Backbone exponents of the two-dimensional q-state Potts model: a Monte Carlo investigation

Backbone and shortest-path exponents of the two-dimensional Q-state Potts model

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional Q-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents d_{B}=1.7320(3) and 1.794(2) for Q=2,3, respectively. However, for large Q, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n^{2}. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops d_{B}=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+sqrt[3],4, respectively, and d_{min}=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+sqrt[3],4, respectively. Our estimates significantly improve over the existing results for both d_{B} and d_{min}. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of n for the leading correction exponent.

Loop-Cluster Coupling and Algorithm for Classical Statistical Models

Potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the q-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with q-flow variables and formulate an LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the q-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model but also brings new insights into the rich geometric structures of the FK clusters.

Asymptotic correlation functions in the Q-state Potts model: A universal form for point group C_{4v}

Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the C_{4v} symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the Q-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with the asymptotic form above the critical temperature, we reproduce the exact solution of the the anisotropic correlation length (ACL) of the Ising model (Q=2) within a five-digit accuracy. For Q=3 and 4, we obtain numerical evidence that the asymptotic form is applicable to their correlation functions and the ACLs. Furthermore, we successfully apply it to the bond percolation problem which corresponds to the Q→1 limit. From the calculated ACLs, the equilibrium crystal shapes (ECSs) are derived via duality and Wulff's construction. Regarding Q as a continuous variable, we find that the ECS of the Q-state Potts model is essentially the same as those of the Ising models on the Union Jack and 4-8 lattices, which are represented in terms of a simple algebraic curve of genus 1.

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as K_{c}(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C_{1} follows a single-variable function as P(C_{1},L)dC_{1}=P[over ̃](x)dx with x≡C_{1}/L^{d_{F}} (L is the linear size), where the fractal dimension d_{F} is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution P[over ̃](x) in three dimensions, and attributed to the different approaching behaviors for K→K_{c}+0^{±}. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d_{min}(3D)=1.25936(12) and d_{min}(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d_{B}(3D)=2.1673(15) and d_{B}(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d_{min} and d_{B} either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

No-Enclave Percolation Corresponds to Holes in the Cluster Backbone

The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of such holes give the same size-distribution exponent τ=1.82(1) as found for the NEP model. An argument is given that τ=1+d_{B}/2≈1.822 for backbone holes, where d_{B} is the backbone dimension. On the other hand, a model of simple holes within a percolation cluster yields τ=1+d_{f}/2=187/96≈1.948, where d_{f} is the fractal dimension of the cluster, and this value is consistent with the experimental results of gel collapse of Sheinman et al., which give τ=1.91(6). This suggests that the gel clusters are of the universality class of percolation cluster holes. Both models give a discontinuous maximum hole size at p_{c}, signifying explosive percolation behavior.

Recursive percolation

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and find compelling numerical evidence that it can be repeated recursively any number n of generations. In two dimensions, we determine the percolation thresholds up to n=5. The corresponding critical clusters become more and more compact as n increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any n. This family of exponents differs from previously known universality classes, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.

Nonlinear scaling analysis approach of agent-based Potts financial dynamical model

A financial agent-based price model is developed and investigated by one of statistical physics dynamic systems-the Potts model. Potts model, a generalization of the Ising model to more than two components, is a model of interacting spins on a crystalline lattice which describes the interaction strength among the agents. In this work, we investigate and analyze the correlation behavior of normalized returns of the proposed financial model by the power law classification scheme analysis and the empirical mode decomposition analysis. Moreover, the daily returns of Shanghai Composite Index and Shenzhen Component Index are considered, and the comparison nonlinear analysis of statistical behaviors of returns between the actual data and the simulation data is exhibited.

Geometric structure of percolation clusters

We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).

Percolation with long-range correlated disorder

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent H, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on H.

Shortest-path fractal dimension for percolation in two and three dimensions

We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension d(min) for percolation in two and three dimensions, using the Leath-Alexandrowicz method which grows a cluster from an active seed site. A variety of quantities are sampled as a function of the chemical distance, including the number of activated sites, a measure of the radius, and the survival probability. By finite-size scaling, we determine d(min)=1.13077(2) and 1.3756(6) in two and three dimensions, respectively. The result in two dimensions rules out the recently conjectured value d(min)=217/192 [Deng et al., Phys. Rev. E 81, 020102(R) (2010)].

Conductivity of Coniglio-Klein clusters

We performed numerical simulations of the q-state Potts model to compute the reduced conductivity exponent t/ν for the critical Coniglio-Klein clusters in two dimensions, for values of q in the range [1,4]. At criticality, at least for q<4, the conductivity scales as C(L) ~ L(-t/ν), where t and ν are, respectively, the conductivity and correlation length exponents. For q=1, 2, 3, and 4, we followed two independent procedures to estimate t/ν. First, we computed directly the conductivity at criticality and obtained t/ν from the size dependence. Second, using the relation between conductivity and transport properties, we obtained t/ν from the diffusion of a random walk on the backbone of the cluster. From both methods, we estimated t/ν to be 0.986 ± 0.012, 0.877 ± 0.014, 0.785 ± 0.015, and 0.658 ± 0.030, for q=1, 2, 3, and 4, respectively. We also evaluated t/ν for noninteger values of q and propose the conjecture 40 gt/ν = 72 + 20 g - 3g(2) for the dependence of the reduced conductivity exponent on q, in the range 0 ≤ q ≤ 4, where g is the Coulomb gas coupling.

Some geometric critical exponents for percolation and the random-cluster model

We introduce several infinite families of critical exponents for the random-cluster model and present scaling arguments relating them to the k -arm exponents. We then present Monte Carlo simulations confirming these predictions. These exponents provide a convenient way to determine k -arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension d(min) in two dimensions: d(min)=[over ?](g+2)(g+18)/(32 g) , where g is the Coulomb-gas coupling, related to the cluster fugacity q via q=2+2 cos(gpi/2) with 2< or =g< or =4 .

Dilute Potts model in two dimensions

We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.

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) .

Percolation between vacancies in the two-dimensional Blume-Capel model

Using suitable Monte Carlo methods and finite-size scaling, we investigate the Blume-Capel model on the square lattice. We construct percolation clusters by placing nearest-neighbor bonds between vacancies with a variable bond probability p(b) . At the tricritical point, we locate the percolation threshold of these vacancy clusters at p(bc) =0.706 33 (6) . At this point, we determine the fractal dimension of the vacancy clusters as Xf =0.1308 (5) approximately equal to 21/160, and the exponent governing the renormalization flow in the p(b) direction as y(p) =0.426 (2) approximately equal to 17/40 . For bond probability p(b) > p(bc) , the vacancy clusters maintain strong critical correlations; the fractal dimension is Xf =0.0750 (2) approximately equal to 3/40 and the leading correction exponent is y(p) =-0.45 (2) approximately equal to -19/40 . The above values fit well in the Kac table for the tricritical Ising model. These vacancy clusters have much analogy with those consisting of Ising spins of the same sign, although the associated quantities rho and magnetization m are energylike and magnetic quantities, respectively. However, along the critical line of the Blume-Capel model, the vacancies are more or less uniformly distributed over the whole lattice. In this case, no critical percolation correlations are observed in the vacancy clusters, at least in the physical region p(b) < or = 1 .

Simulation algorithms for the random-cluster model

We compare the performance of Monte Carlo algorithms for the simulation of the random-cluster representation of the q-state Potts model for continuous values of q. In particular we consider a local bond update method, a statistical reweighting method of percolation configurations, and a cluster algorithm, all of which generate Boltzmann statistics. The dynamic exponent z of the cluster algorithm appears to be quite small, and to assume the values of the Swendsen-Wang algorithm for q = 2 and 3. The cluster algorithm appears to be much more efficient than our versions of the other two methods for the simulation of the random-cluster model. The higher efficiency of the cluster method with respect to the local method is primarily due to the fact that the computer time usage of the local method increases more rapidly with system size; the difference between the dynamic exponents is less important.

Red-bond exponents of the critical and the tricritical Ising model in three dimensions

Using the Wolff and geometric cluster algorithms and finite-size scaling analysis, we investigate the critical Ising and the tricritical Blume-Capel models with nearest-neighbor interactions on the simple-cubic lattice. The sampling procedure involves the decomposition of the Ising configuration into geometric clusters, each of which consists of a set of nearest-neighboring spins of the same sign connected with bond probability p. These clusters include the well-known Kasteleyn-Fortuin clusters as a special case for p=1-exp(-2K) , where K is the Ising spin-spin coupling. Along the critical line K=Kc , the size distribution of geometric clusters is investigated as a function of p . We observe that, unlike in the case of two-dimensional tricriticality, the percolation threshold in both models lies at pc =1-exp(-2Kc) . Further, we determine the corresponding red-bond exponents as yr =0.757(2) and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture yr =1/2 for the latter model.

Magnetic and backbone exponents of the percolation and Ising models in three dimensions

We investigate random-cluster representations of the q=1 - and 2-state Potts models in three dimensions, i.e., the bond-percolation and the Ising model, respectively. Using a recently developed sampling technique, we determine the probabilities C1 (r) and C2 (r) that a pair of lattice sites at a distance r are connected by at least one and two mutually independent paths, respectively. The scaling behavior of C1 and C2 at criticality is governed by the magnetic and the backbone scaling dimension X(h) and X(b) , respectively. From a finite-size analysis of the numerical data, we determine X(h) =0.4768 (7) and X(b) =1.125 (3) for the percolation and X(h) =0.5178 (7) and X(b) =0.829 (4) for the Ising model.

Spontaneous edge order and geometric aspects of two-dimensional Potts models

Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.