Short-range correlations in percolation at criticality

The bond percolation problem is studied by the Monte Carlo method on a two-dimensional square lattice of 2 X lo6 bonds. Through the inclusion of a ghost field h, we obtain the generating function (the percolation analogue of the Gibbs free energy), percolation probability (the analogue of the spontaneous magnetisation), and mean cluster size ('isothermal susceptibility') as functions of two 'thermodynamic' variables, c = ( p , -p ) / p c and h. We discuss the non-trivial problems associated with the identification of the singular parts of these functions. We demonstrate that scaling holds for all three 'thermodynamic' functions within a rather large 'scaling region'.

Journal of Physics A: Mathematical and General, 1978

The bond percolation problem is studied by the Monte Carlo method on a two-dimensional square lattice of 2 X lo6 bonds. Through the inclusion of a ghost field h, we obtain the generating function (the percolation analogue of the Gibbs free energy), percolation probability (the analogue of the spontaneous magnetisation), and mean cluster size ('isothermal susceptibility') as functions of two 'thermodynamic' variables, c = ( p , -p ) / p c and h. We discuss the non-trivial problems associated with the identification of the singular parts of these functions. We demonstrate that scaling holds for all three 'thermodynamic' functions within a rather large 'scaling region'.

1998

Extensive Monte-Carlo simulations were performed to study bond percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic (b.c.c.) lattices, using an epidemic kind of approach. These simulations provide very precise values of the critical thresholds for each of the lattices: p c (s.c.) = 0.248 812 6 ± 0.000 000 5, p c (f.c.c.) = 0.120 163 5 ± 0.000 001 0, and p c (b.c.c.) = 0.180 287 5 ± 0.000 001 0. For p close to p c , the results follow the expected finite-size and scaling behavior, with values for the Fisher exponent τ (2.189 ± 0.002), the finite-size correction exponent Ω (0.64 ± 0.02), and the scaling function exponent σ (0.445 ± 0.01) confirmed to be universal. PACS numbers(s): 64.60Ak, 05.70.Jk Typeset using REVT E X 1 I. INTRODUCTION Percolation theory is used to describe a variety of natural physical processes, which have been discussed in detail by Stauffer and Aharony [1] and Sahimi [2]. In two-dimensional percolation, either exact values or precise estimates are known for the critical thresholds and other related coefficients and exponents [3-6].

Physical Review B, 1984

A method for generalizing bond-percolation problems to include the possibility of infinite-range (equivalent-neighbor) bonds is presented. On Bravais lattices the crossover from nonclassical to classical (mean-field) percolation criticality in the presence of such bonds is described. The Cayley tree with nearest-neighbor and equivalent-neighbor bonds is solved exactly, and a nonuniversal line of percolation transitions with exponents dependent on nearest-neighbor bond occupation probability is observed. Points of logarithmic and exponential singularity are also encountered, and the behavior is interpreted as dimensional reduction due to the breaking of translational invariance by bonds of Cayley-tree connectivity.

2001

Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on L_1× L_2 planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratio L_1/L_2. We calculate the probability for the appearance of n percolating clusters, W_n, the percolating probabilities, P, the average fraction of lattice bonds (sites) in the percolating clusters, <c^b>_n (<c^s>_n), and the probability distribution function for the fraction c of lattice bonds (sites), in percolating clusters of subgraphs with n percolating clusters, f_n(c^b) (f_n(c^s)). Using a small number of nonuniversal metric factors, we find that W_n, P, <c^b>_n (<c^s>_n), and f_n(c^b) (f_n(c^s)) for random lattices, duals of random lattices, and square lattices have the same universal fi...

Communications in Theoretical Physics, 2015

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 bo...

Physical Review B, 1980

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Communications in Mathematical Physics, 1990

The triangle condition for percolation states that ]Γ τ(0,x) τ(x,y) χ,y-τ (y, 0) is finite at the critical point, where τ (x, y) is the probability that the sites x and y are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thê /-dimensional hypercubic lattice, if d is sufficiently large, and (ii) in more than six dimensions for a class of "spread-out" models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values (y = β = 1, δ = Δ t = 2, ί ^ 2) and that the percolation density is continuous at the critical point. We also prove that v 2 = 1/2 in (i) and (ii), where v 2 is the critical exponent for the correlation length.

Physical Review E, 2003

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 to 13 dimensions. For d < 6 they are preliminary, for d ≥ 6 they are between 20 to 10 4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to ≈ 30 and more. Based on these data we propose a new scaling law for finite cluster size corrections.

We introduce anisotropic bond percolation in which there exist different occupation probabilities for bonds placed in different coordinate directions. We study in detail a d-dimensional hypercubical lattice, with probabilities p I for bonds within (d -1)-dimensional layers perpendicular to the z direction, and p11= Rp, for bonds parallel to z. For this model, we calculate low-density series for the mean size S, in both two and three dimensions for arbitrary values of the anisotropy parameter R. We find that in the limit 1/R + 0, the model exhibits crossover between 1 and d-dimensional critical behaviour, and that the mean-size function scales in 1/R. From both exact results and series analysis, we derive that the crossover exponent (=&) is 1 for all d, and that the divergence of successive derivatives of S with respect to 1 / R increases with a constant gap equal to 1 in two and three dimensions. In the opposite limit R + 0, crossover between d -1 and d-dimensional order occurs, and from our analysis of the three-dimensional series it appears that here the crossover exponent &-I is not equal to the two-dimensional mean-size exponent. This feature is in contrast with the corresponding situation in thermal critical phenomena where 6 d -l does equal the susceptibility exponent in two dimensions. Finally, our analysis appears to confirm that the value of the mean-size exponent is independent of anisotropy in accordance with universality.