Site percolation thresholds for Archimedean lattices

Influence of Li Content on the Topological Inhibition of Oxygen Loss in Li-Rich Cathode Materials

Lithium-rich layer oxide cathodes are promising energy storage materials due to their high energy densities. However, the oxygen loss during cycling limits their practical applications. Here, the essential role of Li content on the topological inhibition of oxygen loss in lithium-rich cathode materials and the relationship between the migration network of oxygen ions and the transition metal (TM) component are revealed. Utilizing first-principles calculations in combination with percolation theory and Monte Carlo simulations, it is found that TM ions can effectively encage the oxidized oxygen species when the TM concentration in TM layer exceeds 5/6, which hinders the formation of a percolating oxygen migration network. This study demonstrates the significance of rational compositional design in lithium-rich cathodes for effectively suppressing irreversible oxygen release and enhancing cathode cycling performance.

Universality of random-site percolation thresholds for two-dimensional complex noncompact neighborhoods

The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds p_{c} for random-site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence p_{c}∝(ζ/K)^{-γ} with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and p_{c}∝ζ^{-γ} with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑_{i}z_{i}r_{i} stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number z_{i}, and the distance r_{i} to sites in the ith coordination zone from the central site. The number K indicates lattice connectivity, that is, K=3, 4, and 6 for the honeycomb, square, and triangular lattice, respectively.

Engineering SYK Interactions in Disordered Graphene Flakes under Realistic Experimental Conditions

We model interactions following the Sachdev-Ye-Kitaev (SYK) framework in disordered graphene flakes up to 300 000 atoms in size (∼100  nm in diameter) subjected to an out-of-plane magnetic field B of 5-20 Tesla within the tight-binding formalism. We investigate two sources of disorder: (i) irregularities at the system boundaries, and (ii) bulk vacancies-for a combination of which we find conditions that could be favorable for the formation of the phase with Sachdev-Ye-Kitaev features under realistic experimental conditions above the liquid helium temperature.

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.

Supramolecular Tiling of a Conformationally Flexible Precursor

Supramolecular self-assembly offers a possible pathway for nanopatterning and functionality. In particular, molecular tiling such as trihexagonal tiling (also known as the Kagome lattice) has promising chemical and physical properties. Distorted Kagome lattices are not well understood due to their complexity, and studies of their controllable fabrication are few. Here, by employing a conformationally flexible precursor, 2,4,6-tris(3-bromophenyl)-1,3,5-triazine (mTBPT), we demonstrate two-dimensional distorted Kagome lattice p3, (333) by supramolecular self-assembly and achieve tuning of the metastable phases, including the homochiral porous network and distorted Kagome lattice p3, (333) by steering deposition rates on a cold Ag(111) substrate. By a combination of scanning tunneling microscopy and density functional theory calculations, the distorted Kagome lattice is energetically unfavorable but can be trapped at a high deposition rate, and the process mainly depends on surface kinetics. This work using conformationally flexible mTBPT molecules provides a pathway for the controllable growth of different phases, including metastable Kagome lattices.

Percolation thresholds of randomly rotating patchy particles on Archimedean lattices

We study the percolation of randomly rotating patchy particles on 11 Archimedean lattices in two dimensions. Each vertex of the lattice is occupied by a particle, and in each model the patch size and number are monodisperse. When there are more than one patches on the surface of a particle, they are symmetrically decorated. As the proportion χ of the particle surface covered by the patches increases, the clusters connected by the patches grow and the system percolates at the threshold χ_{c}. We combine Monte Carlo simulations and the critical polynomial method to give precise estimates of χ_{c} for disks with one to six patches and spheres with one to two patches on the 11 lattices. For one-patch particles, we find that the order of χ_{c} values for particles on different lattices is the same as that of threshold values p_{c} for site percolation on these lattices, which implies that χ_{c} for one-patch particles mainly depends on the geometry of lattices. For particles with more patches, symmetry becomes very important in determining χ_{c}. With the estimates of χ_{c} for disks with one to six patches, using analyses related to symmetry, we are able to give precise values of χ_{c} for disks with an arbitrary number of patches on all 11 lattices. The following rules are found for patchy disks on each of these lattices: (1) as the number of patches n increases, values of χ_{c} repeat in a periodic way, with the period n_{0} determined by the symmetry of the lattice; (2) when mod(n,n_{0})=0, the minimum threshold value χ_{min} appears, and the model is equivalent to site percolation with χ_{min}=p_{c}; and (3) disks with mod(n,n_{0})=m and n_{0}-m (m<n_{0}/2) share the same χ_{c} value. The results can be useful references for studying the connectivity of patchy particles on two-dimensional lattices at finite temperatures.

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.

Supramolecular tessellations by the exo-wall interactions of pagoda[4]arene

Supramolecular tessellation has gained increasing interest in supramolecular chemistry for its structural aesthetics and potential applications in optics, magnetics and catalysis. In this work, a new kind of supramolecular tessellations (STs) have been fabricated by the exo-wall interactions of pagoda[4]arene (P4). ST with rhombic tiling pattern was first constructed by P4 itself through favorable π···π interactions between anthracene units of adjacent P4. Notably, various highly ordered STs with different tiling patterns have been fabricated based on exo-wall charge transfer interactions between electron-rich P4 and electron-deficient guests including 1,4-dinitrobenzene, terephthalonitrile and tetrafluoroterephthalonitrile. Interestingly, solvent modulation and guest selection played a crucial role in controlling the molecular arrangements in the co-crystal superstructures. This work not only proves that P4 is an excellent macrocyclic building block for the fabrication of various STs, but also provides a new perspective and opportunity for the design and construction of supramolecular two-dimensional organic materials.

Supramolecular tessellation has gained increasing interest in supramolecular chemistry for its structural aesthetics and potential applications in optics, magnetics and catalysis. Here, the authors expand the examples of molecular building blocks for supramolecular tessellation and fabricate supramolecular tessellations using the exo-wall interactions of pagoda[4]arene.

Percolation Theory Reveals Biophysical Properties of Virus-like Particles

The viral protein containers that encapsulate a virus' genetic material are repurposed as virus-like particles in a host of nanotechnology applications, including cargo delivery, storage, catalysis, and vaccination. These viral architectures have evolved to sit on the knife's edge between stability, to provide adequate protection for their genetic cargoes, and instability, to enable their efficient and timely release in the host cell environment upon environmental cues. By introducing a percolation theory for viral capsids, we demonstrate that the geometric characteristics of a viral capsid in terms of its subunit layout and intersubunit interaction network are key for its disassembly behavior. A comparative analysis of all alternative homogeneously tiled capsid structures of the same stoichiometry identifies evolutionary drivers favoring specific viral geometries in nature and offers a guide for virus-like particle design in nanotechnology.

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.

Critical polynomials in the nonplanar and continuum percolation models

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

How a supercooled liquid borrows structure from the crystal

Using computer simulations, we establish that the structure of a supercooled binary atomic liquid mixture consists of common neighbor structures similar to those found in the equilibrium crystal phase, a Laves structure. Despite the large accumulation of the crystal-like structure, we establish that the supercooled liquid represents a true metastable liquid and that liquid can "borrow" the crystal structure without being destabilized. We consider whether this feature might be the origin of all instances of liquids with a strongly favored local structure.

Supramolecular Tessellations via Pillar[n]arenes-Based Exo-Wall Interactions

Supramolecular tessellation is a newly emerging and promising area in supramolecular chemistry because of its unique structural aesthetics and potential applications. Herein, we investigate the "exo-wall" interactions of pillar[n]arenes and prepare fantastic hexagonal supramolecular tessellations based on perethylated pillar[6]arenes (EtP6) with electron-deficient molecules 1,5-difluoro-2,4-dinitrobenzene (DFN) and tetrafluoro-1,4-benzoquinone (TFB). The crystal structures clearly confirm that EtP6 can form highly ordered hexagonal 2D tiling patterns with DFN/TFB as linkers through cocrystallization. Moreover, the self-assembled packing arrangements in the ultimate cocrystal superstructures can be adjusted under different crystallization conditions. This work not only explores the rare exo-wall interactions based on pillar[n]arenes but also reports the fabrication of supramolecular tessellations based on pillararenes for the first time, showing a new perspective in supramolecular chemistry.

Long paths and connectivity in 1-independent random graphs

A probability measure μ on the subsets of the edge set of a graph G is a 1-independent probability measure (1-ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1-ipm μ , denote byG μ the associated random graph model. Letℳ 1 , ⩾ p ( G ) denote the collection of 1-ipms μ on G for which each edge is included inG μ with probability at least p. For G = Z 2 , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all μ ∈ ℳ 1 , ⩾ p ( G ) ,G μ almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p ⋆. We also determine the 1-independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f 1, G (p), the infimum over all μ ∈ ℳ 1 , ⩾ p ( G ) of the probability thatG μ is connected. We determine f 1, G (p) exactly when G is a path, a complete graph and a cycle of length at most 5.

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}.

Deactivation Kinetics of Solid Acid Catalyst with Laterally Interacting Protons

Differences in catalyst deactivation kinetics in solid acid catalysis are studied with catalyst models that allow for lateral interaction between protons. Deactivation of a solid acid catalyst with laterally interacting protons induces inhomogeneity of proton reactivity that develops with time. As a consequence, product selectivity changes and deactivation will accelerate. This is demonstrated by simulations of the deactivation kinetics of the alkylation reaction of propylene with isobutane. The effect of lateral interactions between protons arises because initial catalyst deactivation is not caused by pore blocking or coke deposition but by a molecular mechanism where protons are consumed due to the formation of stable nonreactive carbenium ions. High selectivity to alkylate requires a catalyst with protons of high reactivity. When protons become consumed by formation of stable deactivating carbenium ions, initially reactive protons are converted into protons of lower reactivity. The latter only catalyze deactivating oligomerization reactions. Simulations that compare the deactivation kinetics of a catalyst model with laterally interacting protons and a catalyst model that contains protons of similar but different reactivity, but that do not laterally interact, are compared. These simulations demonstrate that the lateral interaction catalyst model is initially more selective but also has a lower stability. Catalyst deactivation of the alkylation reaction occurs through two reaction channels. One reaction channel is due to oligomerization of reactant propylene. The other deactivation reaction channel is initiated by deprotonation of intermediate carbenium ions that increase alkene concentration. By consecutive reactions, this also leads to deactivation. The hydride transfer reaction competes with oligomerization reactions. It is favored by strongly acid sites that also suppress the deprotonation reaction. Within the laterally interacting proton catalyst model, when reactive protons become deactivated, weakly reactive protons are generated that only catalyze the deactivating alkene oligomerization and consecutive reactions. This rapid formation of the weakly reactive protons is the cause of decreasing selectivity with reaction time and increased rate of deactivation. Solutions of the mean field kinetic equations as well as stochastic simulations are presented. Comparative simulations with a reduced number of neighbors of the protons illustrate decreased deactivation rates when the proton density decreases. Island formation of adsorbed reaction intermediates on the catalyst surface is observed in stochastic kinetics simulations. When alkylation selectivity is high, this island formation increases the rate of catalyst deactivation in comparison to the rate of deactivation according to the mean field studies. A nonlinear dynamics model of proton dynamics is provided, which shows that the differences between stochastic and mean field simulations are due to frustrated proton state percolation.

Fire forbids fifty-fifty forest

Recent studies have interpreted patterns of remotely sensed tree cover as evidence that forest with intermediate tree cover might be unstable in the tropics, as it will tip into either a closed forest or a more open savanna state. Here we show that across all continents the frequency of wildfires rises sharply as tree cover falls below ~40%. Using a simple empirical model, we hypothesize that the steepness of this pattern causes intermediate tree cover (30‒60%) to be unstable for a broad range of assumptions on tree growth and fire-driven mortality. We show that across all continents, observed frequency distributions of tropical tree cover are consistent with this hypothesis. We argue that percolation of fire through an open landscape may explain the remarkably universal rise of fire frequency around a critical tree cover, but we show that simple percolation models cannot predict the actual threshold quantitatively. The fire-driven instability of intermediate states implies that tree cover will not change smoothly with climate or other stressors and shifts between closed forest and a state of low tree cover will likely tend to be relatively sharp and difficult to reverse.

Construction of molecular regular tessellations on a Cu(111) surface

Through thermal treatment, three regular molecular tessellations are constructed on Cu(111) with a linear DOD precursor.

Honeycomb lattices with defects

In this paper, we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using an alternative computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.

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.

Site trimer percolation on square lattices

Percolation of site trimers (k-mers with k=3) is investigated in a detailed way making use of an analytical model based on renormalization techniques in this problem. Results are compared to those obtained here by means of extensive computer simulations. Five different deposition possibilities for site trimers are included according to shape and orientation of the depositing objects. Analytical results for the percolation threshold p(c) are all close to 0.55, while numerical results show a slight dispersion around this value. A comparison with p(c) values previously reported for monomers and dimers establishes the tendency of p(c) to decrease as k increases. Critical exponent ν was also obtained both by analytical and numerical methods. Results for the latter give values very close to the expected value 4/3 showing that this percolation case corresponds to the universality class of random percolation.

Ising antiferromagnet on the Archimedean lattices

Geometric frustration effects were studied systematically with the Ising antiferromagnet on the 11 Archimedean lattices using the Monte Carlo methods. The Wang-Landau algorithm for static properties (specific heat and residual entropy) and the Metropolis algorithm for a freezing order parameter were adopted. The exact residual entropy was also found. Based on the degree of frustration and dynamic properties, ground states of them were determined. The Shastry-Sutherland lattice and the trellis lattice are weakly frustrated and have two- and one-dimensional long-range-ordered ground states, respectively. The bounce, maple-leaf, and star lattices have the spin ice phase. The spin liquid phase appears in the triangular and kagome lattices.

A comprehensive review on the molecular dynamics simulation of the novel thermal properties of graphene

This review summarizes state-of-the-art progress in the molecular dynamics simulation of the novel thermal properties of graphene.

Distribution of entanglement in large-scale quantum networks

The concentration and distribution of quantum entanglement is an essential ingredient in emerging quantum information technologies. Much theoretical and experimental effort has been expended in understanding how to distribute entanglement in one-dimensional networks. However, as experimental techniques in quantum communication develop, protocols for multi-dimensional systems become essential. Here, we focus on recent theoretical developments in protocols for distributing entanglement in regular and complex networks, with particular attention to percolation theory and network-based error correction.

Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly

The tessellation of the Euclidean plane by regular polygons has been contemplated since ancient times and presents intriguing aspects embracing mathematics, art, and crystallography. Significant efforts were devoted to engineer specific 2D interfacial tessellations at the molecular level, but periodic patterns with distinct five-vertex motifs remained elusive. Here, we report a direct scanning tunneling microscopy investigation on the cerium-directed assembly of linear polyphenyl molecular linkers with terminal carbonitrile groups on a smooth Ag(111) noble-metal surface. We demonstrate the spontaneous formation of fivefold Ce-ligand coordination motifs, which are planar and flexible, such that vertices connecting simultaneously trigonal and square polygons can be expressed. By tuning the concentration and the stoichiometric ratio of rare-earth metal centers to ligands, a hierarchic assembly with dodecameric units and a surface-confined metal-organic coordination network yielding the semiregular Archimedean snub square tiling could be fabricated.

Percolation thresholds on planar Euclidean relative-neighborhood graphs

In the present article, statistical properties regarding the topology and standard percolation on relative neighborhood graphs (RNGs) for planar sets of points, considering the Euclidean metric, are put under scrutiny. RNGs belong to the family of "proximity graphs"; i.e., their edgeset encodes proximity information regarding the close neighbors for the terminal nodes of a given edge. Therefore they are, e.g., discussed in the context of the construction of backbones for wireless ad hoc networks that guarantee connectedness of all underlying nodes. Here, by means of numerical simulations, we determine the asymptotic degree and diameter of RNGs and we estimate their bond and site percolation thresholds, which were previously conjectured to be nontrivial. We compare the results to regular 2D graphs for which the degree is close to that of the RNG. Finally, we deduce the common percolation critical exponents from the RNG data to verify that the associated universality class is that of standard 2D percolation.

Universality of the Ising and the S=1 model on Archimedean lattices: a Monte Carlo determination

The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.

Numerical evidence against a conjecture on the cover time of planar graphs

We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time τ(G(N)) of a planar graph G(N) of N vertices and maximal degree d is lower bounded by τ(G(N))≥C(d)N(lnN)2 with C(d)=(d/4π)tan(π/d), with equality holding for some geometries. We tested this conjecture on the regular honeycomb (d=3), regular square (d=4), regular elongated triangular (d=5), and regular triangular (d=6) lattices, as well as on the nonregular Union Jack lattice (dmin=4, dmax=8). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violate the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.

Spangolite: an s = 1/2 maple leaf lattice antiferromagnet?

Spangolite, Cu(6)Al(SO(4))(OH)(12)Cl·3H(2)O, is a hydrated layered copper sulfate mineral. The Cu(2+) ions of each layer form a systematically depleted triangular lattice which approximates a maple leaf lattice. We present details of the crystal structure, which suggest that in spangolite this lattice actually comprises two species of edge linked trimers with different exchange parameters. However, magnetic susceptibility measurements show that despite the structural trimers, the magnetic properties are dominated by dimerization. The high temperature magnetic moment is strongly reduced below that expected for the six s = 1/2 in the unit cell.

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis

In the preceding paper, one of us (F. Y. Wu) considered the Potts model and bond and site percolation on two general classes of two-dimensional lattices, the triangular-type and kagome-type lattices, and obtained closed-form expressions for the critical frontier with applications to various lattice models. For the triangular-type lattices Wu's result is exact, and for the kagome-type lattices Wu's expression is under a homogeneity assumption. The purpose of the present paper is twofold: First, an essential step in Wu's analysis is the derivation of lattice-dependent constants A,B,C for various lattice models, a process which can be tedious. We present here a derivation of these constants for subnet networks using a computer algorithm. Second, by means of a finite-size scaling analysis based on numerical transfer matrix calculations, we deduce critical properties and critical thresholds of various models and assess the accuracy of the homogeneity assumption. Specifically, we analyze the q -state Potts model and the bond percolation on the 3-12 and kagome-type subnet lattices (n×n):(n×n) , n≤4 , for which the exact solution is not known. Our numerical determination of critical properties such as conformal anomaly and magnetic correlation length verifies that the universality principle holds. To calibrate the accuracy of the finite-size procedure, we apply the same numerical analysis to models for which the exact critical frontiers are known. The comparison of numerical and exact results shows that our numerical values are correct within errors of our finite-size analysis, which correspond to 7 or 8 significant digits. This in turn infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher. Finally, we also obtained the exact percolation thresholds for site percolation on kagome-type subnet lattices (1×1):(n×n) for 1≤n≤6 .

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions

We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is rigorous and based on an established result without the need of a uniqueness assumption, thus firmly establishing all derived results. For kagome-type lattices the exact critical frontier is not known. We derive a closed-form expression for the Potts critical frontier by making use of a homogeneity assumption. The closed-form expression is unique, and we apply it to a host of problems including site, bond, and mixed site-bond percolations on various lattices. It yields exact thresholds for site percolation on kagome, martini, and other lattices and is highly accurate numerically in other applications when compared to numerical determination.

Mechanical properties of methyl functionalized graphene: a molecular dynamics study

Molecular dynamics simulations have been performed to study the mechanical properties of methyl (CH(3)) functionalized graphene. It is found that the mechanical properties of functionalized graphene greatly depend on the location, distribution and coverage of CH(3) radicals on graphene. Surface functionalization exhibits a much stronger influence on the mechanical properties than edge functionalization. For patterned functionalization on graphene surfaces, the radicals arranged in lines perpendicular to the tensile direction lead to larger strength deterioration than those parallel to the tensile direction. For random functionalization, the elastic modulus of graphene decreases gradually with increasing CH(3) coverage, while both the strength and fracture strain show a sharp drop at low coverage. When CH(3) coverage reaches saturation, the elastic modulus, strength and fracture strain of graphene drop by as much as 18%, 43% and 47%, respectively.

Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations

The site percolation threshold for the random Voronoi network is determined numerically, with the result pc=0.714 10+/-0.000,02 , using Monte Carlo simulation on periodic systems of up to 40,000 sites. The result is very close to the recent theoretical estimate pc approximately 0.7151 of Neher For the bond threshold on the Voronoi network, we find pc=0.666, 931+/-0.000,005 implying that, for its dual, the Delaunay triangulation pc=0.333 069+/-0.000 005 . These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3, respectively, but support the conjecture of Wierman that, for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin pi/18 approximately 0.3473 .

Early-stage waves in the retinal network emerge close to a critical state transition between local and global functional connectivity

A novel, biophysically realistic model for early-stage, acetylcholine-mediated retinal waves is presented. In this model, neural excitability is regulated through a slow after-hyperpolarization (sAHP) operating on two different temporal scales. As a result, the simulated network exhibits competition between a desynchronizing effect of spontaneous, cell-intrinsic bursts, and the synchronizing effect of synaptic transmission during retinal waves. Cell-intrinsic bursts decouple the retinal network through activation of the sAHP current, and we show that the network is capable of operating at a transition point between purely local and global functional connectedness, which corresponds to a percolation phase transition. Multielectrode array recordings show that, at this point, the properties of retinal waves are reliably predicted by the model. These results indicate that early spontaneous activity in the developing retina is regulated according to a very specific principle, which maximizes randomness and variability in the resulting activity patterns.

Percolation transitions in two dimensions

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

Site percolation on planar Phi(3) random graphs

In this paper, site percolation on random Phi(3) planar graphs is studied by Monte Carlo numerical techniques. The method consists in randomly removing a fraction q = 1-p of vertices from graphs generated by Monte Carlo simulations, where p is the occupation probability. The resulting graphs are made of clusters of occupied sites. By measuring several properties of their distribution, it is shown that percolation occurs for an occupation probability above a percolation threshold p(c) = 0.7360(5) . Moreover, critical exponents are compatible with those analytically known for bond percolation.

Correlation-space description of the percolation transition in composite microstructures

We explore the percolation threshold shift as short-range correlations are introduced and systematically varied in binary composites. Two complementary representations of the correlations are developed in terms of the distribution of phase bonds or, alternatively, using a set of appropriate short-range order parameters. In either case, systematic exploration of the correlation space reveals a boundary that separates percolating from nonpercolating structures and permits empirical equations that identify the location of the threshold for systems of arbitrary short-range correlation states. Two- and three-dimensional site lattices with two-body correlations, as well as a two-dimensional hexagonal bond network with three-body correlations, are explored. The approach presented here should be generalizable to more complex correlation states, including higher-order and longer-range correlations.

Rigorous confidence intervals for critical probabilities

We use the method of Balister, Bollobás, and Walters [Random Struct. Algorithms 26, 392 (2005)] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

Predictions of bond percolation thresholds for the kagomé and Archimedean (3, 12(2)) lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3, 12(2)) lattices. We present two different methods: one which provides an approximation to the inhomogeneous kagomé and bond problems, and the other which gives estimates of for the homogeneous kagomé (0.524 408 8...) and (3, 12(2)) (0.740 421 2...) problems that, respectively, agree with numerical results to five and six significant figures.

Diffusion on Archimedean lattices

We consider random diffusive motion of classical particles over the edges of Archimedean lattices. The diffusion coefficient is obtained by using periodic orbit theory. We also study deterministic motion over a honeycomb lattice without the possibility for an immediate return to the preceding node, controlled by a tent map with the golden ratio slope. Numerical analysis is performed to confirm the theoretical results.

Generalized cell-dual-cell transformation and exact thresholds for percolation

Suggested by Scullard's recent star-triangle relation for correlated bond systems, we propose a general "cell-dual-cell" transformation, which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated. We directly verify Scullard's site percolation thresholds, and derive the bond thresholds for his "martini" lattice (pc = 1/square root 2) and the "A" lattice (pc = 0.625457..., solution to p5 - 4p4 + 3p3 + 2p2 - 1 = 0). We also present a precise Monte Carlo test of the site threshold for the "A" lattice.

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation

I construct a two-dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined. The primary lattice studied here, the "martini lattice," is a hexagonal lattice with every second site transformed into a triangle. The site threshold of this lattice is found to be 0.764826..., i.e., the solution to p4 - 3p3 + 1 = 0, while the others have (square root 5 - 1)/2 (the inverse of the golden ratio) and 1/square root 2. This last solution suggests a possible approach to establishing the bound for the hexagonal site threshold, pc < 1/square root 2. To derive these results, I solve a correlated bond problem on the hexagonal lattice by use of the star-triangle transformation and then, by a particular choice of correlations derived from a site-to-bond transformation, solve the site problem on the martini lattice.

Improved site percolation threshold universal formula for two-dimensional matching lattices

A universal formula is proposed for predicting the site percolation threshold of two-dimensional matching lattices. The formula is slightly more accurate for these lattices than the formulas of Galam and Mauger, based on a comparison over a class of 38 lattices, and does not require two universality classes for two-dimensional lattices. The formula is constructed from the Galam-Mauger square root formula for site thresholds, by a modification which makes it consistent with the theoretical relationship between percolation thresholds of matching pairs of lattices. In the framework for evaluation of universal formulas introduced by Wierman and Naor, the formula is currently the best performing universal formula for site thresholds on this class of lattices.

Monte Carlo study of the site-percolation model in two and three dimensions

We investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size L(d-1) x infinity, where L specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at p(c) =0.592 746 5 (4) , and determine two universal quantities as Q(gbc) =0.930 34 (1) and Q(gsc) =0.793 72 (3) , which are associated with bulk and surface correlations, respectively. These values agree well with the exact values 2(-5/48) and 2(-1/3) , respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at p(c) =0.311 607 7 (4) . We further determine the bulk thermal and magnetic exponents as y(t) =1.1437 (6) and y(h) =2.5219 (2) , respectively, and the surface magnetic exponent at the ordinary phase transition as y (o)(hs) =1.0248 (3) .

Criteria for evaluation of universal formulas for percolation thresholds

Several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature. The existing universal formulas have been found to have substantial errors in their predictions for some lattices. This paper proposes a set of desirable criteria for universal formulas to satisfy, and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature. The analysis is limited to lattices in two dimensions.

Square-lattice site percolation at increasing ranges of neighbor bonds

We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same pc = 0.592... . The fourth-nearest neighbors (5N) give pc = 0.298... . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN), (NN+4N), (NN+NNN+4N), and (NN+5N). The calculated associated thresholds are respectively pc = 0.407..., 0.337..., 0.288..., and 0.234... . The existing Galam-Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.