Bond Percolation Model Research Articles

Polyimide Aerogels by Ring-Opening Metathesis Polymerization (ROMP)

Polyimide aerogel monoliths are prepared by ring-opening metathesis polymerization (ROMP) of a norbornene end-capped diimide, bis-NAD, obtained as the condensation product of nadic anhydride with 4,4′-methylenedianiline. The density of the material was varied in the range of 0.13−0.66 g cm−3 by varying the concentration of bis-NAD in the sol. Wet gels experience significant shrinkage, relative to their molds (28%−39% in linear dimensions), but the final aerogels retain high porosities (50%−90% v/v), high surface areas (210−632 m2 g−1, of which up to 25% is traced to micropores), and pore size distributions in the mesoporous range (20−33 nm). The skeletal framework consists of primary particles 16−17 nm in diameter, assembling to form secondary aggregates (by SANS and SEM) 60−85 nm in diameter. At lower densities (e.g., 0.26 g cm−3), secondary particles are mass fractals (Dm = 2.34 ± 0.03) turning to closed-packed surface fractal objects (DS = 3.0) as the bulk density increases (≥0.34 g cm−3), suggesting a change in the network-forming mechanism from diffusion-limited aggregation of primary particles to a space-filling bond percolation model. The new materials combine facile one-step synthesis with heat resistance up to 200 °C, high mechanical compressive strength and specific energy absorption (168 MPa and 50 J g−1, respectively, at 0.39 g cm−3 and 88% ultimate strain), low speed of sound (351 m s−1 at 0.39 g cm−3) and styrofoam-like thermal conductivity (0.031 W m−1 K−1 at 0.34 g cm−3 and 25 °C); hence, they are reasonable multifunctional candidate materials for further exploration as thermal/acoustic insulation at elevated temperatures.

Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

A generalized star-triangle transformation and a concept of triangle-duality have been introduced recently in the physics literature to predict exact percolation threshold values of several lattices. Conditions for the solution of bond percolation models are investigated, and an infinite class of lattice graphs for which exact bond percolation thresholds may be rigorously determined as the solution of a polynomial equation are identified. This class is naturally described in terms of hypergraphs, leading to definitions of planar hypergraphs and self-dual planar hypergraphs. There exist infinitely many self-dual planar 3-uniform hypergraphs, and, as a consequence, there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.

Magneto-transport properties of Fe-doped LSMO manganites

Magneto-transport measurements have been performed on La 0.67Sr 0.33Mn 1− x Fe x O 3 ( x = 0.0, 0.05 and 0.07) manganites synthesized by solid state route. The overall nature of AC susceptibility is found to be frequency independent. With the substitution of Mn by Fe, the transport properties dramatically change suppressing the double-exchange interaction. This in turn weakens the ferromagnetism and consequently decreases the paramagnetic to ferromagnetic transition temperature ( T c ). However, DC electrical resistivity of the samples show board hump at metal–insulator transition temperature ( T MI ) in contrary with sharp transition at T c and T MI < T c indicating decoupling of electrical and magnetic properties. Resistivity behavior of these samples can be well explained by the consideration of bond percolation model which is basically developed from effective approximation method.

A Theory of Aggregate Demand Fluctuations and Asset Bubbles

We look at the financial markets as represented by a network of agents similar to bond percolation models in physics or epidemiology models. We aim to figure out how an agent based network model can cause perturbations that can cause failures of the traditional economic theory, specifically the AD-AS model of New Keynesian Theory. This model is utilized for its dual uses of having an inbuilt correction mechanism that is similar to what markets display and real world correlations in micro behaviour. We analyse the threshold conditions that result in boundaries beyond which an idea can spread throughout the network and look at the various factors that contribute towards such information percolation. It allows us a look into the micro-behaviour of agents involved in the markets and helps us look at a specific role of the market as a provider of feedback on actions taken. It further gives us a good insight into the roles needed to be played by policymakers and market regulators and gives a few modest proposals that could serve as a warning mechanism against future bubbles and Great Depressions. We look into such a system here to model how bubbles form in economic systems as a consequence of its current architecture and information transmission.

Model investigation on the probability of QGP formation at different centralities in relativistic heavy ion collisions

The quantitative dependence of quark-gluon plasma (QGP)-formation probability (${P}_{\mathrm{QGP}}$) on the centrality of $\mathrm{Au}\ensuremath{-}\mathrm{Au}$ collisions is studied using a bond percolation model. The ${P}_{\mathrm{QGP}}$ versus the maximum distance ${S}_{\mathrm{max}}$ for a bond to form is calculated from the model for various nuclei and the ${P}_{\mathrm{QGP}}$ at different centralities of $\mathrm{Au}\ensuremath{-}\mathrm{Au}$ collisions for the given ${S}_{\mathrm{max}}$ are obtained therefrom. The experimental data of the nuclear modification factor ${R}_{\mathit{AA}}({p}_{T})$ for the most central $\mathrm{Au}\ensuremath{-}\mathrm{Au}$ collisions at $\sqrt{{s}_{\mathit{NN}}}=200$ and $130$ GeV are utilized to transform ${S}_{\mathrm{max}}$ to $\sqrt{{s}_{\mathit{NN}}}$. The ${P}_{\mathrm{QGP}}$ for different centralities of $\mathrm{Au}\ensuremath{-}\mathrm{Au}$ collisions at these two energies are thus obtained, which is useful for correctly understanding the centrality dependence of the experimental data.

Bond percolation with attenuation in high dimensional Voronoĭ tilings

AbstractLet $\cal{P}$ be the set of points in a realization of a uniform Poisson process in ℝn. The set $\cal{P}$ determines a Voronoĭ tiling of ℝn. Construct an infinite graph $\cal{G}$ with vertex set $\cal{P}$ and edges joining vertices when the corresponding Voronoĭ cells share a (n − 1)‐dimensional boundary face. We consider bond percolation models on $\cal{G}$ obtained by declaring each edge xy of $\cal{G}$ open independently with probability p(∥x − y∥), depending only on the Euclidean distance ∥x − y∥ between the vertices. We give some sufficient conditions on p(t) that ensures that an infinite connected component (i.e., percolation) occurs, or does not occur. In particular, we show that for p(t) = p is a constant, there is a phase transition at a critical probability p = pc(n), where 2−n(5nlog n)−1 ≤ pc(n) ≤ C2−n $\sqrt{n}\;{\rm log} \; n$. We also show that if p(t) = e−λt then there is a phase transition at a critical parameter λ = λc(n), where λc(n) = (log e2 + o(1))n/2rn, where rn is the radius of the n‐dimensional sphere that, on average, contains a single point of $\cal{P}$. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Liquid property of sQGP obtained from a bond percolation model with color confinement

Liquid properties of the strongly coupled quark–gluon plasma (sQGP) during the intermediate stage and after the end of crossover from hadronic matter to sQGP are studied using a bond percolation model. A method to calculate the pair distribution function with respect to the chemical distance is developed. The shape of the corresponding pair distribution functions turns out to exhibit the character of liquid.

Multilevel unipolar resistance switching in TiO2 thin films

We report on multilevel switching behavior in the unipolar resistance switching of TiO2 thin films. Multiple metastable states were observed during the reset process by measuring I-V curves. As observed using a conducting atomic force microscope, the multilevel resistance switching was accompanied by decreases in area and in the conductance of the top surface conducting regions. These experimental observations at both the macroscopic and microscopic levels could be explained by using the “random circuit breaker network” model, which is a dynamic bond percolation model.

Conduction in rectangular quasi-one-dimensional and two-dimensional random resistor networks away from the percolation threshold

In this study we investigate electrical conduction in finite rectangular random resistor networks in quasione and two dimensions far away from the percolation threshold p(c) by the use of a bond percolation model. Various topologies such as parallel linear chains in one dimension, as well as square and triangular lattices in two dimensions, are compared as a function of the geometrical aspect ratio. In particular we propose a linear approximation for conduction in two-dimensional systems far from p(c), which is useful for engineering purposes. We find that the same scaling function, which can be used for finite-size scaling of percolation thresholds, also applies to describe conduction away from p(c). This is in contrast to the quasi-one-dimensional case, which is highly nonlinear. The qualitative analysis of the range within which the linear approximation is legitimate is given. A brief link to real applications is made by taking into account a statistical distribution of the resistors in the network. Our results are of potential interest in fields such as nanostructured or composite materials and sensing applications.

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.

Heterogeneous bond percolation on multitype networks with an application to epidemic dynamics

Considerable attention has been paid, in recent years, to the use of networks in modeling complex real-world systems. Among the many dynamical processes involving networks, propagation processes-in which the final state can be obtained by studying the underlying network percolation properties-have raised formidable interest. In this paper, we present a bond percolation model of multitype networks with an arbitrary joint degree distribution that allows heterogeneity in the edge occupation probability. As previously demonstrated, the multitype approach allows many nontrivial mixing patterns such as assortativity and clustering between nodes. We derive a number of useful statistical properties of multitype networks as well as a general phase transition criterion. We also demonstrate that a number of previous models based on probability generating functions are special cases of the proposed formalism. We further show that the multitype approach, by naturally allowing heterogeneity in the bond occupation probability, overcomes some of the correlation issues encountered by previous models. We illustrate this point in the context of contact network epidemiology.

INVASION PERCOLATION IN THE PRESENCE OF NANOPORES

Site invasion percolation (IP) processes are combined with bond percolation model, to study the effects of size restriction on low capillary number immiscible displacement in heterogeneous nanoporous media. Both cases of compressible (NTIP) and incompressible defender fluid (TIP) are considered. It is found that in site IP the value of mass uptake increases with the size of invader particles, if the latter is not greater than a critical value. This occurs when the accessible porosity of the medium decreases as the size of fluid particles increases. We also investigate the effect of nanopore's concentration on the mass and the anisotropy of sample spanning cluster as well as the critical exponent of trap numbers.

Corner percolation on ℤ2 and the square root of 17

We consider a four-vertex model introduced by Bálint Tóth: a dependent bond percolation model on ℤ2 in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability ℙ(diameter of the cycle of the origin >n)≈n−γ and the expectation $\mathbb{E}$ (length of a typical cycle with diameter n)≈nδ, with $\gamma=(5-\sqrt{17})/4=0.219\ldots$ and $\delta=(\sqrt{17}+1)/4=1.28\ldots$. The value of δ comes from a singular sixth order ODE, while the relation γ+δ=3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive Brownian motion, whose level sets have Hausdorff dimension 3/2. We also include many open problems, for example, on the conformal invariance of certain linear entropy models.

Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

We have obtained the exact asymptotics of the determinant det 1 ⩽ r , s ⩽ L [ ( r + s − 2 r − 1 ) + exp ( i θ ) δ r , s ] . Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to relative order L −14 to the leading term. This determinant is known to give weighted enumerations of cyclically symmetric plane partitions, weighted enumerations of certain families of vicious walkers and it has been conjectured to be proportional to the one point function of the O ( 1 ) loop model on a cylinder of circumference L. We apply our result to the loop model and give exact expressions for the asymptotics of the average of the number of loops surrounding a point and the fluctuation in this number. For the related bond percolation model at the critical point, we give exact expressions for the asymptotics of the probability that a point is on a cluster that wraps around a cylinder of even circumference and the probability that a point is on a cluster spanning a cylinder of odd circumference.

Second look at the spread of epidemics on networks

In an important paper, Newman [Phys. Rev. E66, 016128 (2002)] claimed that a general network-based stochastic Susceptible-Infectious-Removed (SIR) epidemic model is isomorphic to a bond percolation model, where the bonds are the edges of the contact network and the bond occupation probability is equal to the marginal probability of transmission from an infected node to a susceptible neighbor. In this paper, we show that this isomorphism is incorrect and define a semidirected random network we call the epidemic percolation network that is exactly isomorphic to the SIR epidemic model in any finite population. In the limit of a large population, (i) the distribution of (self-limited) outbreak sizes is identical to the size distribution of (small) out-components, (ii) the epidemic threshold corresponds to the phase transition where a giant strongly connected component appears, (iii) the probability of a large epidemic is equal to the probability that an initial infection occurs in the giant in-component, and (iv) the relative final size of an epidemic is equal to the proportion of the network contained in the giant out-component. For the SIR model considered by Newman, we show that the epidemic percolation network predicts the same mean outbreak size below the epidemic threshold, the same epidemic threshold, and the same final size of an epidemic as the bond percolation model. However, the bond percolation model fails to predict the correct outbreak size distribution and probability of an epidemic when there is a nondegenerate infectious period distribution. We confirm our findings by comparing predictions from percolation networks and bond percolation models to the results of simulations. In the Appendix, we show that an isomorphism to an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model.

CARDY'S FORMULA FOR CERTAIN MODELS OF THE BOND-TRIANGULAR TYPE

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so-called Cardy–Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.

A percolation study of electrical properties of reservoir rocks

This paper focuses on theory studies of rock electrical properties using percolation theory. The results show that classical site and bond percolation models can well describe the characteristics of the pore connectivity and the effects on the electrical resistivity. Because of the over-idealized simplification, classical site and bond percolation models cannot well show the effects of the wettability on rock resistivity. Invasion percolation (IP) model can conveniently describe the micropores around main void space and the effects of the wettability. Therefore, it can give the results agreeing with the experimental data. At high water saturation, rock electrical properties show the Archie behavior. At low water saturation, the non-Archie behavior appears. For water-wet rock with irregular grooves and crevices, the I vs. S w curve shows downwards trend. For oil-wet sample (non-shaly sandstone, without conductive grain), the curve shows upward trend.

Incorporating variability into an approximation formula for bond percolation thresholds of planar periodic lattices

Approximation formulas to predict values for bond percolation thresholds of periodic graphs make use of certain features of lattice graphs such as dimension and average degree. We show that a relationship exists between the average and second-moment of the degree of a graph and the average degree of its line graph. Using this relationship together with the well-known bond-to-site transformation between the bond percolation model on a graph and the site percolation model on its line graph, we create a new approximation formula that improves the accuracy of bond percolation threshold predictions.

Geometrical aspects of isoscaling

The property of isoscaling in nuclear fragmentation is studied using a simple bond percolation model with “isospin” added as an extra degree of freedom. It is shown, first, that with the assumption of fair sampling and with homogeneous probabilities it is possible to solve the problem analytically. Second, numerical percolations of hundreds of thousands of grids of different sizes and with different N to Z ratios confirm this prediction with remarkable agreement. It is thus concluded that isoscaling emerges even in the simple case of a classical non-interacting system such as two-species percolation under the assumption of fair sampling; if put in the nomenclature of the minimum information theory, isoscaling in percolation appears to require nothing more than the existence of equiprobable configurations in maximum entropy states.

Finite-size scaling of energylike quantities in percolation

We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, Nc and Nb, respectively. These quantities include C(2c) = (<N(2)c> - <Nc>2)/Ld and C(cb) = (<NcNb> - <Nc><Nb>)/Ld, where L is the linear system size and d is the spatial dimensionality. In statistical models with thermal fluctuations, C(2c) and C(cb) are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of C(2c) and C(cb) is described by the thermal critical exponent y(t)-d. We also measure quantity kappa b = 2<NbS2> / <S2> - <NbS4> / <S4> - <Nb> and an analogous quantity kappa c for Nc, where S2 and S4 are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that kappa b and kappa c diverge as L(y)t for L --> infinity. The analysis of the data of kappa b and kappa c yields y(t) = 1.145(2) for the three-dimensional percolation, in good agreement with existing results.