Scaling and universality in the spanning probability for percolation

Percolation transition and bimodal density distribution in hydrogen fluoride

Hydrogen-bond networks in associating fluids can be extremely robust and characterize the topological properties of the liquid phase, as in the case of water, over its whole domain of stability and beyond. Here, we report on molecular dynamics simulations of hydrogen fluoride (HF), one of the strongest hydrogen-bonding molecules. HF has more limited connectivity than water but can still create long, dynamic chains, setting it apart from most other small molecular liquids. Our simulation results provide robust evidence of a second-order percolation transition of HF's hydrogen bond network occurring below the critical point. This behavior is remarkable as it underlines the presence of two different cohesive mechanisms in liquid HF, one at low temperatures characterized by a spanning network of long, entangled hydrogen-bonded polymers, as opposed to short oligomers bound by the dispersion interaction above the percolation threshold. This second-order phase transition underlines the presence of marked structural heterogeneity in the fluid, which we found in the form of two liquid populations with distinct local densities.

Bridge percolation: electrical connectivity of discontinued conducting slabs by metallic nanowires

The properties of nanostructured networks of conductive materials have been extensively studied under the lens of percolation theory. In this work, we introduce a novel type of local percolation phenomenon used to investigate the conduction properties of a new hybrid material that combines sparse metallic nanowire networks and fractured conducting thin films on flexible substrates. This original concept could potentially lead to the design of a novel composite transparent conducting material. Using a complementary approach including formal analytical derivations, Monte Carlo simulations and electrical circuit representation for the modelling of bridged-percolating nanowire networks, we unveil the key relations between linear crack density, nanowire length and network areal mass density that ensure electrical percolation through the hybrid. The proposed theoretical model provides key insights into the conduction mechanism associated with the original concept of bridge percolation in random nanowire networks.

Percolation in a simple cubic lattice with distortion

Site percolation in a distorted simple cubic lattice is characterized numerically employing the Newman-Ziff algorithm. Distortion is administered in the lattice by systematically and randomly dislocating its sites from their regular positions. The amount of distortion is tunable by a parameter called the distortion parameter. In this model, two occupied neighboring sites are considered connected only if the distance between them is less than a predefined value called the connection threshold. It is observed that the percolation threshold always increases with distortion if the connection threshold is equal to or greater than the lattice constant of the regular lattice. On the other hand, if the connection threshold is less than the lattice constant, the percolation threshold first decreases and then increases steadily as distortion is increased. It is shown that the variation of the percolation threshold can be well explained by the change in the fraction of occupied bonds with distortion. The values of the relevant critical exponents of the transition strongly indicate that percolation in regular and distorted simple cubic lattices belong to the same universality class. It is also demonstrated that this model is intrinsically distinct from the site-bond percolation model.

Statistics of percolating clusters in a model of photosynthetic bacteria

In photosynthetic organisms the energy of the illuminating light is absorbed by the antenna complexes and transmitted by the excitons to the reaction centers (RCs). The energy of light is either absorbed by the RCs, leading to their "closing" or is emitted through fluorescence. The dynamics of the light absorption is described by a simple model developed for exciton migration that involves the exciton hopping probability and the exciton lifetime. During continuous illumination the fraction of closed RCs x continuously increases, and at a critical threshold x_{c}, a percolation transition takes place. Performing extensive Monte Carlo simulations, we study the properties of the transition in this correlated percolation model. We measure the spanning probability in the vicinity of x_{c}, as well as the fractal properties of the critical percolating cluster, both in the bulk and at the surface.

Critical polynomials in the nonplanar and continuum percolation models

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Mechanical Heterogeneity in Tissues Promotes Rigidity and Controls Cellular Invasion

We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity is introduced into the cell shape index (p_{0}) that tunes the stiffness at a single-cell level. The addition of heterogeneity can always enhance the mechanical rigidity of the epithelial layer by increasing its shear modulus, hence making it more rigid. There is an excellent scaling collapse of our data as a function of a single scaling variable f_{r}, which accounts for the overall fraction of rigid cells. We identify a universal threshold f_{r}^{*} that demarcates fluid versus solid tissues. Furthermore, this rigidity onset is far below the contact percolation threshold of rigid cells. These results give rise to a separation of rigidity and contact percolation processes that leads to distinct types of solid states. We also investigate the influence of heterogeneity on tumor invasion dynamics. There is an overall impedance of invasion as the tissue becomes more rigid. Invasion can also occur in an intermediate heterogeneous solid state that is characterized by significant spatial-temporal intermittency.

Percolation of sites not removed by a random walker in d dimensions

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of N=uL^{d} steps on a d-dimensional hypercubic lattice of size L^{d} (with periodic boundaries). We systematically explore dependence of the probability Π_{d}(L,u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u. The concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated-their correlations decaying with the distance r as 1/r^{d-2} (in d>2). On increasing L, the percolation probability Π_{d}(L,u) approaches a step function, jumping from 1 to 0 when u crosses a percolation threshold u_{c} that is close to 3 for all 3≤d≤6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν=2/(d-2). There is no percolation threshold at the lower critical dimension of d=2, with the percolation probability approaching a smooth function Π_{2}(∞,u)>0.

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as K_{c}(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C_{1} follows a single-variable function as P(C_{1},L)dC_{1}=P[over ̃](x)dx with x≡C_{1}/L^{d_{F}} (L is the linear size), where the fractal dimension d_{F} is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution P[over ̃](x) in three dimensions, and attributed to the different approaching behaviors for K→K_{c}+0^{±}. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d_{min}(3D)=1.25936(12) and d_{min}(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d_{B}(3D)=2.1673(15) and d_{B}(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d_{min} and d_{B} either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

Machine learning of phase transitions in the percolation and XY models

In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.

Widom line, dynamical crossover, and percolation transition of supercritical oxygen via molecular dynamics simulations

Supercritical oxygen, a cryogenic fluid, is widely used as an oxidizer in jet propulsion systems and is therefore of paramount importance in gaining physical insights into processes such as transcritical and supercritical vaporization. It is well established in the scientific literature that the supercritical state is not homogeneous but, in fact, can be demarcated into regions with liquid-like and vapor-like properties, separated by the "Widom line." In this study, we identified the Widom line for oxygen, constituted by the loci of the extrema of thermodynamic response functions (heat capacity, volumetric thermal expansion coefficient, and isothermal compressibility) in the supercritical region, via atomistic molecular dynamics simulations. We found that the Widom lines derived from these response functions all coincide near the critical point until about 25 bars and 15-20 K, beyond which the isothermal compressibility line begins to deviate. We also obtained the crossover from liquid-like to vapor-like behavior of the translational diffusion coefficient, shear viscosity, and rotational relaxation time of supercritical oxygen. While the crossover of the translational diffusion coefficient and shear viscosity coincided with the Widom lines, the rotational relaxation time showed a crossover that was largely independent of the Widom line. Further, we characterized the clustering behavior and percolation transition of supercritical oxygen molecules, identified the percolation threshold based on the fractal dimension of the largest cluster and the probability of finding a cluster that spans the system in all three dimensions, and found that the locus of the percolation threshold also coincided with the isothermal compressibility Widom line. It is therefore clear that supercritical oxygen is far more complex than originally perceived and that the Widom line, dynamical crossovers, and percolation transitions serve as useful routes to better our understanding of the supercritical state.

Architectural Engineering of Nanowire Network Fine Pattern for 30 μm Wide Flexible Quantum Dot Light-Emitting Diode Application

Replacing rigid metal oxides with flexible alternatives as a next-generation transparent conductor is important for flexible optoelectronic devices. Recently, nanowire networks have emerged as a new type of transparent conductor and have attracted wide attention because of their all-solution-based process manufacturing and excellent flexibility. However, the intrinsic percolation characteristics of the network determine that its fine pattern behavior is very different from that of continuous films, which is a critical issue for their practical application in high-resolution devices. Herein, a simple optimization approach is proposed to address this issue through the architectural engineering of the nanowire network. The aligned and random silver nanowire networks are fabricated and compared in theory and experimentally. Remarkably, network performance can be notably improved with an aligned structure, which is helpful for external quantum efficiency and the luminance of quantum dot light-emitting diodes (QLEDs) when the network is applied as the bottom-transparent electrode. More importantly, the advantage introduced by network alignment is also of benefit to fine pattern performance, even when the pattern width is narrowed to 30 μm, which leads to improved luminescent properties and lower failure rates in fine QLED strip applications. This paradigm illuminates a strategy to optimize nanowire network based transparent conductors and can promote their practical application in high-definition flexible optoelectronic devices.

Percolation of the site random-cluster model by Monte Carlo method

We propose a site random-cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random-cluster model, we measure several quantities, such as the wrapping probability Re, the percolating cluster density P∞, and the magnetic susceptibility per site χp, as well as two exponents, such as the thermal exponent yt and the fractal dimension yh of the percolating cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5, and 4, the numerical estimation of the exponents yt and yh are consistent with the theoretical values. The universalities of the site random-cluster model and the bond random-cluster model are completely identical. For larger values of q, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.

Early warning signals of desertification transitions in semiarid ecosystems

The identification of early warning signals for regime shifts in ecosystems is of crucial importance given their impact in terms of economic and social effects. We present here the results of a theoretical study on the desertification transition in semiarid ecosystems under external stress. We performed numerical simulations based on a stochastic cellular automaton model, and we studied the dynamics of the vegetation clusters in terms of percolation theory, assumed as an effective tool for analyzing the geometrical properties of the clusters. Focusing on the role played by the strength of external stresses, measured by the mortality rate m, we followed the progressive degradation of the ecosystem for increasing m, identifying different stages: first, the fragmentation transition occurring at relatively low values of m, then the desertification transition at higher mortality rates, and finally the full desertification transition corresponding to the extinction of the vegetation and the almost complete degradation of the soil, attained at the maximum value of m. For each transition we calculated the spanning probabilities as functions of m and the percolation thresholds according to different spanning criteria. The identification of the different thresholds is proposed as an useful tool for monitoring the increasing degradation of real-world finite-size systems. Moreover, we studied the time fluctuations of the sizes of the biggest clusters of vegetated and nonvegetated cells over the entire range of mortality values. The change of sign in the skewness of the size distributions, occurring at the fragmentation threshold for the biggest vegetation cluster and at the desertification threshold for the nonvegetated cluster, offers new early warning signals for desertification. Other new and robust indicators are given by the maxima of the root-mean-square deviation of the distributions, which are attained respectively inside the fragmentation interval, for the vegetated biggest cluster, and inside the desertification interval, for the nonvegetated cluster.

Morphological study of elastic-plastic-brittle transitions in disordered media

We use a spring lattice model with springs following a bilinear elastoplastic-brittle constitutive behavior with spatial disorder in the yield and failure thresholds to study patterns of plasticity and damage evolution. The elastic-perfectly plastic transition is observed to follow percolation scaling with the correlation length critical exponent ν≈1.59, implying the universality class corresponding to the long-range correlated percolation. A quantitative analysis of the plastic strain accumulation reveals a dipolar anisotropy (for antiplane loading) which vanishes with increasing hardening modulus. A parametric study with hardening modulus and ductility controlled through the spring level constitutive response demonstrates a wide spectrum of behaviors with varying degree of coupling between plasticity and damage evolution.

Two-dimensional percolation at the free water surface and its relation with the surface tension anomaly of water

The percolation temperature of the lateral hydrogen bonding network of the molecules at the free water surface is determined by means of molecular dynamics computer simulation and identification of the truly interfacial molecules analysis for six different water models, including three, four, and five site ones. The results reveal that the lateral percolation temperature coincides with the point where the temperature derivative of the surface tension has a minimum. Hence, the anomalous temperature dependence of the water surface tension is explained by this percolation transition. It is also found that the hydrogen bonding structure of the water surface is largely model-independent at the percolation threshold; the molecules have, on average, 1.90 ± 0.07 hydrogen bonded surface neighbors. The distribution of the molecules according to the number of their hydrogen bonded neighbors at the percolation threshold also agrees very well for all the water models considered. Hydrogen bonding at the water surface can be well described in terms of the random bond percolation model, namely, by the assumptions that (i) every surface water molecule can form up to 3 hydrogen bonds with its lateral neighbors and (ii) the formation of these hydrogen bonds occurs independently from each other.

Microscopic origin of the surface tension anomaly of water

We investigate the hydrogen bonding percolation threshold of water molecules at the surface of the liquid-vapor interface. We show that the percolation temperature agrees within statistical accuracy with the high-temperature inflection point of the water surface tension. We associate the origin of this surface tension anomaly of water with the sudden breakup of the hydrogen-bonding network in the interfacial molecular layer.

Flexible transparent conductive materials based on silver nanowire networks: a review

The class of materials combining high electrical or thermal conductivity, optical transparency and flexibility is crucial for the development of many future electronic and optoelectronic devices. Silver nanowire networks show very promising results and represent a viable alternative to the commonly used, scarce and brittle indium tin oxide. The science and technology research of such networks are reviewed to provide a better understanding of the physical and chemical properties of this nanowire-based material while opening attractive new applications.

Electrical percolation in quasi-two-dimensional metal nanowire networks for transparent conductors

We simulate the conductivity of quasi-two-dimensional mono- and polydisperse rod networks having rods of various aspect ratios (L/D = 25-800) and rod densities up to 100 times the critical density and assuming contact-resistance dominated transport. We report the rod-size dependence of the percolation threshold and the density dependence of the conductivity exponent over the entire L/D range studied. Our findings clarify the range of applicability for the popular widthless-stick description for physical networks of rodlike objects with modest aspect ratios and confirm predictions for the high-density dependence of the conductivity exponent obtained from modest-density systems. We also propose a heuristic extension to the finite-width excluded area percolation model to account for arbitrary distributions in rod length and validate this solution with numerical results from our simulations. These results are relevant to nanowire films that are among the most promising candidates for high performance flexible transparent electrodes.

Continuum percolation thresholds in two dimensions

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality.

Corrected finite-size scaling in percolation

This Rapid Communication proposes a comprehensive scaling theory for percolation, which clarifies the intrinsic nature of finite-size scaling and effectively addresses the finite-size effects. This theory applies to extensive systems, including especially the explosive percolation. It is suggested that explosive percolation shares the same scaling law as normal percolation, but may suffer from more severe finite-size effects. Remarkably, in contrast to previous studies, relying on the framework of our theory, the present Rapid Communication suggests that for all systems, the universal scaling functions do not depend on the boundary conditions.

Potts and percolation models on bowtie lattices

We give the exact critical frontier of the Potts model on bowtie lattices. For the case of q = 1, the critical frontier yields the thresholds of bond percolation on these lattices, which are exactly consistent with the results given by Ziff et al. [J. Phys. A 39, 15083 (2006)]. For the q = 2 Potts model on a bowtie A lattice, the critical point is in agreement with that of the Ising model on this lattice, which has been exactly solved. Furthermore, we do extensive Monte Carlo simulations of the Potts model on a bowtie A lattice with noninteger q. Our numerical results, which are accurate up to seven significant digits, are consistent with the theoretical predictions. We also simulate the site percolation on a bowtie A lattice, and the threshold is s(c) = 0.5479148(7). In the simulations of bond percolation and site percolation, we find that the shape-dependent properties of the percolation model on a bowtie A lattice are somewhat different from those of an isotropic lattice, which may be caused by the anisotropy of the lattice.

Crossing on hyperbolic lattices

We divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p. We consider the {7,3} (heptagonal), enhanced or extended binary tree (EBT), the EBT-dual, and the {5,5} (pentagonal) lattices. We find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p_{l} to the upper p_{u} critical values. We find bounds and estimates for the values of p_{l} and p_{u} for these lattices and identify the self-duality point p corresponding to where the crossing probability equals 1/2. Comparison is made with recent numerical and theoretical results.

Finite-size scaling in asymmetric systems of percolating sticks

We investigate finite-size scaling in percolating widthless stick systems with variable aspect ratios in an extensive Monte Carlo simulation study. A generalized scaling function is introduced to describe the scaling behavior of the percolation distribution moments and probability at the percolation threshold. We show that the prefactors in the generalized scaling function depend on the system aspect ratio and exhibit features that are generic to the whole class of the percolating systems. In particular, we demonstrate the existence of a characteristic aspect ratio for which percolation probability at the threshold is scale invariant and definite parity of the prefactors in the generalized scaling function for the first two percolation probability moments.

Correction-to-scaling exponent for two-dimensional percolation

We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Ω ≤ 72/91, ω = DΩ ≤ 3/2, and Δ₁ = νω ≤ 2, based upon Cardy's result for the crossing probability on an annulus. The upper bounds are consistent with many previous measurements of site percolation on square and triangular lattices and new measurements for bond percolation, suggesting that they are exact. They also agree with exponents for hulls proposed recently by Aharony and Asikainen, based upon results of den Nijs. A corrections scaling form evidently applicable to site percolation is also found.

Effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory

We investigate the effects of anisotropy on the finite-size scaling of connectivity and conductivity of continuum percolation in three dimensions. We consider a system of size X×Y×Z in which cubic bodies of size a×b×c are placed randomly. We define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction P (averaged over the realizations), about the isotropic universal curves will be a function of the two aspect ratios. This is accounted by considering an apparent percolation threshold in each direction which leads to 50% of realizations connecting in that direction. We find the aspect ratios' dependency of the apparent threshold and investigate the finite-size scaling transformations for the mean connected fraction and its associated fluctuations. Moreover, we apply a single phase pressure solver to determine the conductivity of various realizations of the system. Finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling.

Finite-size scaling in stick percolation

This work presents the generalization of the concept of universal finite-size scaling functions to continuum percolation. A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems. The percolation threshold of high precision is determined for isotropic widthless stick systems as Ncl2=5.637 26+/-0.000 02 , with Nc as the critical density and l as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor A=0.106 910+/-0.000 009 , the spanning probability of stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolation.

Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation

When a two-dimensional Ising ferromagnet is quenched from above the critical temperature to zero temperature, the system eventually converges to either a ground state or an infinitely long-lived metastable stripe state. By applying results from percolation theory, we analytically determine the probability to reach the stripe state as a function of the aspect ratio and the form of the boundary conditions. These predictions agree with simulation results. Our approach generally applies to coarsening dynamics of nonconserved scalar fields in two dimensions.

Percolation and critical O(n) loop configurations

We study a percolation problem based on critical loop configurations of the O(n) loop model on the honeycomb lattice. We define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop, and investigate the bond-percolation properties of these dual clusters. The universal properties at the percolation threshold are argued to match those of Kasteleyn-Fortuin random clusters in the critical Potts model. This relation is checked numerically by means of cluster simulations of several O(n) models in the range 1<or=n<or=2. The simulation results include the percolation threshold for several values of n, as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold. Our numerical results for the exponents are in agreement with existing Coulomb-gas results for the random-cluster model, which confirms the relation between both models. We discuss the renormalization flow of the bond-dilution parameter p as a function of n, and provide an expression that accurately describes a line of unstable fixed points as a function of n, corresponding with the percolation threshold. Furthermore, the renormalization scenario indicates the existence, in a p versus n diagram, of another line of fixed points at p=1, which is stable with respect to p.

Percolation threshold parameters of fluids

Extensive Monte Carlo simulations on three qualitatively different model supercritical fluids (square-well fluid, Lennard-Jonesium, and primitive water) have been performed to examine percolation threshold parameters for continuum (correlated) models and their relation to general results valid for random lattice models; random-site percolation simple-cubic lattice has therefore been considered as well. Two different bond criteria, the configurational and self-bound ones, defining a cluster have been used. In addition to the percolation threshold occupation probability pc and the percolation threshold fluid density rhoc, the correlation length exponent nu and the wrapping probability at the percolation threshold Rw,c have also been evaluated. It is found that parameters nu and Rw,c exhibit not only strong temperature dependence but also, unlike the case of lattice systems, dependence on the nature of the system considered and the employed definition of the cluster.

Pseudo-random-number generators and the square site percolation threshold

Selected pseudo-random-number generators are applied to a Monte Carlo study of the two-dimensional square-lattice site percolation model. A generator suitable for high precision calculations is identified from an application specific test of randomness. After extended computation and analysis, an ostensibly reliable value of p_{c}=0.59274598(4) is obtained for the percolation threshold.

Percolation transition in fluids: scaling behavior of the spanning probability functions

We examine different spanning probability functions (wrapping and crossing) near the percolation threshold of a supercritical square-well fluid and determine the threshold values of these probabilities, which may be universal for all fluids. It is shown that for a continuous system, over a wide range of system size, the wrapping probabilities can be described by universal scaling functions, whereas the crossing probabilities do not show such universal behavior over the same range of system size. The obtained universal functions for the wrapping probabilities can be used for an estimation of the percolation threshold in fluids in general. The results for the crossing probabilities allow us then to characterize large clusters in real fluids.

Percolation Transition in Supercritical Water: A Monte Carlo Simulation Study

Computer simulations of water have been performed on the canonical ensemble at 15 different molecular number densities, ranging from 0.006 to 0.018 A-3, along the supercritical isotherm of 700 K, in order to characterize the percolation transition in the system. It is found that the percolation transition occurs at a somewhat higher density than what is corresponding to the supercritical extension of the boiling line. We have shown that the fractal dimension of the largest cluster and the probability of finding a spanning cluster are the most appropriate properties for the location of the true percolation threshold. Thus, percolation transition occurs when the fractal dimension of the largest cluster reaches 2.53, and the probability of finding a cluster that spans the system in at least one dimension and in all the three dimensions reaches 0.97 and 0.65, respectively. On the other hand, the percolation threshold cannot be accurately located through the cluster size distribution, as it is distorted by appearance of clusters crossing the finite simulated system even far below the percolation threshold. The structure of the largest water cluster is dominated by a linear, chainlike arrangement, which does not change noticeably until the largest cluster becomes infinite.

Formation of mesoscopic water networks in aqueous systems

Formation of the macroscopically-infinite hydrogen-bonded water network in various aqueous systems occurs via 3D percolation transition when the probability of finding a spanning water cluster exceeds 95%. As a result, in a wide interval of water content below the percolation threshold, rarefied quasi-2D water networks span over the mesoscopic length scale. Formation and topology of spanning water networks, which affect various properties of aqueous systems, can be described within the framework of the percolation theory.

Effect of anisotropy on finite-size scaling in percolation theory

We investigate the effects of anisotropy on finite-size scaling of site percolation in two dimensions. We consider a lattice of size n(x) x n(y). We define an aspect ratio omega=n(x)/n(y) and consider the mean connected fraction P (averaged over the realizations) as a function of the site occupancy probability (p), the system size (n(x)), and this aspect ratio. It is clear that there is an easy direction for percolation, which is in the short direction (i.e., y if omega>1) and a difficult direction which is along the long axis. We define an apparent percolation threshold in each direction as the value of p when 50% of realizations connect in that direction. We show that standard finite-size scaling applies if we use this apparent threshold. We also find a finite-size scaling for the fluctuations about this mean connected fraction.

Tails of the crossing probability

The scaling of the tails of the probability of a system to percolate only in the horizontal direction pi(hs) was investigated numerically for the correlated site-bond percolation model (q -state Potts model) for q=1 , 2, 3, 4 (where q is the number of spin states). We have to demonstrate that the crossing probability pi(hs) (p) far from the critical point p(c) has the shape pi(hs) (p) similar to D exp [cL (p- p(c) )(nu) ] where nu is the correlation length index, and p=1-exp (-beta) is the probability of a bond to be closed. For the tail region the correlation length is smaller than the lattice size. At criticality the correlation length reaches the sample size and we observe crossover to another scaling pi(hs) (p) similar to A exp {-b [L (p- p(c) )(nu)](x)}. Here x is a scaling index describing the central part of the crossing probability.

Formation of Spanning Water Networks on Protein Surfaces via 2D Percolation Transition

The formation of spanning hydrogen-bonded water networks on protein surfaces by a percolation transition is closely connected with the onset of their biological activity. To analyze the structure of the hydration water at this important threshold, we performed the first computer simulation study of the percolation transition of water in a model protein powder and on the surface of a single protein molecule. The formation of an infinite water network in the protein powder occurs as a 2D percolation transition at a critical hydration level, which is close to the values observed experimentally. The formation of a spanning 2D water network on a single rigid protein molecule can be described by adapting the cluster analysis of conventional percolation studies to the characterization of the connectivity of the hydration water on the surface of finite objects. Strong fluctuations of the surface water network are observed close to the percolation threshold. Our simulations also furnish a microscopic picture for understanding the specific values of the experimentally observed hydration levels, where different steps of increasing mobility in the hydrated powder are observed.

Universality of the crossing probability for the Potts model for q=1, 2, 3, 4

The universality of the crossing probability pi(hs) of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for q=2, 3, 4 and for percolation q=1. We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction pi(hs) has the universal form pi(hs)=A(q)Q(z) for q=1,2,3,4 as a function of the scaling variable z=[b(q)L(1/nu(q))[p-p(c)(q,L)]](zeta(q)). Here, p=1-exp(-beta) is the probability of a bond to be closed, A(q) is the nonuniversal crossing amplitude, b(q) is the nonuniversal metric factor, nu(q) is the correlation length index, and zeta(q) is the additional scaling index. The universal function Q(x) approximately equal exp(-/z/). The nonuniversal scaling factors were found numerically.

Percolation on two- and three-dimensional lattices

In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.

Convergence of threshold estimates for two-dimensional percolation

Using a recently introduced algorithm for simulating percolation in microcanonical (fixed-occupancy) samples, we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary, specifically for site percolation on a square lattice. We show that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90+/-0.02. For the median and cell-to-cell estimates of the percolation threshold we verify that convergence does not depend on this exponent, having instead a slightly faster convergence with a trivial analytic leading exponent.

Fast Monte Carlo algorithm for site or bond percolation

We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.

Universal scaling functions for bond percolation on planar-random and square lattices with multiple percolating clusters

Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on L1xL2 planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratios L(1)/L(2). We calculate the probability for the appearance of n percolating clusters, W(n); the percolating probabilities P; the average fraction of lattice bonds (sites) in the percolating clusters, <c(b)>(n) (<c(s)>(n)), and the probability distribution function for the fraction c of lattice bonds (sites), in percolating clusters of subgraphs with n percolating clusters, f(n)(c(b)) [f(n)(c(s))]. Using a small number of nonuniversal metric factors, we find that W(n), P, <c(b)>(n) (<c(s)>(n)), and f(n)(c(b)) [f(n)(c(s))] for random lattices, duals of random lattices, and square lattices have the same universal finite-size scaling functions. We also find that nonuniversal metric factors are independent of boundary conditions and aspect ratios.

Aspect-ratio dependence of percolation probability in a rectangular system

I investigate site percolation on a rectangular system (aspect ratio a) of a square lattice for a given occupation probability p (not restricted to p(c)) using computer simulations. The dependence of the percolation probability R on a is shown and analyzed on the basis of a modified finite-size scaling function. A method for evaluating R without statistical simulations is proposed for given conditions (longitudinal dimension L, a, and p) of the system.

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.