Universality of the ratio of the critical amplitudes of the magnetic susceptibility in a two-dimensional ising model with nonmagnetic impurities

The behavior of the magnetic susceptibility of a two-dimensional Ising model with nonmagnetic impurities is investigated numerically. A new method for determining the critical amplitudes and critical temperature is developed. The results of a numerical investigation of the ratio of the critical amplitudes of the magnetic susceptibility are presented. It is shown that the ratio of the critical amplitudes is universal right up to impurity concentrations q ≤ 0.25 (the percolation point of a square lattice is q _c = 0.407254). The behavior of the effective critical exponent γ(q) of the magnetic susceptibility is discussed. Apparently, a transition from Ising-type universal behavior to percolation behavior should occur in a quite narrow concentration range near the percolation point of the lattice.     

On the gauge Potts model and the plaquette percolation problem

It is shown that the ar 1, 0 limit of the Potts gauge model describes plaquette percolation as the analogous limit of the spin model describes bond percolation. These results further strengthen the connection between gauge theories and random surfaces. Moreover, further generalizations to other types of gauge theories are presented.     

Bond percolation for homogeneous two-dimensional lattices

Bond percolation is studied for the three homogeneous two-dimensional lattices: square lattice (SL), triangular lattice (TL) and honeycomb lattice (HL). An expanding cell technique is used to obtain percolation thresholds and other relevant information for different cell sizes. We extend the analysis as to include slightly asymmetric cells in addition to the usual symmetric cells to get more points in the scaling analysis. Exact percolation functions are obtained for each size. Then, the percolation threshold is obtained by means of two complementary methods: one based on the well-known renormalization techniques and the other one introduced here which is based upon determining the inflection point of the percolation curves. A comparison of the results obtained by these two methods is performed. The study includes iterations to extrapolate numerical results towards the thermodynamic limit. Critical exponents ν, β and γ are obtained. Values are compared with numerical results and expected theoretical estimations; present results show agreement and even improvement (in the case of γ) with respect to some numeric values available in the literature. Comparison tables are provided.     

Percolation thresholds of a group of anisotropic three-dimensional fracture networks

Percolation thresholds (average number of connections per object) of two models of anisotropic three-dimensional (3D) fracture networks made of mono-disperse hexagons have been calculated numerically. The first model is when the fracture networks are comprised of two groups of fractures that are distributed in an anisotropic manner about two orthogonal mean directions, i.e., Z- and X-directions. We call this model bipolar anisotropic fracture network (BFN). The second model is when three groups of fractures are distributed about three orthogonal mean directions, that is Z-, X-, and Y-directions. In this model three families of fractures about three orthogonal mean directions are oriented in 3D space. We call this model tripolar anisotropic fracture network (TFN). The finite-size scaling method is used to predict the infinite percolation thresholds. The effect of anisotropicity on percolation thresholds in X-, Y-, and Z-directions is investigated. We have revealed that as the anisotropicity of networks increases, the percolation thresholds in X-, Y-, and Z-directions span the range of 2.3 to 2.0, where 2.3 and 2.0 are extremums of percolation thresholds for isotropic and non-isotropic orthogonal fracture networks, respectively.     

Monte carlo simulation of the two-spin facilitated model above the thermodynamic transition temperature

Results of a Monte Carlo simulation of the two-spin facilitated model proposed by Fredrickson and Andersen above the thermodynamic transition temperature without external field on the square lattice are presented. The model is a kinetic Ising model with a special constraint to its kinetic process and was designed to simulate viscous fluid. A time correlation function for uniform magnetization is analyzed using the idea of finite-size scaling and percolation length. For the system size that was simulated, our data fit into a scaling plot adopting the percolation length as a length scale, and we obtain the dynamical critical exponent z ∼ 4, which is different from the usual value z ∼ 2.     

The nature of percolation ‘channels’

A relation is found between the percolation probability, P^(s)(x), for site percolation, the conductance, G(x), of certain random networks, and the spin-stiffness coefficient, D(x), of dilute Heisenberg ferromagnet. Numerical results and critical exponents for these quantities near their common threshold are reported. These results demonstrate that the percolation ‘channels’ in which spin waves occur cannot be regarded as one or two dimensional.     

Long-range connections, quantum magnets and dilute contact processes

In this article, we briefly review the critical behaviour of a long-range percolation model in which any two sites are connected with a probability that falls off algebraically with the distance. The results of this percolation transition are used to describe the quantum phase transitions in a dilute transverse Ising model at the percolation threshold p_c of the long-range connected lattice. In the similar spirit, we propose a new model of a contact process defined on the same long-range diluted lattice and explore the transitions at p_c. The long-range nature of the percolation transition allows us to evaluate some critical exponents exactly in both the above models. Moreover, mean field theory is valid for a wide region of parameter space. In either case, the strength of Griffiths McCoy singularities are tunable as the range parameter is varied.     

The correct extension of the Fortuin-Kasteleyn result to plaquette percolation

The standard s-state Potts model is generalized to a model defined of laquettes of $\text{Zz}{}^{\mathrm{d}}$, and the s → 1, 0 limits are shown to correspond to plaquette percolation and percolation of plaquette trees, respectively. The model is further extended to r-cells (cubes, hypercubes, etc.) on $\text{Z}{}^{\mathrm{d}}$ and the s → 1, 0 limits are exhibited.     

Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

This Letter is focused on the impact of network topology on the site percolation. Specifically, we study how the site percolation threshold depends on the network dimensions (topological d and fractal D), degree of connectivity (quantified by the mean coordination number $Z_{\infty }$), and arrangement of bonds (characterized by the connectivity index Q also called the ramification exponent). Using the Fisher's containment principle, we established exact inequalities between percolation thresholds on fractal networks contained in the square lattice. The values of site percolation thresholds on some fractal lattices were found by numerical simulations. Our findings suggest that the most relevant parameters to describe properly the values of site percolation thresholds on fractal networks contained in square lattice (Sierpiński carpets and Cantor tartans) and based on the square lattice (weighted planar stochastic fractal and Cantor lattices) are the mean coordination number and ramification exponent, but not the fractal dimension. Accordingly, we propose an empirical formula providing a good approximation for the site percolation thresholds on these networks. We also put forward an empirical formula for the site percolation thresholds on d-dimensional simple hypercubic lattices.     

Concentration dependence of the anomalous Hall effect in Fe/SiO2 granular films below the percolation threshold

The anomalous Hall effect is studied on Fe_x(SiO₂)_1?x nanocomposite films with x<0.7 in the vicinity of the percolation transition (x _c ≈0.6). It is found that, as the transition is approached from the side of metallic conduction, the Hall angle nonmonotonically varies, passing through a minimum. A qualitative model for describing the concentration dependence of the anomalous Hall effect is proposed. The model is based on that of the conductivity of a two-phase system near the percolation threshold [9, 10]. The anomalous Hall effect is governed by two conduction channels: one of them (a conducting network) is formed by large metal clusters that are separated by narrow dielectric interlayers below the percolation threshold, and the other is represented by the dielectric part of the medium containing Fe grains; in this part of the medium, the anomalous Hall effect occurs through the interference of amplitudes from the tunneling junctions in a set of three grains. It is shown that, at x<x _c , the network may give rise to a “shunting” effect, which makes the effective Hall voltage even less than the Hall voltage of the dielectric component.     

Influence of the shape of inclusions on the percolation thresholds in 2D models of composites

The influence of the shape of inclusions on the conductivity of composites, including critical concentration N _c (percolation threshold), is considered using 2D models as an example. A relationship between constant N _I , which characterizes effective conductivity ?? _e for low concentrations N of inclusions and percolation threshold N _c is established.     

Similarities between the lattice gas model and some other models of nuclear multifragmentation

We continue our development of the nuclear lattice gas model by exploring links and similarities with other theoretical approaches to nuclear multifragmentation: the percolation model and the statistical multifragmentation model. It is shown that there exists a limit where the lattice gas model reduces to the percolation model. The similarity between the lattice gas model and the statistical multifragmentation model is more indirect and we utilize the equations of state in the two models. By using the law of partial pressures we obtain P-ϱ diagrams for the statistical multifragmentation model and find that these are remarkably similar to those obtained in the lattice gas model via an exact evaluation of the nuclear partition function on the lattice. For completeness, we also compute the P-ϱ diagram for a system obeying pure classical molecular dynamics with a simple two-body force.     

Percolation on the square lattice in aL×M geometry

A study of the site percolation model on the square lattice in aL×M geometry at critically is presented. ForL?M one observes the growth of numerous percolation colation clusters in theL-direction in contrast to the absence of percolation in theM-direction. Consequently, relevant properties of these clusters such us for example the average number of clusters (N _CL ), the cluster length distribution (P(l,L), withl=cluster length in theM direction) and average cluster length (l _CL ), are studied by means of the Monte Carlo technique and analyzed on the basis of finite-size scaling arguments. The following behavior is found:N _CL ?(3/8) (L/M)^?δ, with δ=1; andl _CL ?2.0L. Also the distributionP(l, L) is of the exponential-exponential type and their characteristic exponents are evaluated.     

Influence of the size distribution of granules and of their attractive interaction on the percolation threshold in granulated alloys

Numerical simulation was applied to study the influence of the size distribution of granules and the interaction between them on the percolation threshold in granulated metal-insulator alloys. An alloy model was considered in which metal granules have two characteristic sizes, l and L (with L>l), and the size distribution of granules of greater size L having an average value of approximately L ₀ is described by a normal distribution with a standard deviation d, by a step function with a halfwidth d, or by a delta function. A model with attraction between granules and mechanisms of trapping of an additional granule by an already developed cluster with a characteristic trapping range R was also considered. The percolation threshold significantly grows with the ratio L ₀/l and with R for both two-and three-dimensional cases and tends to flattening at large L/l or R. The calculated results make it possible to explain the high percolation threshold observed for the majority of granulated alloys.     

Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory

We consider the scaling limit of the two-dimensional q-state Potts model for q ⩽ 4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one-and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit q → 1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.     

Percolation on Interdependent Networks with a Fraction of Antagonistic Interactions

Recently, the percolation transition has been characterized on interacting networks both in presence of interdependent interactions and in presence of antagonistic interactions. Here we characterize the phase diagram of the percolation transition in two Poisson interdependent networks with a percentage q of antagonistic nodes. We show that this system can present a bistability of the steady state solutions, and both discontinuous and continuous phase transitions. In particular, we observe a bistability of the solutions in some regions of the phase space also for a small fraction of antagonistic interactions 0<q<0.4. Moreover, we show that a fraction q>q _c =2/3 of antagonistic interactions is necessary to strongly reduce the region in phase-space in which both networks are percolating. This last result suggests that interdependent networks are robust to the presence of antagonistic interactions. Our approach can be extended to multiple networks, and to complex boolean rules for regulating the percolation phase transition.     

Connectivity Patterns in Loop Percolation I: the Rationality Phenomenon and Constant Term Identities

Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice \({\mathbb{Z}^2}\) whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the “cylindrical geometry” as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.     

Estimation of Critical Point in Branching Reactions: Further Examination of Gel Point Formula

The theory of gel point in real polymer solutions is examined with the empirical correlation between the reciprocal of the percolation threshold and the coordination number given by the percolation theory. Applying a larger value of the relative frequency of cyclization, an excellent agreement is obtained between the present theory and the percolation result. This suggest that while the ring distribution on lattices is similar to that in real systems, ring production is more frequent in the lattice model than in real systems. To confirm this conjecture, we derive the ring distribution function of the lattice model as a limiting case of d→∞, and show that the solution is in fact identical to the asymptotic formula of C→∞ in real systems except for the coefficient C, which has a maximum at d = 5, in support of the above conjecture. To examine the validity of the asymptotic solution for the lattice model, we apply it to the critical point problem of the percolation theory, showing that the solution works well in high dimensions greater than six.     

Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice ?? by ? bonds connecting the same adjacent vertices, thereby yielding the lattice ?? _? . This relation is used to calculate the bond percolation threshold on ?? _? . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d??2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N???? limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N????.