Continuum Nonsimple Loops and 2D Critical Percolation

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE ₆(the Stochastic Loewner Evolution with parameter κ=6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation “exploration process.” In this paper we use that and other results to construct what we argue is the fullscaling limit of the collection of allclosed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in Bbb R²is constructed inductively by repeated use of chordal SLE ₆. These loops do not cross but do touch each other—indeed, any two loops are connected by a finite “path” of touching loops.     

Percolation transition in an irreversible-surface reaction model

Monte Carlo simulation studies of percolation transition in a surface reaction model describing the oxidation of carbon mono-oxide on a catalytic surface are presented. The percolation transition for adsorbed oxygen atoms occurs below the poisoning transition where carbon mono-oxide completely covers the surface of the catalyst and takes place for an oxygen coverage of about 0.525 which is close to the percolation transition in an Ising lattice gas with nearest-neighbor attractive interactions. In several respects the oxygen clusters near the percolation threshold resemble those of the Ising lattice gas near its critical point.     

The influence of inhomogeneous properties of a system on the percolation process in two-dimensional space

The percolation process in a two-dimensional inhomogeneous lattice is studied by the Monte Carlo method. The inhomogeneous lattice is simulated by a random distribution of inhomogeneities differing in size and number. The influence of inhomogeneities on the parameters (critical concentration, average number of sites in finite clusters, percolation probability, critical exponents, and fractal dimension of an infinite cluster) characterizing the percolation in the system is analyzed. It is demonstrated that all these parameters essentially depend on the linear size of inhomogeneities and their relative area.     

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     

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.     

Critical exponents for one-dimensional percolation clusters

For d=1, percolation clusters follow a scaling law with critical exponents σ=1 and τ=2. For the limit d→1, critical exponents can differ from their d=1 values, a complication which can already be studied in the simple Bethe lattice solution for cluster numbers.     

Killing the Hofstadter butterfly,one bond at a time

Electronic bands in a square lattice when subjected to a perpendicular magnetic field form the Hofstadter butterfly pattern. We study the evolution of this pattern as a function of bond percolation disorder (removal or dilution of lattice bonds). With increasing concentration of the bonds removed, the butterfly pattern gets smoothly decimated. However, in this process of decimation, bands develop interesting characteristics and features. For example, in the high disorder limit, some butterfly-like pattern still persists even as most of the states are localized. We also analyze, in the low disorder limit, the effect of percolation on wavefunctions (using inverse participation ratios) and on band gaps in the spectrum. We explain and provide the reasons behind many of the key features in our results by analyzing small clusters and finite size rings. Furthermore, we study the effect of bond dilution on transverse conductivity (σ _xy ). We show that starting from the clean limit, increasing disorder reduces σ _xy to zero, even though the strength of percolation is smaller than the classical percolation threshold. This shows that the system undergoes a direct transition from a integer quantum Hall state to a localized Anderson insulator beyond a critical value of bond dilution. We further find that the energy bands close to the band edge are more stable to disorder than at the band center. To arrive at these results we use the coupling matrix approach to calculate Chern numbers for disordered systems. We point out the relevance of these results to signatures in magneto-oscillations.     

Breakdown of universality in directed spiral percolation

Directed spiral percolation (DSP), percolation under both directional and rotational constraints, is studied on the triangular lattice in two dimensions (2D). The results are compared with that of the 2D square lattice. Clusters generated in this model are generally rarefied and have chiral dangling ends on both the square and triangular lattices. It is found that the clusters are more compact and less anisotropic on the triangular lattice than on the square lattice. The elongation of the clusters is in a different direction than the imposed directional constraint on both the lattices. The values of some of the critical exponents and fractal dimension are found considerably different on the two lattices. The DSP model then exhibits a breakdown of universality in 2D between the square and triangular lattices. The values of the critical exponents obtained for the triangular lattice are not only different from that of the square lattice but also different form other percolation models.Received: 12 March 2004, Published online: 23 July 2004PACS: 02.50.-r Probability theory, stochastic processes, and statistics - 64.60.-i General studies of phase transitions - 72.80.Tm Composite materials     

The affection on the nature of percolation by concentration of pentagon–heptagon defects in graphene lattice

In this paper the percolation behavior with a specific concentration of the defects was discussed on the twodimensional graphene lattice. The percolation threshold is determined by a numerical method with a high degree of accuracy. This method is also suitable for locating the percolation critical point on other crystalline structures. Through investigating the evolution of the largest cluster size and the cluster sizes distribution, we find that under various lattice sizes and concentrations of pentagon-heptagon defects there is no apparent change for the percolation properties in graphene lattice.     

Viscosity critical behaviour at the gel point in a 3d lattice model

Within a recently introduced model based on the bond-fluctuation dynamics, we study the viscoelastic behaviour of a polymer solution at the gelation threshold. We here present the results of the numerical simulation of the model on a cubic lattice: the percolation transition, the diffusion properties and the time autocorrelation functions have been studied. From both the diffusion coefficients and the relaxation times critical behaviour a critical exponent k for the viscosity coefficient has been extracted: the two results are comparable within the errors giving , in close agreement with the Rouse model prediction and with some experimental results. In the critical region below the transition threshold the time autocorrelation functions show a long-time tail which is well fitted by a stretched exponential decay. Received 20 December 1999 and Received in final form 18 February 2000     

Conductivity exponent in three-dimensional percolation by diffusion based on molecular trajectory algorithm and blind-ant rules

Diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method, respectively. The classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods. Our Monte Carlo simulations by the simple lattice random walk method give the conductivity exponent μ/ν=2.32±0.02 for diffusion on the incipient infinite clusters and μ/ν=2.21±0.03 for diffusion on a percolating lattice above the threshold. However, while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold, the result is found to be μ/ν=2.26±0.02. In addition, it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk.     

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

     

There is more to be learned from the Lorentz model

The classical Lorentz model for charged noninteracting point particles in a perpendicular magnetic field is reconsidered in 2D. We show that the standard Boltzmann equation is not valid for this model, even in the Grad limit. We construct a generalized Boltzmann equation which is, and solve the corresponding initial value problem exactly. By an independent calculation, we find the same solution by directly constructing the Green function from the dynamics of the model in the Grad limit. From this solution an expression for the diffusion tensor, valid for arbitrary short-range forces, is derived. For hard disks we calculate the diffusion tensor explicitly. Away from the Grad limit a percolation problem arises. We determine numerically the percolation threshold and the corresponding geometric critical exponents. The numerical evidence strongly suggests that this continuum percolation model is in the universality class of 2D lattice percolation. Although we have explicitly determined a number of limiting properties of the model, several intriguing open problems remain.It is with great pleasure we include this paper in the issue honoring Matthieu Ernst, who not only shares our love for kinetic theory, but who also contributed so much to its modern development.     

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.     

The strange man in random networks of automata

We have performed computer simulations of Kauffman’s automata on several graphs, such as the regular square lattice and invasion percolation clusters, in order to investigate phase transitions, radial distributions of the mean total damage (dynamical exponent) and propagation speeds of the damage when one adds a damaging agent, nicknamed “strange man”. Despite the increase in the damaging efficiency, we have not observed any appreciable change of the transition threshold to chaos neither for the short-range nor for the small-world case on the square lattices when the strange man is added, in comparison to when small initial damages are inserted in the system. Particularly, we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters (high boundary-saturation clusters). It is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters.     

Clusters and Ising droplets in the antiferromagnetic lattice gas

A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if P_b = 1 - exp(-|K|/2). Along this line these clusters called “droplets” diverge with Ising exponents.     

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE ₆ paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).Research partially supported by the U.S. NSF under grant DMS-01-04278.     

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.     

Random walk on topologically biased anisotropic percolation clusters and transport properties

Unbiased random walks are performed on topologically biased anisotropic percolation clusters (APC). Topologically biased APCs are generated using suitable anisotropic percolation models. New walk dimensions are found to characterize the anisotropic behaviour of the unbiased random walk on the biased topology. Critical properties of electro and magneto conductivities are characterized estimating respective dynamical critical exponents. A dynamical scaling theory relating dynamical and static critical exponents has been developed. The dynamical critical exponents satisfy the scaling relations within error bar.     

Infinite Canonical Super-Brownian Motion and Scaling Limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on when these objects are critical, mean-field and infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation above 6 dimensions, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.