is evolving extremely quickly, with the introduction of powerful new methods and the development of increasingly sophisticated techniques, and there have been a number of very significant breakthroughs in the area in recent years. The workshop emphasized several of these recent breakthroughs, which include applications of probabilistic techniques to extremal questions, and of combinatorial techniques to areas of discrete probability theory, such as random matrices and planar percolation.

Percolation and electrical conduction in random systems of curved linear objects on a plane: Computer simulations along with a mean-field approach

Using computer simulations, we have studied the percolation and the electrical conductance of two-dimensional, random percolating networks of curved, zero-width metallic nanowires. We mimicked the curved nanowires using circular arcs. The percolation threshold decreased as the aspect ratio of the arcs increased. Comparison with published data on the percolation threshold of symmetric quadratic Bézier curves suggests that when the percolation of slightly curved wires is simulated, the particular choice of curve to mimic the shape of real-world wires is of little importance. Considering the electrical properties, we took into account both the nanowire resistance per unit length and the junction (nanowire/nanowire contact) resistance. Using mean-field approximation (MFA), we derived the total electrical conductance of nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo numerical simulations. For our random homogeneous and isotropic systems of conductive curved wires, the electric conductance decreased as the wire shape changed from a stick to a ring when the wire length remained fixed.

Biased Random Walk on Spanning Trees of the Ladder Graph

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight c for the (vertical) rungs. Now take a random walk on that spanning tree with a bias beta >1 to the right. In contrast to other random graphs considered in the literature (random percolation clusters, Galton–Watson trees) this one allows for an explicit analysis based on a decomposition of the graph into independent pieces. We give an explicit formula for the speed of the biased random walk as a function of both the bias beta and the edge weight c. We conclude that the speed is a continuous, unimodal function of beta that is positive if and only if beta < beta _c^{(1)} for an explicit critical value beta _c^{(1)} depending on c. In particular, the phase transition at beta _c^{(1)} is of second order. We show that another second order phase transition takes place at another critical value beta _c^{(2)}<beta _c^{(1)} that is also explicitly known: For beta <beta _c^{(2)} the times the walker spends in traps have second moments and (after subtracting the linear speed) the position fulfills a central limit theorem. We see that beta _c^{(2)} is smaller than the value of beta which achieves the maximal value of the speed. Finally, concerning linear response, we confirm the Einstein relation for the unbiased model (beta =1) by proving a central limit theorem and computing the variance.

Electrical conductance of two-dimensional random percolating networks based on mixtures of nanowires and nanorings: A mean-field approach along with computer simulation.

We have studied the electrical conductance of two-dimensional (2D) random percolating networks of zero-width metallic nanowires (a mixture of rings and sticks). We took into account the nanowire resistance per unit length and the junction (nanowire-nanowire contact) resistance. Using a mean-field approximation (MFA) approach, we derived the total electrical conductance of these nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo (MC) numerical simulations. The MC simulations were focused on the case when the circumferences of the rings and the lengths of the wires were equal. In this case, the electrical conductance of the network was found to be almost insensitive to the relative proportions of the rings and sticks, provided that the wire resistance and the junction resistance were equal. When the junction resistance dominated over the wire resistance, a linear dependency of the electrical conductance of the network on the proportions of the rings and sticks was observed.

Self-dual quasiperiodic percolation.

How does the percolation transition behave in the absence of quenched randomness? To address this question, we study two nonrandom self-dual quasiperiodic models of square-lattice bond percolation. In both models, the critical point has emergent discrete scale invariance, but none of the additional emergent conformal symmetry of critical random percolation. From the discrete sequences of critical clusters, we find fractal dimensions of D_{f}=1.911943(1) and D_{f}=1.707234(40) for the two models, significantly different from D_{f}=91/48=1.89583... of random percolation. The critical exponents ν, determined through a numerical study of cluster sizes and wrapping probabilities on a torus, are also well below the ν=4/3 of random percolation. While these new models do not appear to belong to a universality class, they demonstrate how the removal of randomness can fundamentally change the critical behavior.

Critical Percolation on Temporal High-Speed Railway Networks

Deeply understanding the dynamic operating characteristics of high-speed railway (HSR) systems is of essential significance in theory and practice for the planning, construction, and operational management of HSR systems. In this paper, the HSR system is described as a temporal network, and the evolution of connected clusters in the system is considered as a percolation process. The critical integration time Tc of the percolation process can determine the formation of a globally connected cluster and measure the transport performance of the HSR system. The appearance time of critical edges identified at Tc can significantly affect the reliability of the transport performance of an HSR system. Compared to random percolation in the static HSR network, it can be found that the critical fraction pc of the percolation process in a temporal HSR network is almost always larger. This indicates that the global connectivity and the transport performance of HSR systems is overestimated by the static network abstraction. This paper provides a promising way of understanding the dynamic characteristics of HSR systems, evaluating their transport performance, and improving their reliability.

Random 2D nanowire networks: Finite-size effect and the effect of busbar/nanowire contact resistance on their electrical conductivity

We have studied the resistance of two-dimensional random percolating networks of zero-width metallic nanowires (rings or sticks). We took into account the nanowire resistance per unit length, the junction (nanowire/nanowire contact) resistance, and the busbar/nanowire contact resistance. Using a mean-field approximation (MFA), we derived the total resistance of the nanoring-based networks as a function of their geometrical and physical parameters. We have proposed a way of accounting for the contribution of the busbar/nanowire contact resistance toward the network resistance. The MFA predictions have been confirmed by our Monte Carlo numerical simulations. Our study evidenced that the busbar/nanowire contact resistance has a significant effect on the electrical conductivity when the junction resistance dominates over the wire resistance.

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.

Evaluating strategies for spatial allocation of vaccines based on risk and centrality.

When vaccinating a large population in response to an invading pathogen, it is often necessary to prioritize some individuals to be vaccinated first. One way to do this is to choose individuals to vaccinate based on their location. Methods for this prioritization include strategies that target those regions most at risk of importing the pathogen, and strategies that target regions with high centrality on the travel network. We use a simple infectious disease epidemic model to compare a risk-targeting strategy to two different centrality-targeting strategies based on betweenness centrality and random walk percolation centrality, respectively. We find that the relative effectiveness of these strategies in reducing the total number of infections varies with the basic reproduction number of the pathogen, travel rates, structure of the travel network and vaccine availability. We conclude that when a pathogen has high spreading capacity, or when vaccine availability is limited, centrality-targeting strategies should be considered as an alternative to the more commonly used risk-targeting strategies.

Thresholds and critical exponents of explosive bond percolation on the square lattice

In this paper, we present a Monte Carlo study of four explosive bond percolation models on the square lattice: (i) product rule which suppresses intrabonds (PR-SI), (ii) sum rule which suppresses intrabonds (SR-SI), (iii) product rule which enhances intrabonds (PR-EI) and (iv) sum rule which enhances intrabonds (SR-EI). By performing extensive simulations and finite-size scaling analysis of the wrapping probability [Formula: see text] and a ratio [Formula: see text] for PR-SI, SR-SI, PR-EI, and the composite quantities [Formula: see text] and [Formula: see text] for SR-EI (defined by [Formula: see text] and [Formula: see text] corresponding to two different [Formula: see text]-values, respectively), we determine the thresholds [Formula: see text] of all models with best precision. We also estimate the critical exponents [Formula: see text] and [Formula: see text] for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster [Formula: see text] and the second moment [Formula: see text] of sizes of all clusters. For SR-EI, from [Formula: see text] and [Formula: see text], we only obtain pseudo-critical exponents, which are nonphysical. Precisely at [Formula: see text], we study the critical cluster-size distribution [Formula: see text] (number density of the clusters of size [Formula: see text]) for all models and find that it can be described by [Formula: see text], where [Formula: see text] with [Formula: see text] (fractal dimension) for PR-SI, SR-SI, PR-EI, [Formula: see text] (spatial dimension) for SR-EI, and [Formula: see text] with [Formula: see text] is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of [Formula: see text] (and [Formula: see text] with the help of a scaling relation) for SR-EI. It is found that the exponents for PR-SI and SR-SI are consistent with each other, but the ones for PR-EI and SR-EI are different. Our results disclose two facts: (1) all models investigated here undergo continuous phase transitions, since their behaviors can be described by typical scaling formulas for continuous phase transitions; (2) PR-SI and SR-SI belong to a same universality class, however, PR-EI and SR-EI belong to different universality classes, and all of them differ from which random percolation belongs to. This work provides a testing ground for future theoretical studies.

Explosive dismantling of two-dimensional random lattices under betweenness centrality attacks

In the present paper, we study the robustness of two-dimensional random lattices (Delaunay triangulations) under attacks based on betweenness centrality. Together with the standard definition of this centrality measure, we employ a range-limited approximation known as ℓ-betweenness, where paths having more than ℓ steps are ignored. For finite ℓ, the attacks produce continuous percolation transitions that belong to the universality class of random percolation. On the other hand, the attack under the full range betweenness induces a discontinuous transition that, in the thermodynamic limit, occurs after removing a sub-extensive amount of nodes. This behavior is recovered for ℓ-betweenness if the cutoff is allowed to scale with the linear length of the network faster than ℓ∼L0.91. Our results suggest that betweenness centrality encodes information on network robustness at all scales, and thus cannot be approximated using finite-ranged calculations without losing attack efficiency.