We study N-cluster correlation functions in four- and five-dimensional (4D and 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E 72, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry Ld−1 × ∞, with the linear size up to L = 512 for 4D and 128 for 5D. We determine with a high precision all possible N-cluster exponents, for N = 2 and 3, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with N=2, we obtain the correlation-length critical exponent as 1/ν=1.4610(12) for 4D and 1/ν=1.737(2) for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.
In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly. All these results hold in any dimension, 2≤d≤6. Moreover, in d=2 the critical exponents and universal logarithmic prefactors are obtained exactly. We test these predictions against extensive simulations of critical bond percolation in d=2 and 3, for all correlators up to N=4 (d=2) and N=3 (d=3), finding excellent agreement. In d=3 we further obtain precise numerical estimates for critical exponents and logarithmic prefactors.
In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}, where r is the distance length between distinct sites and d=1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_{c} at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^{-3}, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for C_{c} are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the ɛ-expansion used with the introduction of a suitably defined effective dimension d_{eff} relating the long-range model with a short-range one in dimension d_{eff}. We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with σ>0.
Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d − 1 × of arbitrary dimensionality d . Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision.
We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds,
Nc and
Nb, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where
L is the linear system size and
d is the spatial dimensionality. In statistical models with thermal fluctuations,
C2c and
Ccb are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of
C2c and
Ccb is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for
Nc, where
S2 and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as
Lyt for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields
yt=1.145(2) for the three-dimensional percolation, in good agreement with existing results.
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.