The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.
The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.
Finite-size scaling (FSS) for a critical phase transition (t=0) states that within a window of size ∣t∣∼L−1/ν, the scaling behavior of any observable Q in a system of linear size L asymptotically follows a scaling form as Q(t,L)=LYQQ~(tL1/ν), where ν is the correlation-length exponent, YQ is an FSS exponent and Q~(x) is a function of the scaled distance-to-criticality x≡tL1/ν. We systematically study the asymptotic scaling behavior of Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with t→0 and ∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as ∣t∣∼L−λ with λ<1/ν, the FSS becomes λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
Abstract In this chapter we apply the scaling theories developed in the one-dimensional system and the infinite-dimensional system to systems of finite dimensions. The lowest dimension with an interesting behavior is two dimensions. Here, we introduce effective ways to measure the cluster number density n(s,p) n ( s , p ) in two dimension. We develop the scaling theory for n(s,p) n ( s , p ) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for n(s,p) n ( s , p ) to derive expressions for the density of the spanning cluster, P, and the average cluster size, S. Finally, we demonstrate how the scaling theory provides scaling relations, that is, relations between exponents, and bounds for the values of the critical exponents.
Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...
The dynamic structure factor of very large percolating clusters (bond and site), in two dimensions and three dimensions are calculated using the spectral moments method. Interactions are represented by the scalar model. Numerical results are presented and interpreted in terms of scaling arguments given by Alexander, Courtens and Vacher (ACV) in 1993. Concerning the q lambda <<1 limit, our results confirm and supplement previous numerical work and are in agreement with the scaling behaviour theoretically deduced by ACV. Concerning the q lambda >>1 limit, we have shown that the dynamic structure factor complies with the very nice asymptotic behaviour g(q, omega )=q y H(q lambda ( omega )) where the scaling function H(x) is of power-law form x - tau ' in agreement with the theory. However, our results indicate a scaling behaviour with exponents that differ from those deduced by theory. The values obtained for tau ' are 1.20 for the two-dimensional bond and site percolating lattices, and 1.00 for the three-dimensional percolating lattices. Comparisons with previous simulations are reported.
Calculating, by a Monte Carlo technique, the conductivities of small, two-dimensional, square, bond-percolation networks (4*4 to 50*50) at a limited number of probability values, from the critical region to full conductivity, the authors have been able to show, by an original interpolation technique, that in the critical region the conductivity function does indeed obey a universal, Fisher-type, finite-size scaling function. They further show that this technique permits them to deduce values of the critical exponents for both the conductivity and the correlation length, even from calculations on relatively small networks.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep, the percolation probability P, and the probability Wn for the appearance of n percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L. Using the HMCSM, we calculate the existence probability Ep and the percolation probability P for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep and P in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep and P have universal finite-size scaling functions. This implies that the critical Ep is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn for site and bond percolation on sc lattices have universal finite-size scaling functions.