Square-lattice site percolation at increasing ranges of neighbor bonds

Site percolation threshold of composite square lattices and its agroecology applications

We analyze the percolation threshold of square lattices comprising a combination of sites with regular and extended neighborhoods. We found that the percolation threshold of these composed systems smoothly decreases with the fraction of sites with extended neighbors. This behavior can be well-fitted by a Tsallis q-Exponential function. We found a relation between the fitting parameters and the differences in the gyration radius among neighborhoods. We also compared the percolation threshold with the critical susceptibility of nearest and next-to-nearest neighbor monoculture plantations vulnerable to the spread of phytopathogen. Notably, the critical susceptibility in monoculture plantations can be described as a linear combination of two composite systems. These results allow the refinement of mathematical models of phytopathogen propagation in agroecology. In turn, this improvement facilitates the implementation of more efficient computational simulations of agricultural epidemiology that are instrumental in testing and formulating control strategies.

Unexpected Coexisting Solid Solutions in the Quasi-Binary Ag(II) F2 /Cu(II) F2 Phase Diagram

High-temperature solid-state reaction between orthorhombic AgF2 and monoclinic CuF2 (y=0.15, 0.3, 0.4, 0.5) in a fluorine atmosphere resulted in coexisting solid solutions of Cu-poor orthorhombic and Cu-rich monoclinic phases with stoichiometry Ag1-x Cux F2 . Based on X-ray powder diffraction analyses, the mutual solubility in the orthorhombic phase (AgF2  : Cu) appears to be at an upper limit of Cu concentration of 30 mol % (Ag0.7 Cu0.3 F2 ), while the monoclinic phase (CuF2  : Ag) can form a nearly stoichiometric Cu : Ag=1 : 1 solid solution (Cu0.56 Ag0.44 F2 ), preserving the CuF2 crystal structure. Experimental data and DFT calculations showed that AgF2  : Cu and CuF2  : Ag solid solutions deviate from the classical Vegard's law. Magnetic measurements of Ag1-x Cux F2 showed that the Néel temperature (TN ) decreases with increasing Cu content in both phases. Likewise, theoretical DFT+U calculations for Ag1-x Cux F2 showed that the progressive substitution of Ag by Cu decreases the magnetic interaction strength |J2D | in both structures. Electrical conductivity measurements of Ag0.85 Cu0.15 F2 showed a modest increase in specific ionic conductivity (3.71 ⋅ 10-13 ±2.6 ⋅ 10-15  S/cm) as compared to pure AgF2 (1.85 ⋅ 10-13± 1.2 ⋅ 10-15  S/cm), indicating the formation of a vacancy- or F adatom-free metal difluoride sample.

The spread of infectious diseases from a physics perspective

This article deals with the spread of infectious diseases from a physics perspective. It considers a population as a network of nodes representing the population members, linked by network edges representing the (social) contacts of the individual population members. Infections spread along these edges from one node (member) to another. This article presents a novel, modified version of the SIR compartmental model, able to account for typical network effects and percolation phenomena. The model is successfully tested against the results of simulations based on Monte-Carlo methods. Expressions for the (basic) reproduction numbers in terms of the model parameters are presented, and justify some mild criticisms on the widely spread interpretation of reproduction numbers as being the number of secondary infections due to a single active infection. Throughout the article, special emphasis is laid on understanding, and on the interpretation of phenomena in terms of concepts borrowed from condensed-matter and statistical physics, which reveals some interesting analogies. Percolation effects are of particular interest in this respect and they are the subject of a detailed investigation. The concept of herd immunity (its definition and nature) is intensively dealt with as well, also in the context of large-scale vaccination campaigns and waning immunity. This article elucidates how the onset of herd-immunity can be considered as a second-order phase transition in which percolation effects play a crucial role, thus corroborating, in a more pictorial/intuitive way, earlier viewpoints on this matter. An exact criterium for the most relevant form of herd-immunity to occur can be derived in terms of the model parameters. The analyses presented in this article provide insight in how various measures to prevent an epidemic spread of an infection work, how they can be optimized and what potentially deceptive issues have to be considered when such measures are either implemented or scaled down.

Site percolation in distorted square and simple cubic lattices with flexible number of neighbors

This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance d, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter α, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters α and d, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

Random site percolation on honeycomb lattices with complex neighborhoods

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3-24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number z of sites in the neighborhoods, the site percolation thresholds _p follow the dependency _p ∝ 1 / z, as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of _p corresponding to the same number z of sites in the neighborhoods). An example of a single-value index ζ = ∑ i _{z r}-where _z and _r are the number of sites and radius of the ith coordination zone, respectively-characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence _p ∝ 1 / ζ. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.

Cumulative merging percolation: A long-range percolation process in networks

Percolation on networks is a common framework to model a wide range of processes, from cascading failures to epidemic spreading. Standard percolation assumes short-range interactions, implying that nodes can merge into clusters only if they are nearest neighbors. Cumulative merging percolation (CMP) is a percolation process that assumes long-range interactions such that nodes can merge into clusters even if they are topologically distant. Hence, in CMP clusters do not coincide with the topologically connected components of the network. Previous work has shown that a specific formulation of CMP features peculiar mechanisms for the formation of the giant cluster and allows one to model different network dynamics such as recurrent epidemic processes. Here we develop a more general formulation of CMP in terms of the functional form of the cluster interaction range, showing an even richer phase transition scenario with competition of different mechanisms resulting in crossover phenomena. Our analytic predictions are confirmed by numerical simulations.

Influence of network defects on the conformational structure of nanogel particles: From "closed compact" to "open fractal" nanogel particles

We propose an approach to generate a wide range of randomly branched polymeric structures to gain general insights into how polymer topology encodes a configurational structure in solution. Nanogel particles can take forms ranging from relatively symmetric sponge-like compact structures to relatively anisotropic open fractal structures observed in some nanogel clusters and in some self-associating polymers in solutions, such as aggrecan solutions under physiologically relevant conditions. We hypothesize that this broad "spectrum" of branched polymer structures derives from the degree of regularity of bonding in the network defining these structures. Accordingly, we systematically introduce bonding defects in an initially perfect network having a lattice structure in three and two topological dimensions corresponding to "sponge" and "sheet" structures, respectively. The introduction of bonding defects causes these "closed" and relatively compact nanogel particles to transform near a well-defined bond percolation threshold into "open" fractal objects with the inherent anisotropy of randomly branched polymers. Moreover, with increasing network decimation, the network structure of these polymers acquires other configurational properties similar to those of randomly branched polymers. In particular, the mass scaling of the radius of gyration and its eigenvalues, as well as hydrodynamic radius, intrinsic viscosity, and form factor for scattering, all undergo abrupt changes that accompany these topological transitions. Our findings support the idea that randomly branched polymers can be considered to be equivalent to perforated sheets from a "universality class" standpoint. We utilize our model to gain insight into scattering measurements made on aggrecan solutions.

A Multi-Scale Network with Percolation Model to Describe the Spreading of Forest Fires

Forest fires have been a major threat to forest ecosystems and its biodiversity, as well as the environment in general, particularly in the Mediterranean regions. To mitigate fire spreading, this study aims at finding a fire-break solution for territories prone to fire occurrence. To the effect, here follows a model to map and predict phase transitions in fire regimes (spanning fires vs penetrating fires) based on terrain morphology. The structure consists of a 2-scale network using site percolation and SIR epidemiology rules in a cellular automata to model local fire Dynamics. The target area for the application is the region of Serra de Ossa in Portugal, due to its wildfire incidence. The study considers the cases for a Moore neighbourhood of warm cells of radius 1 and 2 and also considers a heterogeneous terrain with 3 classes of vegetation. Phase transitions are found for different combinations of fire risk for each of these classes and use these values to parametrize the resulting landscape network.

Machine learning for percolation utilizing auxiliary Ising variables

Machine learning for phase transition has received intensive research interest in recent years. However, its application in percolation still remains challenging. We propose an auxiliary Ising mapping method for the machine learning study of the standard percolation as well as a variety of statistical mechanical systems in correlated percolation representation. We demonstrate that unsupervised machine learning is able to accurately locate the percolation threshold, independent of the spatial dimension of system or the type of phase transition, which can be first-order or continuous. Moreover, we show that, by neural network machine learning, auxiliary Ising configurations for different universalities can be classified with a high confidence level. Our results indicate that the auxiliary Ising mapping method, despite its simplicity, can advance the application of machine learning in statistical and condensed-matter physics.

Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions

Extended-range percolation on various regular lattices, including all 11 Archimedean lattices in two dimensions and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds p_{c} for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/p_{c} and z versus -1/ln(1-p_{c}) using the data of d'Iribarne et al. [J. Phys. A 32, 2611 (1999)JPHAC50305-447010.1088/0305-4470/32/14/002] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zp_{c}∼4η_{c}=4.51235, where η_{c} is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zp_{c}∼8η_{c}=2.7351, where η_{c} is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior p_{c}=1/(z-1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zp_{c}-1∼a_{1}z^{-x}, with x=2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21, 56 (2016)1083-648910.1214/16-EJP6] and by Xu et al. [Phys. Rev. E 103, 022127 (2021)2470-004510.1103/PhysRevE.103.022127]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zp_{c} has universal properties, depending only on the dimension of the system and whether site or bond percolation but not on the type of lattice.

Invasion percolation in short-range and long-range disorder background

In the original invasion percolation model, a random number quantifies the role of necks, or generally the quality of pores, ignoring the structure of pores and impermeable regions (to which the invader cannot enter). In this paper, we investigate invasion percolation (IP), taking into account the impermeable regions, the configuration of which is modeled by ordinary and Ising-correlated site percolation (with short-range interactions, SRI), on top of which the IP dynamics is defined. We model the long-ranged correlations of pores by a random Coulomb potential (RCP). By examining various dynamical observables, we suggest that the critical exponents of Ising-correlated cases change considerably only in the vicinity of the critical point (critical temperature), while for the ordinary percolation case the exponents are robust against the occupancy parameter p. The properties of the model for the long-range interactions [LRI (RCP)] are completely different from the normal IP. In particular, the fractal dimension of the external frontier of the largest hole is nearly 4/3 for SRI far from the critical points, which is compatible with normal IP, while it converges to 1.099±0.04 for RCP. For the latter case, the time dependence of our observables is divided into three parts: the power law (short time), the logarithmic (moderate time), and the linear (long time) regimes. The second crossover time is shown to go to infinity in the thermodynamic limit, whereas the first crossover time is nearly unchanged, signaling the dominance of the logarithmic regime. The average gyration radius of the growing clusters, the length of their external perimeter, and the corresponding roughness are shown to be nearly constant for the long-time regime in the thermodynamic limit.

Machine learning predictions of COVID-19 second wave end-times in Indian states

The estimate of the remaining time of an ongoing wave of epidemic spreading is a critical issue. Due to the variations of a wide range of parameters in an epidemic, for simple models such as Susceptible-Infected-Removed (SIR) model, it is difficult to estimate such a time scale. On the other hand, multidimensional data with a large set attributes are precisely what one can use in statistical learning algorithms to make predictions. Here we show, how the predictability of the SIR model changes with various parameters using a supervised learning algorithm. We then estimate the condition in which the model gives the least error in predicting the duration of the first wave of the COVID-19 pandemic in different states in India. Finally, we use the SIR model with the above mentioned optimal conditions to generate a training data set and use it in the supervised learning algorithm to estimate the end-time of the ongoing second wave of the pandemic in different states in India.

Alloying a single and a double perovskite: a Cu+/2+ mixed-valence layered halide perovskite with strong optical absorption

Introducing heterovalent cations at the octahedral sites of halide perovskites can substantially change their optoelectronic properties. Yet, in most cases, only small amounts of such metals can be incorporated as impurities into the three-dimensional lattice. Here, we exploit the greater structural flexibility of the two-dimensional (2D) perovskite framework to place three distinct stoichiometric cations in the octahedral sites. The new layered perovskites AI 4[CuII(CuIInIII)0.5Cl8] (1, A = organic cation) may be derived from a CuI-InIII double perovskite by replacing half of the octahedral metal sites with Cu2+. Electron paramagnetic resonance and X-ray absorption spectroscopy confirm the presence of Cu2+ in 1. Crystallographic studies demonstrate that 1 represents an averaging of the CuI-InIII double perovskite and CuII single perovskite structures. However, whereas the highly insulating CuI-InIII and CuII perovskites are colorless and yellow, respectively, 1 is black, with substantially higher electronic conductivity than that of either endmember. We trace these emergent properties in 1 to intervalence charge transfer between the mixed-valence Cu centers. We further propose a tiling model to describe how the Cu+, Cu2+, and In3+ coordination spheres can pack most favorably into a 2D perovskite lattice, which explains the unusual 1 : 2 : 1 ratio of these cations found in 1. Magnetic susceptibility data of 1 further corroborate this packing model. The emergence of enhanced visible light absorption and electronic conductivity in 1 demonstrates the importance of devising strategies for increasing the compositional complexity of halide perovskites.

Percolation-intercropping strategies to prevent dissemination of phytopathogens on plantations

Phytophthora is one of the most aggressive and worldwide extended phytopathogens that attack plants and trees. Its effects produce tremendous economical losses in agronomy and forestry since no effective fungicide exists. We propose to combine percolation theory with an intercropping sowing configuration as a non-chemical strategy to minimize the dissemination of the pathogen. In this work, we model a plantation as a square lattice where two types of plants are arranged in alternating columns or diagonals, and Phytophthora zoospores are allowed to propagate to the nearest and next-to-nearest neighboring plants. We determine the percolation threshold for each intercropping configuration as a function of the plant's susceptibilities and the number of inoculated cells at the beginning of the propagation process. The results are presented as phase diagrams where crop densities that prevent the formation of a spanning cluster of susceptible or diseased plants are indicated. The main result is the existence of susceptibility value combinations for which no spanning cluster is formed even if every cell in the plantation is sowed. This finding can be useful in choosing a configuration and density of plants that minimize damages caused by Phytophthora. We illustrate the application of the phase diagrams with the susceptibilities of three plants with a high commercial value.

Percolation thresholds on a triangular lattice for neighborhoods containing sites up to the fifth coordination zone

We determine thresholds p_{c} for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_{c} on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ=∑_{i}z_{i}r_{i}^{2}/i (depending on the coordination zone number i, the neighborhood coordination number z, and the square distance r^{2} to sites in ith coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_{c} to ξ. The thresholds roughly follow a power law p_{c}∝ξ^{-γ} with γ≈0.710(19).

Postprocessing techniques for gradient percolation predictions on the square lattice

In this work, we revisit the classic problem of site percolation on a regular square lattice. In particular, we investigate the effect of quantization bias errors on percolation threshold predictions for large probability gradients and propose a mitigation strategy. We demonstrate through extensive computational experiments that the assumption of a linear relationship between probability gradient and percolation threshold used in previous investigations is invalid. Moreover, we demonstrate that, due to skewness in the distribution of occupation probabilities visited the average does not converge monotonically to the true percolation threshold. We identify several alternative metrics which do exhibit monotonic (albeit not linear) convergence and document their observed convergence rates.

Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

By means of extensive Monte Carlo simulation, we study extended-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors up to the eighth nearest neighbors for the square lattice and the ninth nearest neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods of connected sites can be mapped to problems of lattice percolation of extended objects of a given shape, such as disks and spheres, and the thresholds can be related to the continuum thresholds η_{c} for objects of those shapes. This mapping implies zp_{c}∼4η_{c}=4.51235 in two dimensions and zp_{c}∼8η_{c}=2.7351 in three dimensions for large z for circular and spherical neighborhoods, respectively, where z is the coordination number. Fitting our data for compact neighborhoods to the form p_{c}=c/(z+b) we find good agreement with this prediction, c=2^{d}η_{c}, with the constant b representing a finite-z correction term. We also examined results from other studies using this fitting formula. A good fit of the large but finite-z behavior can also be made using the formula p_{c}=1-exp(-2^{d}η_{c}/z), a generalization of a formula of Koza, Kondrat, and Suszcayński [J. Stat. Mech.: Theor. Exp. (2014) P110051742-546810.1088/1742-5468/2014/11/P11005]. We also study power-law fits which are applicable for the range of values of z considered here.

Site percolation thresholds on triangular lattice with complex neighborhoods

We determine thresholds p_c for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are p_c(4NN)=0.192410(43), p_c(3NN+2NN)=0.232008(38), p_c(5NN+4NN)=0.140286(5), p_c(3NN+2NN+NN)=0.215484(19), p_c(5NN+4NN+NN)=0.131792(58), p_c(5NN+4NN+3NN+2NN)=0.117579(41), and p_c(5NN+4NN+3NN+2NN+NN)=0.115847(21). The method is tested on the standard case of site percolation on the triangular lattice, where p_c(NN)=p_c(2NN)=p_c(3NN)=p_c(5NN)=12 is recovered with five digits accuracy p_c(NN)=0.500029(46) by averaging over one thousand lattice realizations only.

Noise induced unanimity and disorder in opinion formation

We propose an opinion dynamics model based on Latané’s social impact theory. Actors in this model are heterogeneous and, in addition to opinions, are characterised by their varying levels of persuasion and support. The model is tested for two and three initial opinions randomly distributed among actors. We examine how the noise (randomness of behaviour) and the flow of information among actors affect the formation and spread of opinions. Our main research involves the process of opinion formation and finding phases of the system in terms of parameters describing noise and flow of the information for two and three opinions available in the system. The results show that opinion formation and spread are influenced by both (i) flow of information among actors (effective range of interactions among actors) and (ii) noise (randomness in adopting opinions). The noise not only leads to opinions disorder but also it promotes consensus under certain conditions. In disordered phase and when the exchange of information is spatially effectively limited, various faces of disorder are observed, including system states, where the signatures of self-organised criticality manifest themselves as scale-free probability distribution function of cluster sizes. Then increase of noise level leads system to disordered random state. The critical noise level above which histograms of opinion clusters’ sizes loose their scale-free character increases with increase of the easy of information flow.

Bond percolation on simple cubic lattices with extended neighborhoods

We study bond percolation on the simple cubic lattice with various combinations of first, second, third, and fourth nearest neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result p_{c}=1/(z-1). Fitting our data and data for additional nearest neighbors, we find p_{c}(z-1)=1+1.224z^{-1/2}.

Site-bond percolation solution to preventing the propagation of Phytophthora zoospores on plantations

We propose a strategy based on the site-bond percolation to minimize the propagation of Phytophthora zoospores on plantations, consisting in introducing physical barriers between neighboring plants. Two clustering processes are distinguished: (i) one of cells with the presence of the pathogen, detected on soil analysis, and (ii) that of diseased plants, revealed from a visual inspection of the plantation. The former is well described by the standard site-bond percolation. In the latter, the percolation threshold is fitted by a Tsallis distribution when no barriers are introduced. We provide, for both cases, the formulas for the minimal barrier density to prevent the emergence of the spanning cluster. Though this work is focused on a specific pathogen, the model presented here can also be applied to prevent the spreading of other pathogens that disseminate, by other means, from one plant to the neighboring ones. Finally, the application of this strategy to three types of commercially important Mexican chili plants is also shown.

A cellular automaton for modeling non-trivial biomembrane ruptures

A novel cellular automaton (CA) for simulating biological membrane rupture is proposed. Constructed via simple rules governing deformation, tension, and fracture, the CA incorporates ideas from standard percolation models and bond-based fracture methods. The model is demonstrated by comparing simulations with experimental results of a double bilayer lipid membrane expanding on a solid substrate. Results indicate that the CA can capture non-trivial rupture morphologies such as floral patterns and the saltatory dynamics of fractal avalanches observed in experiments. Moreover, the CA provides insight into the poorly understood role of inter-layer adhesion, supporting the hypothesis that the density of adhesion sites governs rupture morphology.

The Electrical Properties of Hybrid Composites Based on Multiwall Carbon Nanotubes with Graphite Nanoplatelets

In the present work, we have investigated the concentration dependences of electrical conductivity of monopolymer composites with graphite nanoplatelets or multiwall carbon nanotubes and hybrid composites with both multiwall carbon nanotubes and graphite nanoplatelets. The latter filler was added to given systems in content of 0.24 vol%. The content of multiwall carbon nanotubes is varied from 0.03 to 4 vol%. Before incorporation into the epoxy resin, the graphite nanoplatelets were subjected to ultraviolet ozone treatment for 20 min. It was found that the addition of nanocarbon to the low-viscosity suspension (polymer, acetone, hardener) results in formation of two percolation transitions. The percolation transition of the composites based on carbon nanotubes is the lowest (0.13 vol%).It was determined that the combination of two electroconductive fillers in the low-viscosity polymer results in a synergistic effect above the percolation threshold, which is revealed in increase of the conductivity up to 20 times. The calculation of the number of conductive chains in the composite and contact electric resistance in the framework of the model of effective electrical resistivity allowed us to explain the nature of synergistic effect. Reduction of the electric contact resistance in hybrid composites may be related to a thinner polymer layer between the filler particles and the growing number of the particles which take part in the electroconductive circuit.

Spatial self-organization in hybrid models of multicellular adhesion

Spatial self-organization emerges in distributed systems exhibiting local interactions when nonlinearities and the appropriate propagation of signals are at work. These kinds of phenomena can be modeled with different frameworks, typically cellular automata or reaction-diffusion systems. A different class of dynamical processes involves the correlated movement of agents over space, which can be mediated through chemotactic movement or minimization of cell-cell interaction energy. A classic example of the latter is given by the formation of spatially segregated assemblies when cells display differential adhesion. Here, we consider a new class of dynamical models, involving cell adhesion among two stochastically exchangeable cell states as a minimal model capable of exhibiting well-defined, ordered spatial patterns. Our results suggest that a whole space of pattern-forming rules is hosted by the combination of physical differential adhesion and the value of probabilities modulating cell phenotypic switching, showing that Turing-like patterns can be obtained without resorting to reaction-diffusion processes. If the model is expanded allowing cells to proliferate and die in an environment where diffusible nutrient and toxic waste are at play, different phases are observed, characterized by regularly spaced patterns. The analysis of the parameter space reveals that certain phases reach higher population levels than other modes of organization. A detailed exploration of the mean-field theory is also presented. Finally, we let populations of cells with different adhesion matrices compete for reproduction, showing that, in our model, structural organization can improve the fitness of a given cell population. The implications of these results for ecological and evolutionary models of pattern formation and the emergence of multicellularity are outlined.

Dimer covering and percolation frustration

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.

Nonsynchronous updating in the multiverse of cellular automata

In this paper we study updating effects on cellular automata rule space. We consider a subset of 6144 order-3 automata from the space of 262144 bidimensional outer-totalistic rules. We compare synchronous to asynchronous and sequential updatings. Focusing on two automata, we discuss how update changes destroy typical structures of these rules. Besides, we show that the first-order phase transition in the multiverse of synchronous cellular automata, revealed with the use of a recently introduced control parameter, seems to be robust not only to changes in update schema but also to different initial densities.

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are p(C)(4NN)=0.31160(12),p(C)(4NN+NN)=0.15040(12),p(C)(4NN+2NN)=0.15950(12),p(C)(4NN+3NN)=0.20490(12),p(C)(4NN+2NN+NN)=0.11440(12),p(C)(4NN+3NN+NN)=0.11920(12),p(C)(4NN+3NN+2NN)=0.11330(12), and p(C)(4NN+3NN+2NN+NN)=0.10000(12), where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold p(C)(4NN) is exactly equal to p(C)(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value p(C) to be reached, similar to that obtained with the classical method but ten times faster.

Effective transport properties of random composites: continuum calculations versus mapping to a network

The effective transport properties and percolation of continuum composites have commonly been studied using discrete models, i.e., by mapping the continuum to a lattice or network. In this study we instead directly solve the continuum transport equations for composite microstructures both analytically and numerically, and we extract the continuum percolation threshold and scaling exponents for the two-dimensional square tile system. We especially focus on the role of corner contacts on flux flow and further show that mapping such "random checkerboard" systems to a network leads to a spurious secondary percolation threshold and causes shifts in the critical scaling exponents of the effective transport properties.

Innate visual learning through spontaneous activity patterns

Patterns of spontaneous activity in the developing retina, LGN, and cortex are necessary for the proper development of visual cortex. With these patterns intact, the primary visual cortices of many newborn animals develop properties similar to those of the adult cortex but without the training benefit of visual experience. Previous models have demonstrated how V1 responses can be initialized through mechanisms specific to development and prior to visual experience, such as using axonal guidance cues or relying on simple, pairwise correlations on spontaneous activity with additional developmental constraints. We argue that these spontaneous patterns may be better understood as part of an "innate learning" strategy, which learns similarly on activity both before and during visual experience. With an abstraction of spontaneous activity models, we show how the visual system may be able to bootstrap an efficient code for its natural environment prior to external visual experience, and we continue the same refinement strategy upon natural experience. The patterns are generated through simple, local interactions and contain the same relevant statistical properties of retinal waves and hypothesized waves in the LGN and V1. An efficient encoding of these patterns resembles a sparse coding of natural images by producing neurons with localized, oriented, bandpass structure-the same code found in early visual cortical cells. We address the relevance of higher-order statistical properties of spontaneous activity, how this relates to a system that may adapt similarly on activity prior to and during natural experience, and how these concepts ultimately relate to an efficient coding of our natural world.

Restoring site percolation on damaged square lattices

Restoring site percolation on a damaged square lattice with the nearest neighbor (N2) is investigated using two different strategies. In the first one, a density y of new sites are created on the empty sites with longer range links, either next-nearest neighbor (N3) or next-next-nearest neighbor (N4), but without N2. In the second one, new longer range links N3 or N4 are added to N2 but only for a fraction v of the remaining non-destroyed sites. Starting at p(c)(N2), with a density x of randomly destroyed sites, the values of y(c) and v(c), which restore site percolation, are calculated for both strategies with, respectively, N3 and N4 using Monte Carlo simulations. Results are obtained for the whole range 0 < or = x < or = p(c)(N2).