Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a length-rescaling procedure. For strongly anisotropic systems in 1+1 dimensions, exact results are obtained for the random walk with absorbing boundary conditions, which can be considered as a linearized mean-field approximation for directed percolation. The bond and site directed percolation problem is itself studied numerically via Monte Carlo simulations on the diagonal square lattice with either free or periodic boundary conditions. A scale-invariant critical crossing probability is still obtained, which is a universal function of the effective aspect ratio r eff = c r , where r = L / t z , z is the dynamical exponent and c is a non-universal amplitude.
We consider bond percolation on the square lattice with perfectly correlated random probabilities. According to scaling considerations, mapping to a random walk problem and the results of Monte Carlo simulations the critical behavior of the system with varying degree of disorder is governed by new, random fixed points with anisotropic scaling properties. For weaker disorder both the magnetization and the anisotropy exponents are nonuniversal, whereas for strong enough disorder the system scales into an infinite randomness fixed point in which the critical exponents are exactly known.
The universality of the crossing probability ${\ensuremath{\pi}}_{\mathrm{hs}}$ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for
q=2,3,4 and for percolation
q=1. We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for
q=1,2,3,4 as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed,
A(q) is the nonuniversal crossing amplitude,
b(q) is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.
In this paper, we briefly review our recent results in the calculation of universal scaling functions for site and bond percolation on finite square, plane triangular, and honeycomb lattices.We find that, by choosing an aspect ratio for each lattice and a very small number of nonuniversal metric factors, all scaled data of the existance probability E p and the percolation probability P fallon the same universal scaling functions.We also find that free and periodic boundary conditions share the same nonuniversal metric factors.When the aspect ratio of each lattice is reduced by the same factor, the nonouniversal metric factors remain the same.The probabilities for the "appearance of n, n = 1, 2, 3, ..., percolating clusters for bond and site percolations on various planar lattices also have universal scaling functions.The implications of such results on numerical and experimental studies of critical phenomena are discussed.
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.
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.
A histogram Monte Carlo method is used to evaluate the existence probability
Ep and the percolation probability
P of bond and site percolation on finite square, plane triangular, and honeycomb lattices. We find that, by choosing a very small number of nonuniversal metric factors, all scaled data of
Ep and
P may fall on the same universal scaling functions. We also find that free and periodic boundary conditions share the same nonuniversal metric factors. This study may be extended to many critical systems.