Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Bond percolation for homogeneous two-dimensional lattices

Bond percolation is studied for the three homogeneous two-dimensional lattices: square lattice (SL), triangular lattice (TL) and honeycomb lattice (HL). An expanding cell technique is used to obtain percolation thresholds and other relevant information for different cell sizes. We extend the analysis as to include slightly asymmetric cells in addition to the usual symmetric cells to get more points in the scaling analysis. Exact percolation functions are obtained for each size. Then, the percolation threshold is obtained by means of two complementary methods: one based on the well-known renormalization techniques and the other one introduced here which is based upon determining the inflection point of the percolation curves. A comparison of the results obtained by these two methods is performed. The study includes iterations to extrapolate numerical results towards the thermodynamic limit. Critical exponents ν, β and γ are obtained. Values are compared with numerical results and expected theoretical estimations; present results show agreement and even improvement (in the case of γ) with respect to some numeric values available in the literature. Comparison tables are provided.     

Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term,d _n,1 (k) is expressed by Gauss' hypergeometric series with a variablek. Since the ADBP includes the ordinary directed bond percolation as a special case withk=1, our results give another proof for the Baxter-Guttmann's conjecture thatd _n,1(1) is given by the Catalan number, which was recently proved by Bousquet-Mélou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis byDlog Padé approximations suggests that the critical value depends onk, while asymmetry does not change the critical exponent of percolation probability.     

Percolation of polyatomic species on a square lattice

In this paper, the percolation of (a) linear segments of size k and(b) k-mers of different structures and forms deposited on a square lattice have been studied. In the latter case, site and bond percolation have been examined. The analysis of results obtained by using finite size scaling theory is performed in order to test the universality of the problem by determining the numerical values of the critical exponents of the phase transition occurring in the system. It is also determined that the percolation threshold exhibits a exponentially decreasing function when it is plotted as a function of the k-mer size. The characteristic parameters of that function are dependent not only on the form and structure of the k-mers but also on the properties of the lattice where they are deposited.Received: 3 September 2003, Published online: 23 December 2003PACS: 64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions - 68.35.Rh Phase transitions and critical phenomena - 68.35.Fx Diffusion; interface formation     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

Renormalization group approach to the surface and defect critical behavior in the potts model

A simple real-space renormalization group method with two-terminal clusters is used to treat the critical behavior of Potts ferromagnet with free surface and defect plane on the same footing both for square and cubic lattices. For a square lattice, quite different critical behaviors are found for the cases of line defect and free surface. Whenq is larger than three, like the case ofa line type defect in ‘diamond’ hierarchical lattice, the order parameter on a defect line increases discontinuously at the bulk critical point if the defect interaction is sufficiently strong. This behavior, on the contrary, does not occur on the surface of a semi-infinite plane. For a cubic lattice, the phase diagram and renormalization group flow properties are obtained explicitly for bothq=1 (bond percolation) andq=2 (Ising model). In both cases, our calculations whow that the critical behavior on the surface of a semi-infinite system belongs to a different universality class from the critical behavior on the defect plane of a bulk system.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

The percolation perimeter for two-dimensional directed animals

The percolation perimeter to-size ratio of directed animals is investigated. For the square lattice it is found to be very close to 0.75 and the first correction is found on both the triangular and square lattices to be Bethe-like as for normal lattice animals.     

Self-avoiding walks on percolation clusters at criticality and lattice animals

We use the real-space renormalization group method to study the critical behavior of self-avoiding walks (SAWs) on both site percolation clusters at percolation threshold and site lattice animals in a square lattice. The correlation length exponents of SAWs are found to be on the percolation clusters atp _c and _SAW ^LA =0.804 on lattice animals. These results are compared with Kremer's suggestion of modified Flory formula where is the fractal dimension of the fractal object.     

Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

This Letter is focused on the impact of network topology on the site percolation. Specifically, we study how the site percolation threshold depends on the network dimensions (topological d and fractal D), degree of connectivity (quantified by the mean coordination number $Z_{\infty }$), and arrangement of bonds (characterized by the connectivity index Q also called the ramification exponent). Using the Fisher's containment principle, we established exact inequalities between percolation thresholds on fractal networks contained in the square lattice. The values of site percolation thresholds on some fractal lattices were found by numerical simulations. Our findings suggest that the most relevant parameters to describe properly the values of site percolation thresholds on fractal networks contained in square lattice (Sierpiński carpets and Cantor tartans) and based on the square lattice (weighted planar stochastic fractal and Cantor lattices) are the mean coordination number and ramification exponent, but not the fractal dimension. Accordingly, we propose an empirical formula providing a good approximation for the site percolation thresholds on these networks. We also put forward an empirical formula for the site percolation thresholds on d-dimensional simple hypercubic lattices.     

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.     

Disorder driven roughening transitions of elastic manifolds and periodic elastic media

The simultaneous effect of both disorder and crystal-lattice pinning on the equilibrium behavior of oriented elastic objects is studied using scaling arguments and a functional renormalization group technique. Our analysis applies to elastic manifolds, e.g., interfaces, as well as to periodic elastic media, e.g., charge-density waves or flux-line lattices. The competition between both pinning mechanisms leads to a continuous, disorder driven roughening transition between a flat state where the mean relative displacement saturates on large scales and a rough state with diverging relative displacement. The transition can be approached by changing the impurity concentration or, indirectly, by tuning the temperature since the pinning strengths of the random and crystal potential have in general a different temperature dependence. For D dimensional elastic manifolds interacting with either random-field or random-bond disorder a transition exists for 2<D<4, and the critical exponents are obtained to lowest order in . At the transition, the manifolds show a superuniversal logarithmic roughness. Dipolar interactions render lattice effects relevant also in the physical case of D=2. For periodic elastic media, a roughening transition exists only if the ratio p of the periodicities of the medium and the crystal lattice exceeds the critical value . For p<p _c the medium is always flat. Critical exponents are calculated in a double expansion in and and fulfill the scaling relations of random field models. Received 28 August 1998     

Percolation and cluster distribution. II. layers,variable-range interactions,and exciton cluster model

Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 10⁶ sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents and. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.Supported by NSF Grant DMR76-07832.     

Iterated Fully Coordinated Percolation on a Square Lattice

We study, on a square lattice, an extension to fully coordinated percolation which we call iterated fully coordinated percolation. In fully coordinated percolation, sites become occupied if all four of its nearest neighbors are also occupied. Repeating this site selection process again yields the iterated fully coordinated percolation model. Our results show a large enhancement in the size of highly connected regions after each iteration (from ordinary to fully coordinated and then to iterated fully coordinated percolation); enhancements that are much larger than an extension of correlations by an extra lattice constant might suggest. We also study the universality among the three problems by determining the corresponding static and dynamic critical exponents. Specifically, a new method to directly calculate the walk dimension, d _w , using finite size scaling applied to normal mode analysis is used. This method is applicable to any geometry and requires significantly less computation than previously known calculations to determine d _w .     

On the universality of crossing probabilities in two-dimensional percolation

Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratioa/b. In the limits set by the sample size and by the conventions and on the range of the ratioa/b measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.     

Layered Ising models on triangular and honeycomb lattices

We study inhomogeneous Ising models on triangular and honeycomb lattices. The nearest neighbour couplings can have arbitrary strength and sign such that the coupling distribution is translationally invariant in the direction of one lattice axis, i.e. the models have a layered structure. By using a transfer matrix method we derive closed form expressions for the partition functions and free energies. The critical temperatures are calculated. Phase transitions at a finite critical temperature are universally of Ising type. Models with no phase transition may show different behaviour atT=0, which is explicitly shown for fully frustrated models on square, triangular and honeycomb lattices. Finally, generalizations to layered Ising models on more general lattices are discussed.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Similarity of Percolation Thresholds on the HCP and FCC Lattices

Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (p ^hcp _{c, S} =0.199 255 5±0.000 001 0) and bond (p ^hcp _{c, B} =0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of p _c are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.     

二维次近邻渗流模型

本文采用位置空间重整化群和蒙特-卡罗模拟方法,研究了二维次近邻正方格点渗流问题,计算出其临界值p_c和临界指数α,β,γ,δ,ν等。结果表明:二维考虑次近邻的渗流问题与仅考虑最近邻的渗流问题属于不同的普适类。 关键词：     

Cluster Structure of Collapsing Polymers

In order to better understand the geometry of the polymer collapse transition, we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self-avoiding walk. We argue for this new correlated percolation problem that in two dimensions, and possibly also in three dimensions, a percolation transition takes place at a temperature lower than the collapse transition. Hence this novel transition should be governed by exponents unrelated to the -point exponents. This also implies that there is a temperature range in which the polymer has collapsed, but has no long-range cluster structure. We use Monte Carlo to study the distribution of clusters on the simple cubic and Manhattan lattices. On the Manhattan lattice, where the data are most convincing, we find that the percolation transition occurs at _p =1.461(3), while the collapse transition is known to occur exactly at =1.414.... We propose a finite-size scaling form for the cluster distribution and estimate several of the critical exponents. Regardless of the value of _p , this percolation problem sheds new light on polymer collapse.