Space-filling percolation

Physical Review E, 1997

Boundary effects for invasion percolation are introduced and discussed here. The presence of boundaries determines a set of critical exponents characteristic of the boundary. In this paper we present numerical simulations showing a remarkably different fractal dimension for the region of the percolating cluster near the boundary. In fact, near the surface we find a value of D sur ϭ1.67Ϯ0.03, with respect to the bulk value of D bul ϭ1.87Ϯ0.01. Furthermore, we are able to present a theoretical computation of the fractal dimension near the boundary in fairly good agreement with numerical data.

EPL (Europhysics Letters), 2012

We study a process termed agglomerative percolation (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation. PACS numbers: 64.60.ah, 68.43.Jk, 89.75.Da Percolation is a pervasive concept in statistical physics and an important branch of mathematics [1]. It typifies the emergence of long range connectivity in many systems such as the flow of liquids through porous media [2], transport in disordered media [3], spread of disease in populations [4], resilience of networks to attack [5], formation of gels [6] and even of social groups [7]

Theoretical and Mathematical Physics, 1985

Percolation models in which the centers of defects are distributed randomly in space in accordance with Poisson's law and the shape of each defect is also random are considered. Methods of obtaining rigorous estimates of the critical densities are described. It is shown that the number of infinite clusters can take only three values: 0, 1, or ~. Models in which the defects have an elongated shape and a random orientation are investigated in detail. In the two-dimensional case, it is shown that the critical volume concentration of the defects is proportional to all, where l and a are, respectively, the major and minor axes of the defect; the mean number of (direct) bonds per defect when percolation occurs is bounded.

Physical Review E, 2014

The percolation threshold for flow or conduction through voids surrounding randomly placed spheres is rigorously calculated. With large scale Monte Carlo simulations, we give a rigorous continuum treatment to the geometry of the impenetrable spheres and the spaces between them. To properly exploit finite size scaling, we examine multiple systems of differing sizes, with suitable averaging over disorder, and extrapolate to the thermodynamic limit. An order parameter based on the statistical sampling of stochastically driven dynamical excursions and amenable to finite size scaling analysis is defined, calculated for various system sizes, and used to determine the critical volume fraction φc = 0.0317 ± 0.0004 and the correlation length exponent ν = 0.92 ± 0.05.

1999

We present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions. The results for the fractal support the Rammal and Toulouse proposition that d S (N) / d N x B (N) / S (N). The results for percolation are in excellent agreement with the Alexander and Orbach conjecture that the fracton

Latin American journal of probability and mathematical statistics

We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of co...

Physical Review E, 2009

This work presents the generalization of the concept of universal finite-size scaling functions to continuum percolation. A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems. The percolation threshold of high precision is determined for isotropic widthless stick systems as N c l 2 = 5.637 26Ϯ 0.000 02, with N c as the critical density and l as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor A = 0.106 910Ϯ 0.000 009, the spanning probability of stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolation.

The European Physical Journal B, 2011

We study the percolation properties of the growing clusters model on a 2D square lattice. In this model, a number of seeds placed on random locations on the lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The model exhibits a discontinuous transition for very low values of the seed concentration p and a second, nontrivial continuous phase transition for intermediate p values. Here we study in detail this continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component. Using finite size scaling and large scale Monte Carlo simulations we determine the value of the percolation threshold where the giant component first appears, and the critical exponents that characterize the transition. We find that the transition belongs to a different universality class from the standard percolation transition.

Physica A: Statistical Mechanics and its Applications, 1997

Optical and electrical properties of inhomogeneous media, strongly depend on the percolating properties of one of the component. In many cases, inhomogeneous media present fractal structures near and at the percolation threshold, and in general, transport properties exhibit a very special behaviour in this concentration range. In contrast with the theoretical simulations, the scale invariance ratio of the real systems (granular metal films, porous media...) is limited, which has a direct influence on the percolating properties of the clusters.

Journal of Statistical Mechanics: Theory and Experiment, 2007

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips, and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory, and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to (sin πy) −5/48 {[cos(πy/2)] 1/3 + [sin(πy/2)] 1/3 − 1}. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides, and compare with theory for the density along an edge.