Critical Percolation Research Articles

Percolation and coarsening in the bidimensional voter model.

We study the bidimensional voter model on a square lattice with numerical simulations. We demonstrate that the evolution takes place in two distinct dynamic regimes; a first approach towards critical site percolation and a further approach towards full consensus. We calculate the time dependence of the two growing lengths, finding that they are both algebraic but with different exponents (apart from possible logarithmic corrections). We analyze the morphology and statistics of clusters of voters with the same opinion. We compare these results to the ones for curvature driven two-dimensional coarsening.

The effect of volume exclusion on the formation of DNA minicircle networks: implications to kinetoplast DNA

Kinetoplast DNA (kDNA) is the mitochondrial of DNA of disease causing organisms such as Trypanosoma Brucei (T. Brucei) and Trypanosoma Cruzi (T. Cruzi). In most organisms, KDNA is made of thousands of small circular DNA molecules that are highly condensed and topologically linked forming a gigantic planar network. In our previous work we have developed mathematical and computational models to test the confinement hypothesis, that is that the formation of kDNA minicircle networks is a product of the high DNA condensation achieved in the mitochondrion of these organisms. In these studies we studied three parameters that characterize the growth of the network topology upon confinement: the critical percolation density, the mean saturation density and the mean valence (i.e. the number of mini circles topologically linked to any chosen minicircle). Experimental results on insect-infecting organisms showed that the mean valence is equal to three, forming a structure similar to those found in medieval chain-mails. These same studies hypothesized that this value of the mean valence was driven by the DNA excluded volume. Here we extend our previous work on kDNA by characterizing the effects of DNA excluded volume on the three descriptive parameters. Using computer simulations of polymer swelling we found that (1) in agreement with previous studies the linking probability of two minicircles does not decrease linearly with the distance between the two minicircles, (2) the mean valence grows linearly with the density of minicircles and decreases with the thickness of the excluded volume, (3) the critical percolation and mean saturation densities grow linearly with the thickness of the excluded volume. Our results therefore suggest that the swelling of the DNA molecule, due to electrostatic interactions, has relatively mild implications on the overall topology of the network. Our results also validate our topological descriptors since they appear to reflect the changes in the physical properties of the polymeric chains and at the same time remain faithful to their description of kDNA.

Correlated Percolation, Fractal Structures, and Scale-Invariant Distribution of Clusters in Natural Images.

Natural images are scale invariant with structures at all length scales.We formulated a geometric view of scale invariance in natural images using percolation theory, which describes the behavior of connected clusters on graphs.We map images to the percolation model by defining clusters on a binary representation for images. We show that critical percolating structures emerge in natural images and study their scaling properties by identifying fractal dimensions and exponents for the scale-invariant distributions of clusters. This formulation leads to a method for identifying clusters in images from underlying structures as a starting point for image segmentation.

Fusion rules for the logarithmic N = 1 superconformal minimal models: I. The Neveu–Schwarz sector

It is now well known that non-local observables in critical statistical lattice models, polymers and percolation for example, may be modelled in the continuum scaling limit by logarithmic conformal field theories. Fusion rules for such theories, sometimes referred to as logarithmic minimal models, have been intensively studied over the last ten years in order to explore the representation-theoretic structures relevant to non-local observables. Motivated by recent lattice conjectures, this work studies the fusion rules of the N = 1 supersymmetric analogues of these logarithmic minimal models in the Neveu–Schwarz sector. Fusion rules involving Ramond representations will be addressed in a sequel.

The Role of Percolation Theory in Developing Next Generation Smart Nanomaterials

The incorporation of small volume fractions of nanoscale graphitic particles into varied base materials has been explored across fields ranging from automotive to aerospace to commercial plastics, with the goal of utilizing their enhanced thermal conductivity, electrical conductivity or mechanical strength. Percolation theory has emerged as a useful tool to aid in mapping and predicting the enhancement of properties based on the size and conductivity of incorporated single-walled carbon nanotubes relative to their less conductive base materials. These tools can aid researchers in the development of next generation smart nanomaterials. In this paper, we discuss the use of homogeneous fractions of length- or chirality-sorted single-walled carbon nanotubes (SWNTs) which are incorporated into thin film networks, and cement composites, and are evaluated in terms of their conductivity, mechanical properties and noise spectrum at critical percolation. We demonstrate that, near the percolation threshold, the conductivity of these highly characterized SWNT films exhibits a power law dependence on the network geometrical parameters. We also present our findings on the development of incorporated thin film SWNTs for the development of sensing technology for novel non-destructive failure diagnostic applications. SWNTs are able to be used as benign inclusions, capable of active sensing, when incorporated into cement-based composites for the purpose of detecting crack initiation. As such, we investigate the use of homogeneous length-sorted SWNTs that are randomly distributed in percolated networks capable of being an internal responsive net mechanism. Our findings demonstrate increased microstructure sensitivity of our networks for our shorter length nanotubes near their critical percolation threshold. This shows promise for the development of even more sensitive, embedded piezo-resistive SWNT-based sensors for preemptive failure detection technology.

Size of a polymer chain in an environment of quenched chains

We perform high-coordination three-dimensional (3D) lattice simulations of a single chain of N monomers embedded in matrices of quenched chains, at different concentrations ρ, using pruned-enriched Rosenbluth sampling. The partition function is well-described by the expression, ZN(ρ)=0.051μNNγ−1, where γ≈1.16 is a universal constant, and μ(ρ)=23.39−17.36ρ−10.96ρ2 is the concentration dependent lattice connectivity constant. For sufficiently long chains, N≳50, we find that the radius of gyration R varies nonmonotonically with ρ; R decreases gradually from its unperturbed dimensions R0 until R/R0≈0.9, after which it increases relatively rapidly due to repulsion between monomers. Motivated by the similarity in the shape of the curves, and results on Gaussian chains, we successfully superpose all the simulation data onto a single master curve. Finally, we test the relationship R/R0∼(ρc−ρ)−δ0, suggested by a Flory-type scaling model, where ρc is the critical percolation threshold, and δ0=0.132 is a universal constant. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, 1611–1619

Особливості перколяційного переходу в системах на основі олігогліколів та вуглецевих нанотрубок

The results of researches on the electric conductivity in the percolation transition region of the oligoglycol/nanotubes systems are reported. It is shown that the conductivity can be described in the framework of the critical percolation theory. The critical parameters of percolation transition are found to change, by depending on various factors, and to differ from the predictions of the statistical percolation theory. A relationship between the critical conductivity indices and the fractal dimensionality of a conducting cluster is found. It is demonstrated that the application of a scaling function allows the concentration dependences of conductivity to be described with the help of a unique universal function.

A lower bound on the two-arms exponent for critical percolation on the lattice

We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$. We improve slightly this lower bound in any dimension $d\geq2$. Next, starting only with the hypothesis that $\theta(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.

Inhomogeneous Long-Range Percolation on the Hierarchical Lattice

We study a model for inhomogeneous long-range percolation on the hierarchical lattice Ω N of order N with an ultrametric d . Each vertex x ∈ Ω N is assigned a nonnegative weight W x , where ( W x ) x∈Ω N are i.i.d. random variables. Conditionally on the weights, and given two parameters α ≥ 0, β > 0, the edges are independent and the probability that there is an edge between two vertices x and y is of the form 1 - exp{-α W x W y /β d ( x, y ) }. Conditions on the weight distribution and the parameter β are formulated for the existence of a critical percolation value α c ∈ (0, ∞) such that the resulting graph contains an infinite component when α > α c and no infinite component when α c . Numerical simulations are also provided to validate the theoretical results.

Recursive percolation.

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number n of generations. In two dimensions, we determine the percolation thresholds up to n=5. The corresponding critical clusters become more and more compact as n increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any n. This family of exponents differs from previously known universality classes, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.

Transition from compact to porous films in deposition with temperature-activated diffusion.

We study a thin-film growth model with temperature activated diffusion of adsorbed particles, allowing for the formation of overhangs and pores, but without detachment of adatoms or clusters from the deposit. Simulations in one-dimensional substrates are performed for several values of the diffusion-to-deposition ratio R of adatoms with a single bond and of the detachment probability ε per additional nearest neighbor, respectively, with activation energies are E(s) and E(b). If R and ε independently vary, regimes of low and high porosity are separated at 0.075≤ε(c)≤0.09, with vanishingly small porosity below that point and finite porosity for larger ε. Alternatively, for fixed values of E(s) and E(b) and varying temperature, the porosity has a minimum at T(c), and a nontrivial regime in which it increases with temperature is observed above that point. This is related to the large mobility of adatoms, resembling features of equilibrium surface roughening. In this high-temperature region, the deposit has the structure of a critical percolation cluster due to the nondesorption. The pores are regions enclosed by blobs of the corresponding percolating backbone, thus the distribution of pore size s is expected to scale as s(-τ̃) with τ̃≈1.45, in reasonable agreement with numerical estimates. Roughening of the outer interface of the deposits suggests Villain-Lai-Das Sarma scaling below the transition. Above the transition, the roughness exponent α≈0.35 is consistent with the percolation backbone structure via the relation α=2-d(B), where d(B) is the backbone fractal dimension.

Conformal invariance in three dimensional percolation

.The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric aspects of universality. As a case study we study three dimensional percolation at criticality in bounded domains. Both on discrete and continuous models of critical percolation, we test by numerical experiments the invariance of quantities in finite domains under conformal transformations focusing on crossing probabilities. Our results show clear evidence of the onset of conformal invariance in finite realizations especially for the continuum percolation models. Finally we propose a simple analytical function approximating the crossing probability among two spherical caps on the surface of a sphere and confront it with the numerical results.

Conformal invariance of loop ensembles under Kardar-Parisi-Zhang dynamics

We study scaling properties of the honeycomb fully packed loop ensemble associated with a lozenge tiling model of rough surface, when the latter is driven out of equilibrium by Kardar-Parisi-Zhang (KPZ)–type dynamics. We show numerically that conformal invariance and signatures of critical percolation appear in the stationary KPZ state. In terms of the two-component Coulomb gas description of the Edwards-Wilkinson stationary state, our finding is understood as the invariance of one component under the effect of the non-linear KPZ term. On the other hand, we show a breaking of conformal invariance when the level lines of the other component are considered.

Preparation, structural and optical investigations of ITO nanopowder and ITO/epoxy nanocomposites

Indium Tin Oxide (ITO) nanopowder was synthesized by a sol–gel method. It was characterized by X-ray diffraction (XRD) and transmission electron microscopy. ITO/epoxy nanocomposites (ITO–EP-NCs) were prepared by mechanically dispersing the as-prepared ITO nanopowder into epoxy matrix. The XRD patterns show structural changes depending on ITO content. The interdependence of structural, morphological, optical properties and the dispersed concentration of ITO nanoparticles were investigated. The UV–visible absorption spectra revealed that the ITO–EP-NCs exhibit enhanced UV light absorption properties and wide absorption bandwidth ranging up to 400nm from 2wt% ITO loading. Thus, it indicated that UV and IR-shielding properties have been improved with the incorporation of ITO nanoparticles into the epoxy matrix.The gap energy of epoxy matrix was reduced by adding the ITO-NPs, leading to the improvement of its electrical conductivity. Indeed, the AC electrical conductivity of ITO–EP-NCs showed a critical percolation threshold pc=0.21wt% ITO. For low loading (<2wt% ITO), the ITO–EP-NCs have combined good transparency in the visible range and enhanced electrical conductivity, which are required for optoelectronics devices.

Factorization formulas for [formula omitted] critical percolation, revisited

We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s>0.

Aftershocks and Omori's law in a modified Carlson-Langer model with nonlinear viscoelasticity.

A modified Carlson-Langer model for earthquakes is proposed, which includes nonlinear viscoelasticity. Several aftershocks are generated after the main shock owing to the damping of the additional viscoelastic force. Both the Gutenberg-Richter law and Omori's law are reproduced in a numerical simulation of the modified Carlson-Langer model on a critical percolation cluster of a square lattice.

The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution

The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interface in critical percolation, and hence its scaling limit is the chordal Schramm-Loewner evolution with kappa=6 (SLE_6). Like SLE there are variants of the SKSAW depending on the domain and the initial and terminal points. On the hexagonal lattice these variants have been shown to converge to the corresponding version of SLE_6. We conjecture that the scaling limit of all these variants on any regular lattice is the corresponding version of SLE_6. We test this conjecture for the square lattice by simulating the SKSAW in the full-plane and find excellent agreement with the predictions of full-plane SLE_6.

Mechanical Reinforcement of Polymer Nanocomposites from Percolation of a Nanoparticle Network.

Nanometer-sized particles that are well dispersed in a polymer melt, presumably due to strongly favorable particle-polymer interactions, can form fractal structures via polymer bridging, leading ultimately to a nanoparticle (NP) network analogous to a colloidal gel. The linear viscoelastic response of polymer nanocomposites can be quantitatively predicted by a parameter-free model in which the stress is a simple sum of contributions from the polymer matrix and the fractal NP structure linked by bridging polymer chains. The NP contribution is modeled using critical percolation, while the polymer part is enhanced by the presence of particles, owing to hydrodynamic interactions. The phase diagram at the right shows that small NPs are needed to achieve the stronger reinforcement from glassy bridges at reasonable particle loadings.

Viscoelasticity of Reversible Gelation for Ionomers

Linear viscoelasticity (LVE) of low-ion-content and low-molecular-weight (nonentangled) randomly sulfonated polystyrene shows a sol–gel transition when the average number of ionic groups per chain approaches unity. This transition can be well understood by regarding the number of ionizable sites over a chain as the relevant functionality for cross-linking. For ionomers below but very close to the gel point, the LVE shows power law relaxation similar to gelation of chemical cross-linking. Nevertheless, ionomers near and beyond the gel point also show terminal relaxation not seen in chemically cross-linking systems, which is controlled by ionic dissociation. Careful analysis of the power law region of the frequency dependence of complex modulus close to the gel point shows a change in exponent from ∼1 at high frequency to ∼0.67 at low frequency, which strongly suggests a transition from mean-field to critical percolation known as the Ginzburg point. A mean-field percolation theory by Rubinstein and Semenov ...

Random Walk on Random Infinite Looptrees

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical Galton--Watson tree conditioned to be large. We study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth. We prove that if the offspring distribution of the Galton--Watson process is in the domain of attraction of a stable distribution with index $\alpha\in(1,2]$ then the spectral dimension of the looptree is $2\alpha/(\alpha+1)$.