Continuum Percolation Model Research Articles

Law of large numbers for the largest component in a hyperbolic model of complex networks

We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with so-called complex networks. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power law and $\nu$ controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant $c$ that depends only on $\alpha,\nu$, while all other components are sublinear. We also study how $c$ depends on $\alpha,\nu$. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on $\mathbb{R}^{2}$ that may be of independent interest.

Enhanced Li Ion Conductivity in LiBH4–Al2O3 Mixture via Interface Engineering

A new solid-state Li ion conductor composed of LiBH4 and Al2O3 was synthesized by a simple ball-milling process. The element distribution map obtained by transmission electron microscopy demonstrates that the LiBH4 and Al2O3 are well mixed and form a large interface after ball-milling. The ionic conductivity of the mixture reaches as high as 2 × 10–4 S cm–1 at room temperature when the volume fraction of Al2O3 is approximately 44%. The ionic conductivity of the interface between LiBH4 and Al2O3 was extracted by using a continuum percolation model, which turns out to be about 10–3 S cm–1 at room temperature, being 105 times higher than that of pure LiBH4. This remarkable rise in conductivity is accompanied by the lowered activation energy for the Li ion conduction in the mixture, indicating that the interface layer facilitates Li ion conduction. Near-edge X-ray absorption fine structure analysis reveals the presence of B–O bondings in the mixture, which was not detected by X-ray diffraction. This disruptio...

Ellipses Percolation

We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $\alpha > 0$ associated with the tail decay of the major axis distribution; we only consider distributions $\rho$ satisfying $\rho[r, \infty) \asymp r^{-\alpha}$. We prove that this model presents a double phase transition in $\alpha$. For $\alpha \in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $\alpha \in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $\alpha \in (2, \infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $\alpha = 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $\mathbb{Z}^2$.

Analysis of a more realistic well representation during secondary recovery in 3-D continuum models

The effectiveness of secondary recovery methods in reservoir development studies depends on the knowledge about how fluid-carrying regions (i.e. good-quality rock types) are connected between injection and production wells. To estimate reservoir performance uncertainty, comprehensive simulations on many reservoir model realisations are necessary, which is very CPU consuming and time demanding. Alternatively, we can use much simpler and physically based methods such as percolation approach. Classic percolation assumes connectivity between opposite 2-D faces of a 3-D system; whereas, hydrocarbon production is achieved through active wells that are one-dimensional lines (e.g. vertical, horizontal or deviated wells). The main contribution of this study is to analyse the percolation properties of 3-D continuum percolation models with more realistic well representations during secondary recovery. In particular, the connection of randomly distributed sands (i.e. good-quality rock types) between two lines (representing two wells) located at two corners of the system are modelled by Monte Carlo simulations. Subsequently, the connectivity and conductivity of such a line-to-line well representation is compared with that of face-to-face well representations in the previously published results. The critical percolation properties of those systems as well as the universality concept are also investigated. As there are many rooms for connections in 3-D models, we found that the principal percolation properties will not be altered significantly when the problem with a face-to-face connection is transformed to a line-to-line connection model.

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

We examine the interplay between anisotropy and percolation, i.e. the spontaneous formation of a system spanning cluster in an anisotropic model. We simulate an extension of a benchmark model of continuum percolation, the Boolean model, which is formed by overlapping grains. Here we introduce an orientation bias of the grains that controls the degree of anisotropy of the generated patterns. We analyze in the Euclidean plane the percolation thresholds above which percolating clusters in x- and in y-direction emerge. Only in finite systems, distinct differences between effective percolation thresholds for different directions appear. If extrapolated to infinite system sizes, these differences vanish independent of the details of the model. In the infinite system, the uniqueness of the percolating cluster guarantees a unique percolation threshold. While percolation is isotropic even for anisotropic processes, the value of the percolation threshold depends on the model parameters, which we explore by simulating a score of models with varying degree of anisotropy. To which precision can we predict the percolation threshold without simulations? We discuss analytic formulas for approximations (based on the excluded area or the Euler characteristic) and compare them to our simulation results. Empirical parameters from similar systems allow for accurate predictions of the percolation thresholds (with deviations of <5% in our examples), but even without any empirical parameters, the explicit approximations from integral geometry provide, at least for the systems studied here, lower bounds that capture well the qualitative dependence of the percolation threshold on the system parameters (with deviations of –). As an outlook, we suggest further candidates for explicit and geometric approximations based on second moments of the so-called Minkowski functionals.

Effects of substrate temperature on change in percolation properties of ultra thin dielectric films

AbstractIn this paper, we explored the impact of surface roughness on ultra‐thin dielectrics characteristics. Simple cubic lattice is used as a base for depositing spheres of radius R, designed as a thin film, as the base for simulation of defects generation and deposition. Continuum percolation model, used to generate a rough surface in order to analyse how roughness and temperature influence the conductivity, consists of randomly placed spheres of either one or some distribution of radii, which then form a thin, rough layer in question. It is shown that when the temperature of the substrate is increased material defects tend to form within the film contributing to a faster cluster growth and greater film conductivity. The influence of temperature on average cluster size for homogeneous distribution of defects is assumed according to the applied temperature (depending on the free carrier's diffusion length) is researched. It is evident from the performed simulations that temperature has an important influence on thin film conductivity in several different ways; it can contribute to the increase of the number of spanning clusters or the average size of clusters. This way, defects can either provide starting points for new spanning clusters or can encourage clusters merging. (© 2016 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

We consider a continuum percolation model on $$\mathbb {R}^d$$ , $$d\ge 1$$ . For $$t,\lambda \in (0,\infty )$$ and $$d\in \{1,2,3\}$$ , the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity $$\lambda >0$$ . When $$d\ge 4$$ , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ . We establish that, for $$d=1$$ and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ , there is a non-trivial percolation transition in t, provided $$\lambda $$ and r are chosen properly. The last statement means that $$\lambda $$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when $$d\in \{2,3\}$$ , but finite and dependent on r when $$d\ge 4$$ ). We further show that for all $$d\ge 2$$ , the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.

Connection times in large ad-hoc mobile networks

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation.

Asymptotics of the critical time in Wiener sausage percolation with a small radius

We consider a continuum percolation model on $\R^d$, where $d\geq 4$. The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whose initial points are distributed according to a homogeneous Poisson point process. It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transition in $t$ occuring at a critical time $t_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that $c^{-1}\sqrt{\log(1/r)}\leq t_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t_c(r) \leq c r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.

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.

On homogenized conductivity and fractal structure in a high contrast continuum percolation model

In the previous article (Matsutani et al., 2012) we numerically investigated an electric potential problem with high contrast local conductivities (γ0 and γ1, 0<γ0≪γ1) for a two-dimensional continuum percolation model (CPM). As numerical results, we showed there that the equipotential curves exhibit the fractal structure around the threshold pc and gave an approximated curve representing a relation between the homogenized conductivity and the volume fraction p over [pc,1]. In this article, using the duality of the conductivities and the quasi-harmonic properties, we re-investigate these topics to improve these results. We show that at γ0→0, the quasi-harmonic potential problem in CPM is quasiconformally equivalent to a random slit problem, which leads us to an observation between the conformal property and the fractal structure at the threshold. Further we extend the domain [pc,1] of the approximated curve to [0,1] based on the these results, which is partially generalized to three dimensional case. These curves represent well the numerical results of the conductivities.

Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc &lt; ∞ such that for all λ &gt; λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ &lt; λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λ E , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ E of eavesdropper nodes, there exists a critical intensity λ c &amp;lt; ∞ such that for all λ &amp;gt; λ c the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ &amp;lt; λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

A continuum percolation model for stock price fluctuation as a Lévy process

This paper concerns with two reasons for stock price fluctuation, the instinctive stochastic fluctuation and the fluctuation caused by the spread of information. They are constructed by compound Poisson process and continuum percolation model separately. Combining the two models, the authors get a Levy process for the price fluctuation that can explain the fat-tail phenomenon in stock market. The fat-tails are also presented in numerical simulations.

Tunneling Conductivity and Piezoresistivity of Composites Containing Randomly Dispersed Conductive Nano-Platelets.

In this study, a three-dimensional continuum percolation model was developed based on a Monte Carlo simulation approach to investigate the percolation behavior of an electrically insulating matrix reinforced with conductive nano-platelet fillers. The conductivity behavior of composites rendered conductive by randomly dispersed conductive platelets was modeled by developing a three-dimensional finite element resistor network. Parameters related to the percolation threshold and a power-low describing the conductivity behavior were determined. The piezoresistivity behavior of conductive composites was studied employing a reoriented resistor network emulating a conductive composite subjected to mechanical strain. The effects of the governing parameters, i.e., electron tunneling distance, conductive particle aspect ratio and size effects on conductivity behavior were examined.

Noise sensitivity in continuum percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p c ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

We study the effect of dimensionality on the percolation threshold η(c) of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space R(d). This is done by formulating a scaling relation for η(c) that is based on a rigorous lower bound [Torquato, J. Chem. Phys. 136, 054106 (2012)] and a conjecture that hyperspheres provide the highest threshold, for any d, among all convex hyperparticle shapes (that are not a trivial affine transformation of a hypersphere). This scaling relation also exploits the recently discovered principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive an explicit formula for the exclusion volume v(ex) of a hyperparticle of arbitrary shape in terms of its d-dimensional volume v, surface area s, and radius of mean curvature R[over ¯] (or, equivalently, mean width). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for d=2, Platonic solids, spherocylinders, parallepipeds, and zero-volume plates for d=3 and their appropriate generalizations for d≥4. Using this information, we compute the lower bound and scaling relation for η(c) for this comprehensive set of continuum percolation models across dimensions. We demonstrate that the scaling relation provides accurate upper-bound estimates of the threshold η(c) across dimensions and becomes increasingly accurate as the space dimension increases.

Numerical computations of conductivity over agglomerated continuum percolation models

In order to clarify how the percolation theory governs the conductivities in real materials which consist of small conductive particles, e.g., nanoparticles, with random configurations in an insulator, we numerically investigate the conductivities of continuum percolation models consisting of overlapped particles using the finite difference method as a sequel of our previous article [S. Matsutani, Y. Shimosako, Y. Wang, Int. J. Mod. Phys. C 21 (2010) 709–729]. As the previous article showed the shape effect of each particle by handling different aspect ratios of spheroids, in this article we numerically show influences of the agglomeration of the particles on conductivities after we model the agglomerated configuration by employing a simple numerical algorithm which simulate an agglomerated configuration of particles by a natural parameter. We conclude that the dominant agglomeration effect on the conductivities can be interpreted as the size effect of an analyzed region. We also discuss an effect of shape of the agglomerated clusters on its universal property.

Fractal structure of equipotential curves on a continuum percolation model

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear quasi-equipotential clusters which approximately and locally have the same values as steps and stairs. Since the quasi-equipotential clusters have the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over [0,1]. The fractal dimension in [1.00, 1.246] has a peak at the percolation threshold pc.

The Ising model and critical behavior of transport in binary composite media

We present a general theory for critical behavior of transport in binary composite media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi–static case (frequency dependent) with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures.