Convergence of threshold estimates for two-dimensional percolation

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 of straight slots on a square grid

This work is devoted to the emergence of a connected network of slots (cracks) on a square grid. Accordingly, extensive Monte Carlo simulations and finite-size scaling analysis have been conducted to study the site percolation of straight slots with length l measured in the number of elementary cells of the grid with the edge size L. A special focus was made on the dependence of the percolation threshold p_{C}(l,L) on the slot length l varying in the range 1≤l≤L-2 for the square grids with edge size in the range 50≤L≤1000. In this way, we found that p_{C}(l,L) strongly decreases with increase of l, whereas the variations of p_{C}(l=const,L) with the variation of ratio l/L are very small. Consequently, we acquire the functional dependencies of the critical filling factor and percolation strength on the slot length. Furthermore, we established that the slot percolation model interpolates between the site percolation on square lattice (l=1) and the continuous percolation of widthless sticks (l→∞) aligned in two orthogonal directions. In this regard, we note that the critical number of widthless sticks per unit area is larger than in the case of randomly oriented sticks. Our estimates for the critical exponents indicate that the slot percolation belongs to the same universality class as standard Bernoulli percolation.

Geometric clusters in the overlap of the Ising model

We study the percolation properties of geometrical clusters defined in the overlap space of two statistically independent replicas of a square-lattice Ising model that are simulated at the same temperature. In particular, we consider two distinct types of clusters in the overlap, which we dub soft- and hard-constraint clusters, and which are subsets of the regions of constant spin overlap. By means of Monte Carlo simulations and a finite-size scaling analysis we estimate the transition temperature as well as the set of critical exponents characterizing the percolation transitions undergone by these two cluster types. The results suggest that both soft- and hard-constraint clusters percolate at the critical temperature of the Ising model and their critical behavior is governed by the correlation-length exponent ν=1 found by Onsager. At the same time, they exhibit nonstandard and distinct sets of exponents for the average cluster size and percolation strength.

Postprocessing techniques for gradient percolation predictions on the square lattice

In this work, we revisit the classic problem of site percolation on a regular square lattice. In particular, we investigate the effect of quantization bias errors on percolation threshold predictions for large probability gradients and propose a mitigation strategy. We demonstrate through extensive computational experiments that the assumption of a linear relationship between probability gradient and percolation threshold used in previous investigations is invalid. Moreover, we demonstrate that, due to skewness in the distribution of occupation probabilities visited the average does not converge monotonically to the true percolation threshold. We identify several alternative metrics which do exhibit monotonic (albeit not linear) convergence and document their observed convergence rates.

Percolation threshold of curved linear objects

In this study, we investigate the percolation threshold of curved linear objects, describing them as quadratic Bézier curves. Using Monte Carlo simulations, we calculate the critical number densities of the curves with different curviness. We also obtain the excluded area of the curves. When an excluded area is given, we can find the critical number density of the curves with arbitrary curviness. Apparent conductivity exponents are computed for the curves, and these values are found to be analogous to that of sticks in the percolative region for a junction resistance dominant system. These results can be used to analyze the optoelectrical performance of metal nanowire films because the high-aspect-ratio metal nanowires can be easily curved during coating.

History-dependent percolation in two dimensions

We study the history-dependent percolation in two dimensions, which evolves in generations from standard bond-percolation configurations through iteratively removing occupied bonds. Extensive simulations are performed for various generations on periodic square lattices up to side length L=4096. From finite-size scaling, we find that the model undergoes a continuous phase transition, which, for any finite number of generations, falls into the universality of standard two-dimensional (2D) percolation. At the limit of infinite generation, we determine the correlation-length exponent 1/ν=0.828(5) and the fractal dimension d_{f}=1.8644(7), which are not equal to 1/ν=3/4 and d_{f}=91/48 for 2D percolation. Hence, the transition in the infinite-generation limit falls outside the standard percolation universality and differs from the discontinuous transition of history-dependent percolation on random networks. Further, a crossover phenomenon is observed between the two universalities in infinite and finite generations.

Bond percolation between k separated points on a square lattice

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞, p[over ¯]_{c1}=0.51755(5), p[over ¯]_{c2}=0.53219(5), p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L^{-1/ν_{k}} where ν_{k}=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν_{1}=48/41, ν_{2}=24/23, ν_{3}=16/17, ν_{4}=6/7, etc. We also study three-point correlations in the system and show how for p>p_{c}, the correlation ratio goes to 1 (no net correlation) as L→∞, while at p_{c} it reaches the known value of 1.022.

Standard and inverse bond percolation of straight rigid rods on square lattices

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage p_{j,k} and percolation threshold p_{c,k} are determined for a wide range of k (1≤k≤120). p_{j,k} and p_{c,k} exhibit a decreasing behavior with increasing k, p_{j,k→∞}=0.7476(1) and p_{c,k→∞}=0.0033(9) being the limit values for large k-mer sizes. p_{j,k} is always greater than p_{c,k}, and consequently, the percolation phase transition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold p_{c,k}^{i}, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage p_{j,k}^{i} is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that p_{j,k}^{i}=1-p_{j,k}. The obtained results for p_{c,k}^{i} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤18. For k>18, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds p_{j,k}^{i} is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes (k=1) are much more effective than correlated attacks on groups of close nodes (large k's). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size k considered.

Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

In this article, we investigate explosive bond percolation (EBP) with the product rule, formally known as the Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice. One of the key features of the EBP transition is the delay, compared to the corresponding random bond percolation (RBP), in the onset of the spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, although ultimately proved wrong, that explosive percolation (EP) exhibits a first-order transition. In the case of EP, much effort has been devoted to resolving the issue of its order of transition and almost no effort has been devoted to finding the critical point, critical exponents, etc., to classify it into universality classes. This is in sharp contrast to the situation for classical random percolation. We do not even know all the exponents of EP for a regular planar lattice or for an Erdös-Renyi network. We first find the critical point p_{c} numerically and then obtain all the critical exponents, β, γ, and ν, as well as the Fisher exponent τ and the fractal dimension d_{f} of the spanning cluster. We also compare our results for EBP with those for RBP and find that all the exponents of EBP obey the same scaling relations as do those for RBP. Our findings suggest that EBP is not special except for the fact that the exponent β is unusually small compared to that for RBP.

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.

Universality class of site and bond percolation on multifractal scale-free planar stochastic lattice

In this article, we investigate both site and bond percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. The characteristic property of percolation is that it exhibits threshold phenomena as we find sudden or abrupt jump in spanning probability across p_{c} accompanied by the divergence of some other observable quantities, which is reminiscent of a continuous phase transition. Indeed, percolation is characterized by the critical behavior of percolation strength P(p)∼(p_{c}-p)^{β}, mean cluster size S∼(p_{c}-p)^{-γ}, and the system size L∼(p_{c}-p)^{-ν}, which are known as the equivalent counterpart of the order parameter, susceptibility, and correlation length, respectively. Moreover, the cluster size distribution function n_{s}(p_{c})∼s^{-τ} and the mass-length relation M∼L^{d_{f}} of the spanning cluster also provide useful characterization of the percolation process. We numerically obtain a value for p_{c} and for all the exponents such as β,ν,γ,τ, and d_{f}. We find that, except for p_{c}, all the exponents are exactly the same in both bond and site percolation despite the significant difference in the definition of cluster and other quantities. Our results suggest that the percolation on WPSL belongs to a new universality class, as its exponents do not share the same value as for all the existing planar lattices. Besides, like all other cases, its site and bond type belong to the same universality class.

Dimer covering and percolation frustration

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.

Scaling of the spanning threshold in gradient percolation

A simple and fast way to apply correlations in percolation simulations is to apply a uniform gradient to the occupancy probabilities. For small networks, exact results are presented here for the spanning thresholds in site percolation with a gradient for networks up to 4×4 in two dimensions and 2×2×2 in three dimensions. Numerical results are provided for larger networks that extrapolate to a linear modification of the threshold proportional to the gradient for moderate values of the gradient.

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.

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.

Percolation thresholds of two-dimensional continuum systems of rectangles

The present paper introduces an efficient Monte Carlo algorithm for continuum percolation composed of randomly oriented rectangles. By conducting extensive simulations, we report high-precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios. This paper verifies and extends the excluded area theory. It is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems (except for some special heterogeneous systems where the rectangle lengths differ too much from one another). In terms of the excluded areas, generalized formulas are proposed to effectively predict precise percolation thresholds for all these rectangle systems. This paper is, therefore, helpful for both practical applications and theoretical studies concerning relevant systems.

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.

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.

Threshold of hierarchical percolating systems

Many modern nanostructured materials and doped polymers are morphologically too complex to be interpreted by classical percolation theory. Here, we develop the concept of a hierarchical percolating (percolation-within-percolation) system to describe such complex materials and illustrate how to generalize the conventional percolation to double-level percolation. Based on Monte Carlo simulations, we find that the double-level percolation threshold is close to, but definitely larger than, the product of the local percolation thresholds for the two enclosed single-level systems. The deviation may offer alternative insights into physics concerning infinite clusters and open up new research directions for percolation theory.

Gaussian model of explosive percolation in three and higher dimensions

The Gaussian model of discontinuous percolation, recently introduced by Araújo and Herrmann [Phys. Rev. Lett. 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the simple cubic lattice, in the thermodynamic limit we report a finite jump of the order parameter J=0.415±0.005. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension d(A)=2.5±0.2. The study is extended to hypercubic lattices up to six dimensions and to the mean-field limit (infinite dimension). We find that, in all considered dimensions, the percolation transition is discontinuous. The value of the jump in the order parameter, the maximum of the second moment, and the percolation threshold are analyzed, revealing interesting features of the transition and corroborating its discontinuous nature in all considered dimensions. We also show that the fractal dimension of the external perimeter, for any dimension, is consistent with the one from bridge percolation and establish a lower bound for the percolation threshold of discontinuous models with a finite number of clusters at the threshold.

Tricritical point in explosive percolation

The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.

Scaling behavior of explosive percolation on the square lattice

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based on both moments and wrapping probabilities, yielding p(c)=0.526 565 ± 0.000005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent ν is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P∞ (size of largest cluster), of the susceptibility, and of the second moment of finite clusters, where discontinuities appear at the threshold. The critical cluster-size distribution does not follow a consistent power law for the range of system sizes we study (L ≤ 8192) but may approach a power law with τ>2 for larger L .

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.

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.

Complementary algorithms for graphs and percolation

A pair of complementary algorithms are presented. One of the pair is a fast method for connecting graphs with an edge. The other is a fast method for removing edges from a graph. Both algorithms employ the same tree-based graph representation and so, in concert, can arbitrarily modify any graph. Since the clusters of a percolation model may be described as simple connected graphs, an efficient Monte Carlo scheme can be constructed which uses the algorithms to sweep the occupation probability back and forth between two turning points. This approach concentrates computational sampling time within a region of interest. A high-precision value of p(c) = 0.59274603(9) was thus obtained, by Mersenne twister, for the two-dimensional square site percolation threshold.

Rigorous confidence intervals for critical probabilities

We use the method of Balister, Bollobás, and Walters [Random Struct. Algorithms 26, 392 (2005)] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

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.

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.

Transport properties of incipient gels

We investigate the behavior of the shear viscosity eta(p) and the mass-dependent diffusion coefficient D(m,p) in the context of a simple model that, as the cross link density p is increased, undergoes a continuous transition from a fluid to a gel. The shear viscosity diverges at the gel point according to eta(p) approximately (p(c)-p)(-s) with s approximately 0.65. The diffusion constant shows a remarkable dependence on the mass of the clusters: D(m,p) approximately m(-0.69), not only at p(c) but well into the liquid phase. We also find that the Stokes-Einstein relation Deta proportional, variant k(B)T breaks down already quite far from the gel point.

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.