Regular Two-dimensional Lattice Research Articles

Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices.

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ_{c}, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work [Wang et al., Phys. Rev. E 105, 034118 (2022)2470-004510.1103/PhysRevE.105.034118] found symmetric properties of χ_{c}(n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ_{c} behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i.e., the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ_{c} for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ_{c}(n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.

Chiral organic molecular structures supported by planar surfaces.

We employ the molecular dynamics simulations to study the dynamics of acetanilide (ACN) molecules placed on a flat surface of planar multilayer hexagonal boron nitride. We demonstrate that the ACN molecules, known to be achiral in the three-dimensional space, become chiral after being placed on the substrate. Homochirality of the ACN molecules leads to stable secondary structures stabilized by hydrogen bonds between peptide groups of the molecules. By employing molecular dynamics simulations, we reveal that the structure of the resulting hydrogen-bond chains depends on the isomeric composition of the molecules. If all molecules are homochiral (i.e., with only one isomer being present), they form secondary structures (chains of hydrogen bonds in the shapes of arcs, circles, and spirals). If the molecules at the substrate form a racemic mixture, then no regular secondary structures appear, and only curvilinear chains of hydrogen bonds of random shapes emerge. A hydrogen-bond chain can form a zigzag array only if it has an alternation of isomers. Such chains can create two-dimensional (2D) regular lattices or 2D crystals. The melting scenarios of such 2D crystals depend on density of its coverage of the substrate. At 25% coverage, melting occurs continuously in the temperature interval 295-365K. For a complete coverage, melting occurs at 415-470 K due to a shift of 11% of all molecules into the second layer of the substrate.

ON THE BEHAVIOR OF A TETHERED CYLINDER ARRAY UNDER IRREGULAR WAVES

An interesting option for coastal wave attenuation can be provided by tethered floating breakwaters (see e.g. Agerton, 1976 and Seymour, 1979). This type of structure can be designed to be effective, while determining a very small impact on the environment (Dai, 2018). In principle, a regular lattice of reversed pendula (i.e. the breakwater) can be seen as an approximation to a mechanical metamaterial with interesting properties related to its periodic structure. The tethered floating breakwater we are studying has been designed as a finite lattice of submerged inverse pendula. The objective is to determine the efficiency of a regular two-dimensional lattice of spherical pendula. To isolate the one-dimensional behavior, a single array of reversed cylindrical pendula anchored to the bottom and excited by long crested waves has been tested.

Random sequential adsorption of self-avoiding chains on two-dimensional lattices.

Random sequential adsorption of extended objects deposited on two-dimensional regular lattices is studied. The depositing objects are chains formed by occupying adsorption sites on the substrate through a self-avoiding walk of k lattice steps; these objects are also called "tortuous k-mers." We study how the jamming coverage, θ_{j,k}, depends on k for lattices with different connectivity (honeycomb, square, and triangular). The dependence can be fitted by the function θ_{j,k}=θ_{j,k→∞}+B/k+C/k^{2}, where B and C are found to be shared parameters by the three lattices and θ_{j,k→∞} (>0) is the jamming coverage for infinitely long k-mers for each of them. The jamming coverage is found to have a growing behavior with the connectivity of the lattice. In addition, θ_{j,k} is found to be higher for tortuous k-mers than for the previously reported for linear k-mers in each lattice. The results were obtained by means of numerical simulation through an efficient algorithm whose characteristics are discussed in detail. The computational method introduced here also allows us to investigate the full-time kinetics of the surface coverage θ_{k}(t) [θ_{j,k}≡θ_{k}(t→∞)]. Along this line, different time regimes are identified and characterized.

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.

First-order spatial random coefficient non-negative integer-valued autoregressive (SRCINAR(1,1)) model

In this paper we introduce first-order spatial random coefficient non-negative integer-valued autoregressive (SRCINAR(1,1)) model to fit the spatial count data on a two-dimensional regular lattice. We use a thinning operator in order to define the processes. The mean, variance and autocorrelation function of the model are derived. Three methods (Yule-Walker, Conditional least squares and Conditional maximum likelihood) for the estimation of the parameters of the model are discussed and are compared with respect to the root mean squared error (RMSE), in a simulation study. Finally, SRCINAR(1,1) model is applied to a real data set of striga counts at the ICRISAT Sahelian Center near Niamey (Niger).

Pattern formation in a thermosensitive neural network

Pattern formation and synchronization stability in the network are dependent on cooperation between nodes and the local kinetics as well. Temperature has a significant effect on the membrane potential by regulating the channel conductance and excitability of neurons. Thermistor is a smart sensor and its resistance value changes with temperature. A nonlinear circuit composed of thermistors is effective to detect the changes in surroundings temperature. In this paper, a temperature-induced pattern formation in a two-dimensional regular lattice network composed of thermosensitive neurons is studied. Numerical simulations are performed to research the effect of parameters (coupling amplitude) and temperature spatial gradient distribution of neural network on pattern formation and synchronization stability. The bifurcation analysis of the membrane potential reveals the conversion in electrical activity and pattern selection. The results show that neural membrane potential can present distinct firing patterns with the change of temperature. The electrical synapse coupling between neurons promotes effective signals propagation in the network. It is found that the collective dynamic behavior of the network is changed by taming the coupling intensity. The Hamilton energy is estimated to predict the mode selection in neural activities. The energy can be ordered in the network under the temperature gradient distribution. Our results confirm that electrical synapse coupling and temperature spatial gradient distribution can be controlled to affect the neural activities in nervous system.

Self-Avoiding Trail model in two-dimensional regular lattices: Study of conformational quantities and relation with the Self-Avoiding Walk model

The Self-Avoiding Trail (SAT) is a random walk model represented by an N-step lattice path, with forbidden step overlaps. The SAT is a variant of the Self-Avoiding Walk (SAW) model, where a site is visited only once. There is a consensus that SAT is in the same universality class of SAW, implying that the scaling behavior of conformational quantities, represented by the exponents ν0 and Δ1, is the same in both models. However, in comparison to SAW, SAT results exhibit most pronounced finite-size effects and inconsistent estimates of ν0, depending on the lattice coordination number and conformational quantity. The scaling behavior of the persistence length, λN, is in disagreement with recent estimates for the SAW model. Also, the influence of coordination number on the lattice-dependent constants requires a better understanding. In this study, we revisit the SAT model with data generated by the pivot algorithm up to N=8000 steps, for the hexagonal, square and triangular lattices. Accordingly, we find compelling evidences that ν0 and Δ1 values are the same for the SAT and SAW models, regardless the lattice and conformational quantity used, while the most pronounced finite-size effect comes from the higher amplitude of the correction to scaling terms. Similarly to SAW, we find the λN convergence to lattice dependent constant λ∞(SAW) for the SAT model. Based on the ratio λ∞(SAW)/λ∞(SAT)∝z/2, where z is the lattice coordination number, we find that the factor z/2 quantifies the relation between SAW and SAT lattice dependent constants.

Concurrence Percolation in Quantum Networks.

Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.

A Markov Chain Monte Carlo Algorithm for Spatial Segmentation

Spatial data are very often heterogeneous, which indicates that there may not be a unique simple statistical model describing the data. To overcome this issue, the data can be segmented into a number of homogeneous regions (or domains). Identifying these domains is one of the important problems in spatial data analysis. Spatial segmentation is used in many different fields including epidemiology, criminology, ecology, and economics. To solve this clustering problem, we propose to use the change-point methodology. In this paper, we develop a new spatial segmentation algorithm within the framework of the generalized Gibbs sampler. We estimate the average surface profile of binary spatial data observed over a two-dimensional regular lattice. We illustrate the performance of the proposed algorithm with examples using artificially generated and real data sets.

Self-Avoiding Trail Model in Two-Dimensional Regular Lattices: Study of Conformational Quantities and Relation with the Self-Avoiding Walk Model

Self-Avoiding Trail Model in Two-Dimensional Regular Lattices: Study of Conformational Quantities and Relation with the Self-Avoiding Walk Model

A Study on Application of Markov Random Fields in Image Restoration and its Efficiency

Image restoration has been a popular and active field of research for decades. Images are destroyed when exposed to ‘noise’ which can occur due to physical contact or electrical/electronic interference. Here, Bayesian statistical techniques and Markov random field (MRF) theory were used to restore a black and white binary image corrupted by additive Gaussian noise with zero mean and constant variance. The binary image was used as a Markov random field. An image is comprised of pixels and these pixels have a regular two-dimensional lattice structure. A matrix including ±1 values was generated randomly. It was represented and interpreted as an Ising model in Statistical Mechanics. The probability distribution of the Ising model was used as the prior distribution (a Gibbs or Boltzmann distribution). Likelihood function was obtained by using the random matrix and the observed corrupted image. Markov Chain Monte Carlo (MCMC) method was used to simulate posterior distribution which again turns out to be a Gibbs or Boltzmann distribution. More specifically, Metropolis-Hastings algorithm which is one of the popular MCMC algorithms was used in this simulation. In this study, Peak Signal to Noise Ratio (PSNR) and Mean Squared Error (MSE) methods were used to measure the quality of restored images. MATLAB (R2013a (8.1.0.604)) was used to construct the program in this research work. Finally, high quality images were restored which were almost similar to the original image. A decrease in restored image quality was observed with the increase in noise. When the image size was increased, a higher number of iterations was required to obtain an acceptable level of quality in the restored image.

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents**All series data generated for this paper are available with the preprint at arXiv:1911.05852.

We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the θ-point to be at uc = 2.767 ± 0.002. The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is (Duminil-Copin H and Smirnov S 2014 Ann. Math. 175 1653–65), while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is (Beaton N R et al 2014 Commun. Math. Phys. 326 727–54). We could not help but notice that We discuss the difficulties of trying to prove, or disprove, this possibility. For square lattice ISAWs we find uc = 1.9474 ± 0.001, which is consistent with the best Monte Carlo analysis. We also study bridges and terminally-attached walks (TAWs) on the square lattice at the θ-point. We estimate the exponents to be γb = 0.00 ± 0.03, and γ1 = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction (Duplantier B and Saleur H 1987 Phys. Rev. Lett. 59 539–42; Seno F and Stella A L 1988 Europhys. Lett. 7 605–10; Stella A L et al 1993 J. Stat. Phys. 73 21–46) , albeit for a modified version of the problem, while the former estimate is predicted in [Duplantier B and Guttmann A J 2019 Statistical mechanics of confined polymer networks (in preparation)] to be zero.

Diffusion of innovations in finite networks: Effects of heterogeneity, clustering, and bilingual option on the threshold in the contagion game model

The contagion threshold for diffusion of innovations is defined and calculated in finite graphs (two-dimensional regular lattices, regular random networks (RRNs), and two kinds of scale-free networks (SFNs)) with and without the bilingual option. Without the bilingual option, degree inhomogeneity and clustering enhance the contagion threshold in non-regular networks except for those with an unrealistically small average degree. It is explained by the friendship paradox and detour effect. We found the general boundary of the cost that makes the bilingual option effective. With a low-cost bilingual option, among regular lattices, SFNs, and RRNs, the contagion threshold is largest in regular lattices and smallest in RRNs. The contagion threshold of regular random networks is almost the same as that of the regular trees, which is the minimum among regular networks. We show that the contagion threshold increases by clustering with a low-cost bilingual option.

Effective transport properties of conformal Voronoi-bounded columns via recurrent boundary element expansions

Effective transport properties of heterogeneous structures are predicted by geometric microstructural parameters, but these can be difficult to calculate. Here, a boundary element code with a recurrent series method accurately and efficiently determines the high order parameters of polygonal and conformal prisms in regular two-dimensional lattices and Voronoi tessellations (VT). This reveals that proximity to simpler estimates is associated with centroidal VT (cf. random VT), compactness, and VT structures (cf. similarly compact semiregular lattices).

Critical dynamics of the Kuramoto model on sparse random networks

We consider the Kuramoto model on sparse random networks such as the Erdős–Rényi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical coupling strength more accurate than obtained earlier by finite-size scaling of the stationary order parameter. Our results confirm the compatibility of the correlation-size and the temporal correlation-length exponent with the mean-field universality class. However, the scaling of the order parameter exhibits corrections much stronger than those of the Kuramoto model with all-to-all coupling, making thereby an accurate estimate of the order-parameter exponent hard. We find furthermore that, as a qualitative difference to the model with all-to-all coupling, the effective critical exponents involving the order-parameter exponent, such as the effective decay exponent characterizing the critical desynchronization dynamics show a non-monotonic approach toward the asymptotic value. In the light of these results, the technique of finite-size scaling of limited size data for the Kuramoto model on sparse graphs has to be treated cautiously.

Control of thermal conduction and rectification in a model of complex networks with two asymmetric parts

The controlling of heat flux is an important topic in the field of heat conduction and has potential applications in thermal devices. We here design a model of asymmetric network structure to study this problem in complex networks, which is composed of two parts with different topologies. We find that the heat conduction and rectification can be significantly changed by adjusting two parameters: the network rewiring parameter $\ensuremath{\delta}$ and coupling parameter $\ensuremath{\lambda}$. Two typical network structures are considered. One is composed of a random network and a two-dimensional regular lattice with half by half. Another is composed of two random networks with opposite coupling parameter $\ifmmode\pm\else\textpm\fi{}\ensuremath{\lambda}$. Our numerical simulations show that for the first case, both the network flux and rectification coefficient decrease monotonically with the increase of $\ensuremath{\delta}$ and there is an optimal $\ensuremath{\lambda}$ for the maximum flux. For the second case, the network flux will decrease monotonically while the rectification coefficient increases monotonically, with the increase of $\ensuremath{\lambda}$. A brief theory of phonon spectra is applied to explain the numerical results.

Crystalline splitting of d orbitals in two-dimensional regular optical lattices

In solids, crystal field splitting refers to the lifting of atomic orbital degeneracy by the surrounding ions through the static electric field. Similarly, we show that the degenerated $d$ orbitals, which were derived in the harmonic oscillator approximation, are split into a low-lying $d_{x^2+y^2}$ singlet and a $d_{x^2-y^2/xy}$ doublet by the high-order Taylor polynomials of triangular optical potential. The low-energy effective theory of the orbital Mott insulator at $2/3$ filling is generically described by the Heisenberg-Compass model, where the antiferro-orbital exchange interactions of compass type depend on the bond orientation and are geometrically frustrated in the triangular lattice. While, for the square optical lattice, the degenerated $d$ orbitals are split into a different multiplet structure, i.e. a low-lying $d_{x^2\pm y^2}$ doublet and a $d_{xy}$ singlet, which has its physical origin in the $C_{4v}$ point group symmetry of square optical potential. Our results build a novel bridge between ultracold atom systems and solid-state systems for the investigation of $d$-orbital physics.

Mosaics of topological defects in micropatterned liquid crystal textures.

Topological defects in the orientational order that appear in thin slabs of a nematic liquid crystal, as seen in the standard schlieren texture, behave as a random quasi-two-dimensional system with strong optical birefringence. We present an approach to creating and controlling the defects using air pillars, trapped by micropatterned holes in the silicon substrate. The defects are stabilized and positioned by the arrayed air pillars into regular two-dimensional lattices. We explore the effects of hole shape, lattice symmetry, and surface treatment on the resulting lattices of defects and explain their arrangements by application of topological rules. Last, we show the formation of detailed kaleidoscopic textures after the system is cooled down across the nematic-smectic A phase transition, frustrating the defects and surrounding structures with the equal-layer spacing condition of the smectic phase.

Saturated fully leafed tree-like polyforms and polycubes

We present recursive formulas giving the maximal number of leaves in tree-like polyforms living in two-dimensional regular lattices and in tree-like polycubes in the three-dimensional cubic lattice. We call these tree-like polyforms and polycubes fully leafed. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. In the last part, we concentrate on the particular case of polyforms and polycubes, that we call saturated, which is the family of fully leafed structures that maximize the ratio (number of leaves)/(number of cells). In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4k+1 and the set of tree-like polyominoes of size k. We exhibit a similar bijection between the set of saturated tree-like polycubes of size 41k+28 and a family of polycubes, called 4-trees, of size 3k+2.