Percolation properties of the antiferromagnetic Blume-Capel model in the presence of a magnetic field

The problem of order-order and order-disorder transitions in the system described by the 2D antiferromagnetic Blume-Capel model in the presence of a magnetic field is studied by the Wang and Landau flat-histogram simulation method and by the classical Monte Carlo. Anomalous thermodynamic characteristics in low temperatures indicate different type orderings in finite temperatures. The existence of pure antiferromagnetic phases as well as mixed state is shown by detailed phenomenological analysis of the system. The border lines on the phase diagram between various orderings are determined by the complementary microscopic study of the percolation problem for c(2×2) elementary structures of antiferromagnetic ordered phases. This new approach has also shown a full agreement between the percolation threshold for the cluster of mixed phase and the critical temperature of the ordered system.     

Bounds of percolation thresholds in the enhanced binary tree

By studying its subgraphs, it is argued that the lower critical percolation threshold of the enhanced binary tree (EBT) is bounded as p_c1<0.355059, while the upper threshold is bounded both from above and below by 1/2 according to renormalization-group arguments. We also review a correlation analysis in an earlier work, which claimed a significantly higher estimate of p_c2 than 1/2, to show that this analysis in fact gives a consistent result with this bound. Our result confirms that the duality relation between critical thresholds does not hold exactly for the EBT and its dual, possibly due to the lack of transitivity.     

(2 + 1)D exotic Newton-Hooke symmetry, duality and projective phase

A particle system with a (2 + 1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and supercritical phases (describing 2D isotropic ordinary and exotic oscillators) are separated by the critical phase (one-mode oscillator), and are related by a duality transformation. In the flat limit, the system transforms into a free Galilean exotic particle on the noncommutative plane. The wave equations carrying projective representations of the exotic Newton-Hooke symmetry are constructed.     

Duality Relations for Non-Ohmic Composites, with Applications to Behavior near Percolation

Keller, Dykhne, and others have exploited duality to derive exact results for the effective behavior of two-dimensional Ohmic composites. This paper addresses similar issues in the non-Ohmic context. We focus primarily on three different types of nonlinearity: (a) the weakly nonlinear regime; (b) power-law behavior; and (c) dielectric breakdown. We first make the consequences of duality explicit in each setting. Then we draw conclusions concerning the critical exponents and scaling functions of dual pairs of random non-Ohmic composites near a percolation threshold. These results generalize, unify, and simplify relations previously derived for nonlinear resistor networks. We also discuss some self-dual nonlinear composites. Our treatment is elementary and self-contained; however, we also link it with the more abstract mathematical discussions of duality by Jikov and Kozlov.     

SU(5) gut phase transitions via 't Hooft's duality relation

't Hooft's duality relation is used to investigate the possible dynamical symmetry breaking pattern SU(5)→ SU(4) ? U(1) where the gauge fields of SU(4) may be in one of four possible phases: (i) confinement phase, (ii) Higgs phase, (iii) “self-dual phase”, (iv) Coulomb phase. It is found that the duality relation involving the electric and magnetic free energies is satisfied in all these cases.     

Universality of the ratio of the critical amplitudes of the magnetic susceptibility in a two-dimensional ising model with nonmagnetic impurities

The behavior of the magnetic susceptibility of a two-dimensional Ising model with nonmagnetic impurities is investigated numerically. A new method for determining the critical amplitudes and critical temperature is developed. The results of a numerical investigation of the ratio of the critical amplitudes of the magnetic susceptibility are presented. It is shown that the ratio of the critical amplitudes is universal right up to impurity concentrations q ≤ 0.25 (the percolation point of a square lattice is q _c = 0.407254). The behavior of the effective critical exponent γ(q) of the magnetic susceptibility is discussed. Apparently, a transition from Ising-type universal behavior to percolation behavior should occur in a quite narrow concentration range near the percolation point of the lattice.     

Computing parallel/coincident phase D-brane superpotentials and Type-Ⅱ/F-theory duality

In this paper we study the parallel phase and the coincident phase of D-brane systems with the compactification of one closed modulus. D-brane systems with two phases are described by different 4-folds in terms of Type-Ⅱ/F-theory duality, and the phase transitions are related by the blow-up from a 4-fold with singularities to a 4-fold without. In terms of gauge theory, the phase transition corresponds to the enhancement of gauge group U(1)×U(1)→U(2) connecting the Coulomb branch and the Higgs branch. For the sextic and octic with two D-branes,using mirror symmetry and Type-Ⅱ/F theory duality, A-model superpotentials are obtained from the B-model side for the two phases, and the U(1) Ooguri-Vafa invariants for the parallel phase and U(2) Ooguri-Vafa invariants for the coincident phase are extracted from the A-model superpotential. The difference between the invariants of the two phases is evidence of the phase transition between the Coulomb branch and the Higgs branch.     

Localization criteria and bethe lattices

The problem of electron localization in static, cellularly disordered, tight-binding Bethe lattices is considered. Predictions of a localization criterion due to Economou and Cohen are developed for mobility edges and critical percolation concentrations, and compared with known exact results. The status of the criterion is critically discussed and several general limitations are illustrated. It is concluded that in some respects the criterion is coarsely representative for a range of real lattices but largely for fortuitous reasons.     

Long-range connections, quantum magnets and dilute contact processes

In this article, we briefly review the critical behaviour of a long-range percolation model in which any two sites are connected with a probability that falls off algebraically with the distance. The results of this percolation transition are used to describe the quantum phase transitions in a dilute transverse Ising model at the percolation threshold p_c of the long-range connected lattice. In the similar spirit, we propose a new model of a contact process defined on the same long-range diluted lattice and explore the transitions at p_c. The long-range nature of the percolation transition allows us to evaluate some critical exponents exactly in both the above models. Moreover, mean field theory is valid for a wide region of parameter space. In either case, the strength of Griffiths McCoy singularities are tunable as the range parameter is varied.     

Percolation of polyatomic species on a simple cubic lattice

In the present paper, the site-percolation problem corresponding to linear k-mers (containing k identical units, each one occupying a lattice site) on a simple cubic lattice has been studied. The k-mers were irreversibly and isotropically deposited into the lattice. Then, the percolation threshold and critical exponents were obtained by numerical simulations and finite-size scaling theory. The results, obtained for k ranging from 1 to 100, revealed that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the k-mer size; and (ii) the phase transition occurring in the system belongs to the standard 3D percolation universality class regardless of the value of k considered.     

Percolation analysis of stochastic models of galactic evolution

The stochastic star formation model of galactic evolution can be cast as a problem of directed percolation, the time dimension being that along which the directed bonds exist. We study various aspects of this percolation, those of general interest for the percolation phase transition and those of particular importance for the astrophysical application. Both analytical calculations and computer simulations are provided and the results compared. Among the properties are: value of the percolation threshold, critical indices, percolation probability (star density) near and away from the critical point, local density, cluster sizes, effects of rotation (for disk galaxy models) on the percolation threshold. Astrophysical consequences of some of these properties are discussed, in particular the way in which general phase transition behavior contributes to spiral arm morphology. We look at 1 (space) + 1 (time), 2 + 1 and + 1 dimensions, the 2 + 1 case being of interest for disk galaxies.     

The nature of percolation ‘channels’

A relation is found between the percolation probability, P^(s)(x), for site percolation, the conductance, G(x), of certain random networks, and the spin-stiffness coefficient, D(x), of dilute Heisenberg ferromagnet. Numerical results and critical exponents for these quantities near their common threshold are reported. These results demonstrate that the percolation ‘channels’ in which spin waves occur cannot be regarded as one or two dimensional.     

New critical frontiers for the potts and percolation models

We obtain the critical threshold for a host of Potts and percolation models on lattices having a structure which permits a duality consideration. The consideration generalizes the recently obtained thresholds of Scullard and Ziff for bond and site percolation on the martini and related lattices to the Potts model and to other lattices.     

Dynamic transitions in Domany-Kinzel cellular automata on small-world network

We study Domany-Kinzel cellular automata on small-world network. Every link on a one dimensional chain is rewired and coupled with any node with probability p. We observe that, the introduction of long-range interactions does not remove the critical character of the model and the system still exhibits a well-defined phase transition to absorbing state. In case of directed percolation (DP), we observe a very anomalous behavior as a function of size. The system shows long lived metastable states and a jump in order parameter. This jump vanishes in thermodynamic limit and we recover second-order transition. The critical exponents are not equal to the mean-field values even for large p. However, for compact directed percolation(CDP), the critical exponents reach their mean-field values even for small p.     

Particle nature of light waves in dielectric media

Wave-particle duality is a foundation for modern science. The speed of light waves in dielectric media is less than c. The corresponding particles thus have mass. Combining wave-particle duality with the theory of relativity, an exactly solvable problem was proposed, concerning the transition from photons in vacuum to particles in dielectric media. The rest mass, the momentum, and the total energy of material particles are shown to be the functions of the refractive index of the medium and the wavelength of the incident light. The proposed relationships were applied to study the wavelength-dependent index of refraction of dielectrics and the correlation of the refractive indices of anisotropic crystals, which were confirmed by the experimental results. Variation of the refractive index with wavelength is found to obey the proposed relation. The refractive indices of anisotropic crystals are shown to be the correlated quantities.     

Explosive percolation transition is actually continuous

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.     

Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice ?? by ? bonds connecting the same adjacent vertices, thereby yielding the lattice ?? _? . This relation is used to calculate the bond percolation threshold on ?? _? . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d??2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N???? limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N????.     

Percolation Thresholds in Ternary Polymer Melt Blends with Separate Dispersions of the Inner Phases: Mathematical Model

A mathematical model based on a straightforward geometrical background is developed which enables predictions of a transition of one dispersed phase to a cocontinuous one (i.e., the percolation threshold) on addition of another dispersed phase during melt mixing in ternary polymer blends. The present work concerns only ternary blends with two separate dispersions of the inner phases in which no encapsulation takes place. In addition, in order to simplify the model, one of the inner phases was represented by hard, nondeformable microspheres The expression developed describes well an experimental relationship between the percolation threshold, the concentration above which the former dispersed phase transforms to a continuous one, and concentrations of both inner phases. The results agree well with the experimental data obtained in a previous work.     

Percolation systems away from the critical point

This article reviews some effects of disorder in percolation systems away from the critical density p _c. For densities below p _c, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee-Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singularities in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biased diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field.     

Duality and the θ parameter in Abelian lattice models

The phase diagrams of abelian lattice gauge theories in four dimensions, and spin models in two dimensions, in the presence of a θ-parameter, are constructed using duality arguments. A rich structure involving an infinite number of phases emerges, together with a mechanism for spontaneous CP violation at θ = π.