Critical behavior of a random diode network

We study the percolation properties of a random diode network (RDN) which contains two kinds of directed bonds on a square lattice. This network is a special case of the random insulation-resistor-diode network. Both Monte Carlo simulations and series expansion for the percolation probability show that an estimated critical exponent, beta=0.1794+/-0.008, is different from known values for a conventional insulation-resistor-diode network. RDN belongs to neither the isotropic percolation universality class nor to the directed percolation universality, which we attribute to a difference of symmetry breakdown around the critical point.     

Discrete network models for the low-field Hall effect near a percolation threshold: Theory and simulations

The critical behavior of the weak-field Hall effect near a percolation threshold is studied with the help of two discrete random network models. Many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites. A new algorithm for doing that was developed that is based on the transfer matrix method. The site potentials are used to calculate the bulk effective Hall conductivity and Hall coefficient, as well as some other properties, such as the Ohmic conductivity, the size of the backbone, and the number of binodes. Scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found.School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel     

Criticality of Epidemic Spreading in Mobile Individuals Mediated by Environment

We investigate the critical behaviour of an epidemical model in a diffusive population mediated by a static vector environment on a 2D network. It is found that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. Finite-size and short-time dynamic scaling relations are used to determine the critical population density and the critical exponents characterizing the behaviour near the critical point. The results are compatible with the universality class of directed percolation coupled to a conserved diffusive field with equal diffusion constants.     

Percolation Phase Transitions from Second Order to First Order in Random Networks

We investigate a percolation process where an additional parameter q is used to interpolate between the classical Erd¨os–R′enyi(ER) network model and the smallest cluster(SC) model. This model becomes the ER network at q = 1, which is characterized by a robust second order phase transition. When q = 0, this model recovers to the SC model which exhibits a first order phase transition. To study how the percolation phase transition changes from second order to first order with the decrease of the value of q from 1 to 0, the numerical simulations study the final vanishing moment of the each existing cluster except the N-cluster in the percolation process. For the continuous phase transition,it is shown that the tail of the graph of the final vanishing moment has the characteristic of the convexity. While for the discontinuous phase transition, the graph of the final vanishing moment possesses the characteristic of the concavity.Just before the critical point, it is found that the ratio between the maximum of the sequential vanishing clusters sizes and the network size N can be used to decide the phase transition type. We show that when the ratio is larger than or equal to zero in the thermodynamic limit, the percolation phase transition is first or second order respectively. For our model, the numerical simulations indicate that there exists a tricritical point qcwhich is estimated to be between0.2 qc 0.25 separating the two phase transition types.     

Dynamic transitions in Domany-Kinzel cellular automata on small-world network

We study Domany-Kinzel cellular automata on small-world network. Every link on a one dimensional chain is rewired and coupled with any node with probability p. We observe that, the introduction of long-range interactions does not remove the critical character of the model and the system still exhibits a well-defined phase transition to absorbing state. In case of directed percolation (DP), we observe a very anomalous behavior as a function of size. The system shows long lived metastable states and a jump in order parameter. This jump vanishes in thermodynamic limit and we recover second-order transition. The critical exponents are not equal to the mean-field values even for large p. However, for compact directed percolation(CDP), the critical exponents reach their mean-field values even for small p.     

Universality in structure and elasticity of polymer-nanoparticle gels

We propose a combination of polymer field theory and off-lattice computer simulations to study polymer-bridged gelation in polymer-nanoparticle mixtures. We use this method to study the structure of gels formed in attractive polymer-colloid systems. Our results indicate that such gels exhibit a universal structure with a fractal dimension d(f) approximately 2.5 characteristic of random percolation. By mapping to an affine-network model, the enhancement in elastic moduli is predicted to follow a critical exponent nu(eta) approximately 1.8 characteristic of the resistor network percolation. We analyze selected experimental results to suggest the existence of a universality class corresponding to our results.     

New universality classes in “Percolative” dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.     

On the continuity of the critical value for long range percolation in the exponential case

We show that for a long range percolation model with exponentially decaying connections, the limit of critical values of any sequence of long range percolation models approaching the original model from below is the critical value for the original long range percolation model. As an interesting corollary, this implies that if a long range percolation model with exponential connections is supercritical, then it still percolates even if all long bonds are removed. We also show that the percolation probability is continuous (in a certain sense) in the supercritical regime for long range percolation models with exponential connections.Research supported by a grant from the Swedish National Science Foundation.     

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.     

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.     

Explosive percolation transition is actually continuous

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.     

Continuum Percolation with Unreliable and Spread-Out Connections

We derive percolation results in the continuum plane that lead to what appears to be a general tendency of many stochastic network models. Namely, when the selection mechanism according to which nodes are connected to each other, is sufficiently spread out, then a lower density of nodes, or on average fewer connections per node, are sufficient to obtain an unbounded connected component. We look at two different transformations that spread-out connections and decrease the critical percolation density while preserving the average node degree. Our results indicate that real networks can exploit the presence of spread-out and unreliable connections to achieve connectivity more easily, provided they can maintain the average number of functioningconnections per node.     

Tumor Hypoxia Heterogeneity Affects Radiotherapy: Inverse-Percolation Shell-Model Monte Carlo Simulations

Tumor hypoxia was discovered a century ago, and the interference of hypoxia with all radiotherapies is well known. Here, we demonstrate the potentially extreme effects of hypoxia heterogeneity on radiotherapy and combination radiochemotherapy. We observe that there is a decrease in hypoxia from tumor periphery to tumor center, due to oxygen diffusion, resulting in a gradient of radiative cell-kill probability, mathematically expressed as a probability gradient of occupied space removal. The radiotherapy-induced break-up of the tumor/TME network is modeled by the physics model of inverse percolation in a shell-like medium, using Monte Carlo simulations. The different shells now have different probabilities of space removal, spanning from higher probability in the periphery to lower probability in the center of the tumor. Mathematical results regarding the variability of the critical percolation concentration show an increase in the critical threshold with the applied increase in the probability of space removal. Such an observation will have an important medical implication: a much larger than expected radiation dose is needed for a tumor breakup enabling successful follow-up chemotherapy. Information on the TME’s hypoxia heterogeneity, as shown here with the numerical percolation model, may enable personalized precision radiation oncology therapy.     

Elasticity of stiff polymer networks

We study the elasticity of a two-dimensional random network of rigid rods ("Mikado model"). The essential features incorporated into the model are the anisotropic elasticity of the rods and the random geometry of the network. We show that there are three distinct scaling regimes, characterized by two distinct length scales on the elastic backbone. In addition to a critical rigidity percolation region and a homogeneously elastic regime we find a novel intermediate scaling regime, where the elasticity is dominated by bending deformations.     

Conserved Manna model on Barabasi–Albert scale-free network

In this paper, a conserved Manna model is constructed and studied on Barabasi–Albert scale-free network with degree exponent γ = 3. Numerically I show that the system undergoes an absorbing state phase transition when the particle density is varied. Such a phase transition is characterized by measuring several critical exponents associated with the critical behaviour of the model. It has been found that the critical exponents exhibit mean field values of directed percolation. At the critical point, the spreading exponents have also been estimated. They satisfy the usual scaling relations. The effect of various initial conditions has been investigated and the result found to be independent of initial conditions, contrary to the fact that critical behaviour of such model highly depends on initial conditions when studied on regular lattice. The study confirms that though the Manna model in the lower dimensions exhibits different critical behavior other than DP, in the scale-free network it exhibits similar mean field result of DP class.     

Modeling wireless sensor networks using random graph theory

A critical issue in wireless sensor networks (WSNs) is represented by limited availability of energy within network nodes. Therefore, making good use of energy is necessary in modeling sensor networks. In this paper we proposed a new model of WSNs on a two-dimensional plane using site percolation model, a kind of random graph in which edges are formed only between neighbouring nodes. Then we investigated WSNs connectivity and energy consumption at percolation threshold when a so-called phase transition phenomena happen. Furthermore, we proposed an algorithm to improve the model; as a result the lifetime of networks is prolonged. We analyzed the energy consumption with Markov process and applied these results to simulation.     

Optical percolation in ceramics assisted by porous clusters

We investigated optical transparency in ceramics assisted by disordered porous clusters. The structure and statistical properties of three-dimensional (3D) well porous ceramics is studied. Theoretical model based on the percolation theory and numerical simulations are applied to interpret the observed phase transition from an optically opaque state to a transparent state. The porous ceramic samples were fabricated by the technique of slurry casting. The transmission of optical radiation (optical percolation) over the entire porous samples is observed since the critical concentration of porosity was exceeded. We explain this effect by the rising of the spanning cluster inside of the porous structure that produces a network of porous voids. Our experimental results are in good agreement with the numerical simulations.     

忆阻逾渗导电模型中的初态影响

逾渗网格模型是当前忆阻器件机理分析研究领域的热点之一, 但现有模型缺乏对初态设定的讨论. 本文对逾渗网格模型进行了简化, 并基于此, 通过电压激励步进的方式, 研究了不同初态对单极性忆阻开关元件中逾渗导电通道形成的影响, 分析了形成通道的动态过程以及相应的物理意义, 验证了忆阻开关元件高低阻态的阻值实际表现为高斯分布而非理想双值稳态; 而不同初态条件下, 忆阻开关元件导电通道的形状存在着不同的"树形"结构, 进而影响着其阻值的分布. 研究成果有助于进一步揭示忆阻器尚未明确的导电机理, 为今后对具体不同类型的忆阻元件的初态分析提供指导性作用. 关键词： 忆阻器 开关元件 逾渗模型 初态分析     

Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

Numerical study of currents fluctuations in metal-dielectric composites at the percolation threshold

In this paper, dc current distribution through a 2D square metal-dielectric network is determined and analyzed near the percolation threshold. The current in each link is calculated by solving Kirchhoff’s equations. Using the difference between dielectric and metallic distributions, and the scaling analysis of the peak of large currents, it is found that this peak corresponds to backbone. Furthermore, by examining the critical exponent of current fluctuations, it is found that these fluctuations correspond to the magnetic susceptibility of the network. This correspondence is explained in terms of the fluctuation-dissipation theorem.