Effective boundary extrapolation length to account for finite-size effects in the percolation crossing function

Correction-to-scaling exponent for two-dimensional percolation

We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Ω ≤ 72/91, ω = DΩ ≤ 3/2, and Δ₁ = νω ≤ 2, based upon Cardy's result for the crossing probability on an annulus. The upper bounds are consistent with many previous measurements of site percolation on square and triangular lattices and new measurements for bond percolation, suggesting that they are exact. They also agree with exponents for hulls proposed recently by Aharony and Asikainen, based upon results of den Nijs. A corrections scaling form evidently applicable to site percolation is also found.

Mapping functions and critical behavior of percolation on rectangular domains

The existence probability E_{p} and the percolation probability P of bond percolation on rectangular domains with different aspect ratios R are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of E_{p} and P for such systems with exponents a and b , respectively, found by Watanabe [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower-order approximation of the mapping functions f_{R} and g_{R} for E_{p} and P , respectively; the exponents a and b can be obtained from numerically determined mapping functions f_{R} and g_{R} , respectively.

Exact factorization of correlation functions in two-dimensional critical percolation

By use of conformal field theory, we discover several exact factorizations of higher-order density correlation functions in critical two-dimensional percolation. Our formulas are valid in the upper half-plane, or any conformally equivalent region. We find excellent agreement of our results with high-precision computer simulations. There are indications that our formulas hold more generally.

Modeling 2D and 3D diffusion

Modeling obstructed diffusion is essential to the understanding of diffusion-mediated processes in the crowded cellular environment. Simple Monte Carlo techniques for modeling obstructed random walks are explained and related to Brownian dynamics and more complicated Monte Carlo methods. Random number generation is reviewed in the context of random walk simulations. Programming techniques and event-driven algorithms are discussed as ways to speed simulations.

Superscaling of percolation on rectangular domains

For percolation on (RL)xL two-dimensional rectangular domains with a width L and aspect ratio R, we propose that the existence probability of the percolating cluster E(p)(L,epsilon,R) as a function of L, R, and deviation from the critical point epsilon can be expressed as F(epsilonL(y(t))R(a)), where y(t) identical with1/nu is the thermal scaling power, a is a new exponent, and F is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with a=0.14(1) for R>2. We also propose superscaling for other critical quantities.

Convergence of threshold estimates for two-dimensional percolation

Using a recently introduced algorithm for simulating percolation in microcanonical (fixed-occupancy) samples, we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary, specifically for site percolation on a square lattice. We show that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90+/-0.02. For the median and cell-to-cell estimates of the percolation threshold we verify that convergence does not depend on this exponent, having instead a slightly faster convergence with a trivial analytic leading exponent.

Aspect-ratio dependence of percolation probability in a rectangular system

I investigate site percolation on a rectangular system (aspect ratio a) of a square lattice for a given occupation probability p (not restricted to p(c)) using computer simulations. The dependence of the percolation probability R on a is shown and analyzed on the basis of a modified finite-size scaling function. A method for evaluating R without statistical simulations is proposed for given conditions (longitudinal dimension L, a, and p) of the system.

Finite size scaling for percolation on elongated lattices in two and three dimensions

We derive scaling laws for the percolation properties of an elongated lattice, i.e., those with dimensions of L(d-1)xnL in d dimensions, where n denotes the aspect ratio of the lattice. Based on statistical arguments it is shown that, in the direction of the extension, the percolation threshold scales approximately as ln n(1/a) in both two and three dimensions. Extensive Monte Carlo simulations of the site percolation model confirm this scaling behavior. It is further shown that the density of the incipient infinite cluster at the percolation threshold scales differently in two and three dimensions.

Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals

The bond-percolation process is studied on periodic planar random lattices and their duals. The thresholds and critical exponents of the percolation transition are determined. The scaling functions of the percolating probability, the existence probability of the appearance of percolating clusters, and the mean cluster size are also calculated. The simulation result of the percolation threshold is p(c)=0.3333+/-0.0001 for planar random lattices, and 0.6670+/-0.0001 for the duals of planar random lattices. We conjecture that the exact value of p(c) is 1/3 for a planar random lattice and 2/3 for the dual of a planar random lattice. By taking possible errors into account, the results of our critical exponents agree with the values given by the universality hypothesis. By properly adjusting the metric factors on random lattices and their duals, we demonstrate explicitly that the idea of a universal scaling function with nonuniversal metric factors in the finite-size scaling theory can be extended to random lattices and their duals for the existence probability, the percolating probability, and the mean cluster size.