Percolation as a dynamical phenomenon (original) (raw)
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Structural and dynamical properties of the percolation backbone in two and three dimensions
Physical Review E, 1997
We study structural and dynamical properties of the backbone of the incipient infinite cluster for site percolation in two and three dimensions. We calculate the average mass of the backbone in chemical l space, ͗M B (l)͘ϳl d l B , where d l B is the chemical dimension. We find d l B ϭ1.45Ϯ0.01 in dϭ2 and d l B ϭ1.36Ϯ0.02 in dϭ3. The fractal dimension in r space d f B is obtained from the relation d f B ϭd l B d min , d f B ϭ1.64Ϯ0.02 in dϭ2 and d f B ϭ1.87 Ϯ 0.03 in dϭ3, where d min is the fractal dimension of the shortest path. The distribution function ⌽ B (r,l) is determined, giving the probability of finding two backbone sites at the spatial distance r connected by the shortest path of length l , as well as the related quantity l min B (r,N av), giving the length of the minimal shortest path for two backbone sites at distance r as a function of the number N av of configurations considered. Regarding dynamical properties, we study the distribution functions P B (l ,t) and P B (r,t) of random walks on the backbone, giving the probability of finding a random walker after t time steps, at a chemical distance l , and Euclidean distance r from its starting point, respectively, and their first moments ͗l B (t)͘ϳt 1/d w Bl and ͗r B (t)͘ϳt 1/d w B , from which the fractal dimensions of the random walk d w Bl and d w B are estimated. We find d w Bl ϭ2.28Ϯ0.03 and d w B ϭ2.62Ϯ0.03 in dϭ2 as well as d w Bl ϭ2.25Ϯ0.03 and d w B ϭ3.09Ϯ0.03 in dϭ3. ͓S1063-651X͑97͒00508-4͔
1 ∕ \u3cem\u3ed\u3c/em\u3e Expansion for \u3cem\u3ek\u3c/em\u3e-Core Percolation
2005
The physics of k-core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from orientational ordering in solid ortho-para H2 mixtures to the onset of rigidity in bar-joint networks to dynamical arrest in glass-forming liquids. Unlike ordinary (k=1) and biconnected (k=2) percolation, the mean field k⩾3-core percolation transition is both continuous and discontinuous, i.e., there is a jump in the order parameter accompanied with a diverging length scale. To determine whether or not this hybrid transition survives in finite dimensions, we present a 1∕d expansion for k-core percolation on the d-dimensional hypercubic lattice. We show that to order 1/d3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability. This result suggests that the unusual hybrid nature of the mean field...
Agglomerative percolation in two dimensions
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]
Journal of Statistical Physics, 1978
Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 106 sites. We investigate for the square lattice the variablerange percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents/~ and 7. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent u. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.
Temporally Disordered Bond Percolation on the Directed Square Lattice
Physical Review Letters, 1996
Simple models of directed bond percolation with temporal disorder are introduced and studied via series expansions and Monte Carlo simulations. Series have been derived for the percolation probability on the directed square lattice. Analysis of the series revealed that the critical exponent b and critical point p c change continuously with the strength of the disorder. Monte Carlo simulation confirmed the continuous change of critical exponents. Estimates for the temporal correlationlength exponent n k for weak disorder showed that n k , 2 in apparent violation of the Harris criterion. [S0031-9007 01855-8] PACS numbers: 05.50. + q, 02.50. -r, 05.70.Ln
A Test of Scaling for the Bond Percolation Threshold
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'.
Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation
Physica A-statistical Mechanics and Its Applications, 1999
We present a Monte Carlo study of a linear chain (d = 1) with long-range bonds whose occupancy probabilities are given by pij = p=r ij (06p61; ¿0) where rij = 1; 2; : : : is the distance between sites. The → ∞ ( = 0) corresponds to the ÿrst-neighbor ("mean ÿeld") particular case. We exhibit that the order parameter P∞ equals unity ∀p ¿ 0 for 06 61, presents a familiar behavior (i.e., 0 for p6pc( ) and ÿnite otherwise) for 1 ¡ ¡ 2, and vanishes ∀p ¡ 1 for ¿ 2. Our results conÿrm recent conjecture, namely that the nonextensive region (06 61) can be meaningfully unfolded, as well as uniÿed with the extensive region ( ¿ 1), by exhibiting P∞ as a function of p * where (1 − p * ) = (1 − p) N * (N * ≡ (N 1− =d − 1)=(1 − =d); N being the number of sites of the chain). A corollary of this conjecture, now numerically veriÿed, is that pc˙( − 1) in the → 1 + 0 limit.
2005
This is a survey article to be part of the Encyclopedia of Mathematical Physics, to be published by Elsevier in the beginning of 2006.
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].