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Percolation of Monte Carlo Clusters Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 247 Percolation of Monte Carlo Clusters W.G. Wanzeller1, A. Cucchieri2, T. Mendes2, and G. Krein1 1 Instituto de F´ısica Teorica,´ Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sao˜ Paulo, SP, Brasil 2 Instituto de F´ısica de Sao˜ Carlos, Universidade de Sao˜ Paulo, P.O. Box 369, 13560-970, Sao˜ Carlos, SP, Brasil Received on 15 August, 2003. Percolation theory is of interest in problems of phase transitions in condensed matter physics, and in biology and chemistry. More recently, concepts of percolation theory have been invoked in studies of color deconfinement at high temperatures in Quantum Chromodynamics. In the present paper we briefly review the basic concept of percolation theory, exemplify its application to the Ising model, and present the arguments for a possible relevance of percolation theory to the problem of color deconfinement. 1 Percolation the site will be occupied, and a probability 1 ¡ p that it is empty. A “cluster” is defined as groups of neighboring oc- In order to explain the concept of percolation, let us con- cupied sites. Suppose now that one fixes a value of p and sider Fig. 1. The figure represents pictorially an electrical at each site of the lattice we draw a random number r. If circuit composed of a set of conducting spheres, represented r · p, then one considers that the site is occupied, if r > p, by (²) and a set of insulating spheres, represented by (±). the site is considered empty. Groups of neighboring occu- An electrical current will flow from the top plate to the bot- pied sites are collected to form a cluster. At the end, one tom plate and the bulb (­) will light once enough conduct- checks if there is at least one cluster that has percolated, that ing spheres are present in the circuit. That is, there will be is, if an occupied site from the top of the lattice belonging to a “percolating” electrical current through the system when some cluster is continuously connected to an occupied site a conducting sphere from the top is continuously connected at the bottom of the lattice. Then, the process is repeated to a conducting sphere on the bottom. It is intuitively clear many times. Each complete visit to all sites of the lattice is that percolation will happen once a minimum of conduct- called a “sweep”. Next, we define a percolation probability ing spheres is present in the system, i.e. there is a critical as value for the concentration of conducting spheres for which 1 XN a “percolating cluster” will be formed. hP i = C ; (1) N j j=1 where Cj ´ 1(0) if a percolating cluster has (has not) oc- curred during the sweep j, and N is the number of sweeps. In Fig. 2 we plot hP i for different values of p for L = 100; 250; 400; 600 and 1000, using free boundary condi- tions. One sees that there exists a transition region around a certain value of p where the percolation probability changes from a small value to a value that becomes close to one for large lattices. It is clear that as the lattice becomes larger and larger, the transition region becomes sharper and sharper, with a well defined transition point pc. In the limit of an in- finite lattice, for the present case of a two-dimensional cubic lattice the value of pc is found to be [1] Figure 1. Pictorial representation of percolation. When a continu- ous chain from the top to the bottom is formed as in (b), there will pc = 0:592746 : (2) be a percolating current and the light bulb (­) will turn on. Clearly the numerical simulations, for large lattices, give pc The existence of a critical value for percolation and the very close to this value. To be more precise, using the prin- consequent formation of a percolation cluster can be illus- ciple of finite-size-scaling, our simulations give [2] trated as follows. Let us consider a square lattice of volume L £ L. For each lattice site, we attribute a probability p that pc = 0:5927 § 0:0001 : (3) 248 W.G. Wanzeller et al. 10.000 iteracoes Ising model determines the critical value for the occur- 1 rence of a percolating cluster, with p given by the formula L = 100 c 0.9 L = 400 pc = 1 ¡ exp(¡2J=kTc). The Fortuin-Kasteleyn clusters L = 600 0.8 L = 800 were later used (in 1987) by Swendsen and Wang [5] as L = 250 L = 1000 a very efficient way of performing global spin updates in 0.7 Monte Carlo simulations of spin models. 0.6 In Fig. 3 we show results of our Monte Carlo simulation for the percolation probability hP i, defined in Eq. (1). In the 0.5 <P> figure we show results for lattice sizes L = 100; 400; 600 0.4 and 800. A detailed description of the Swendsen-Wang al- 0.3 gorithm used, the reasons for using it, comparisons with other Monte Carlo algorithms, and the error analysis per- 0.2 formed can be found in Ref. [2]. The exact critical temper- 0.1 ature of the two-dimensional Ising model is (for J = 1) T = 2:269185 0 c . Clearly, our simulations show that the 0.5854 0.5878 0.5902 0.5926 0.595 0.5974 0.5998 percolation temperature, determined with the use of Eq. (5) p shows a good agreement with the exact value. Specifically, we obtained Figure 2. hP i for different lattices sizes. Tc = 2:2691 § 0:0001: (7) σ probalbilidadede percolacao, 5000 sweeps, corr 0.56 2 Percolation theory in the Ising L = 100 L = 400 L = 600 Model 0.49 L = 800 The magnetic phase transition that occurs in the Ising model 0.42 (for dimensions larger than 1) can be related to a percola- 0.35 tion phenomenon. The relation was established in two ma- jor steps. The first step was given in 1972 by Fortuin and 0.28 Kasteleyn [3], who showed that it is possible to rewrite the <P> partition function of the Ising model as a sum over config- 0.21 urations of spin clusters, instead of the usual sum over spin configurations, as follows 0.14 8 9 8 9 0.07 X X <nYij =1 = <nYij =0 = 0 Z = pij±SiSj (1 ¡ pij) ; 2.257 2.2603 2.2635 2.2668 2.27 2.2732 2.2765 2.2797 2.283 : ; : ; T fS=§1g fng <i;j> <i;j> (4) Figure 3. hP i for L = 100; 400; 600; and 800 using the Swensen- where Wang cluster algorithm. pij = 1 ¡ exp(¡2J¯); (5) is the probability that a bond between spins at neighboring sites i and j occurs, and nij = 1(0) indicates that there is 3 Percolation Theory and Deconfine- (there is not) a bond between spins at i and j. Note that a bond may be placed only between spins of the same value. ment in QCD In Eq. (5) ¯ = 1=kT , where k is the Boltzmann constant and As said above, percolation theory is of interest in several J is the spin-spin strength, defined in the Ising Hamiltonian areas of natural science. Our motivation in this theory is as X driven basically by its possible relevance to the problem of H = ¡J S S ; i j (6) color deconfinement in Quantum Chromodynamics (QCD). hiji The connection between the two phenomena can be argued where hiji means nearest neighbors and Si = §1=2 are as follows [6, 8]. spins at site i. A spin cluster can therefore be naturally de- Let us consider the formulation of a pure SU(N) gauge fined as the collection of spins that are continuously bound. theory on an Euclidean lattice with Nσ points in each spatial Note that now the probability of a spin to belong to a clus- direction and N¿ points in the time direction. The partition ter depends on the temperature (and the strength J) as in function at finite temperature can be written as Eq. (5). The next step was given in 1980 by Coniglio and Z Y 2 Klein [4] when they showed, using renormalization group Z(Nσ;N¿ ; g ) = [dU] expf¡S[U]g ; (8) techniques that the magnetic critical temperature Tc of the links Brazilian Journal of Physics, vol. 34, no. 1A, March, 2004 249 where the integration is over link variables U¹, very natural to conjecture that color deconfinement could be a percolation phenomenon. U¹ = exp [¡igaA¹(~x;t)] ; (9) Recently, Satz and Fortunato [6, 8] have undertaken per- colation studies for SU(2) gauge theories. This was made with a the lattice spacing and ¹ the space-time direction, and possible by the use of the Green-Karsch [9] effective theory S[U] is the Wilson action given by which allows to write the partition function of the theory, in · ¸ 2N X 1 the strong coupling limit and neglecting spacelike plaque- S[U] = 1 ¡ Re Tr(UP ) : (10) ttes, as g2 N P P Z Here P means that the sum is done over all plaquettes UP , p X 2 0 where a plaquette is the basic circuit of four U’s [7]. Z» ¦xdLx 1 ¡ Lx exp(¯ LiLj) ; (16) At finite temperatures, the gauge field configurations are ij periodic in the “time” interval [0; ¯]. This periodicity leads to a global symmetry that is important for the phase struc- where Li is the value of the Polyakov loop at site i, the sum ture of the theory at finite temperature.
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