Exact distribution of cluster size and perimeter for two-dimensional percolation

Exact series expansion data of Sykes et al. are used to calculate the average numberc _n and perimeters _n of clusters of sizen20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp _n we find a sharply peaked distribution of perimeterss _n with mean s _n =((1–p _n )/p _c )n+O(n ) and width s _n ² –S _n ²n ^1.6 where1/=0.39. This perimeter s _n should not be interpreted as a cluster surface in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc _n withn is consistent with the postulated asymmetry aboutp _c : logc _n –n forn with1 forp<p _c and1/2 forp>p _c .     

Scaling function for cluster size distribution in two-dimensional site percolation

Exact cluster size distributions of Sykes et al. in the square and triangular lattice for cluster sizes up to 17 are used to extrapolate the scaling function in the site percolation problem. Also the amplitude ratioC ₊/C _- of the second moment is determined.     

Typical cluster size for two-dimensional percolation processes

The typical cluster size for two-dimensional percolation models is discussed. It is shown that, forW ₀={xZ ²0x}, [–lim _n(1/n) logP _p (W ₀=n)]^–1p–p _c ^– aspp _c , provided thatE _p (W ₀²)/E _p (W ₀)p–P _c ^– aspp _c . Furthermore, we introduce a new quantityf _s (p), which may be thought of as the singular part of the free energy, and show thatf _s (p)¦p–p _c ¦^2v provided that the correlation length ¦p–p _c ¦^–v aspp _c .     

The fractal volume of the two-dimensional invasion percolation cluster

We consider both invasion percolation and standard Bernoulli bond percolation on theZ ² lattice. Denote byV andC the invasion cluster and the occupied cluster of the origin, respectively. Let , and     

Hausdorff dimension and fluctuations for the largest cluster at the two-dimensional percolation threshold

Monte Carlo simulation shows the average mass of the largest cluster to increase asL ^1.9 at the percolation threshold inL × L square lattices,L290. This fractal dimension agrees with the finite-size scaling prediction/v for this exponent, in contrast to results of Halley and Thang Mai. The mean-square fluctuations in the mass of the largest cluster diverge with the same exponent/v1.8 as the susceptibility.     

The percolation perimeter for two-dimensional directed animals

The percolation perimeter to-size ratio of directed animals is investigated. For the square lattice it is found to be very close to 0.75 and the first correction is found on both the triangular and square lattices to be Bethe-like as for normal lattice animals.     

On the upper critical dimension of Bernoulli percolation

We derive a set of inequalities for thed-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: $$\begin{gathered} f + v \geqq 1 + \beta _Q , \hfill \\ \mu + v \geqq 1 + \beta _Q , \hfill \\ \zeta \geqq \min \left\{ {1,\frac{{v^, }}{v}} \right\}, \hfill \\ \end{gathered} $$ where the above exponents aref: the flow constant exponent, ν(ν′): the correlation length exponent below (above) threshold, μ: the surface tension exponent, β _Q : the backbone density exponent and ζ: the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponentv defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs of these inequalities with notions of finite-size scaling to derive: $$\max \{ dv,dv^, \} \geqq 1 + \beta _Q ,$$ whered is the lattice dimension. Since β _Q ≧2β, where β is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean-field values; hence an implicit bound on the upper critical dimension:d _c ≧6.     

A microscopic justification of the Wulff construction

We report results about a rigorous microscopic justification of the Wulff construction for the two-dimensional Ising model at low temperatures and under periodic boundary conditions. The idea of the proof is sketched.     

Essential singularity in percolation problems and asymptotic behavior of cluster size distribution

It is rigorously proved that the analog of the free energy for the bond and site percolation problem on in arbitrary dimension (> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.Supported by the Fonds National Suisse de la Recherche Scientifique (to HK).     

Extremum of the percolation cluster surface

The internal and external surfaces of a percolation cluster, as well as the total surface of the entire percolation system, are investigated numerically and analytically. Numerical simulation is carried out using the Monte Carlo method for problems of percolation over lattice sites and bonds on square and simple cubic lattices. Analytic expressions derived by using the probabilistic approach describe the behavior of such surfaces to a high degree of accuracy. It is shown that both the external and total surface areas of a percolation cluster, as well as the total area of the surface of the entire percolation system, have a peak for a certain (different in the general case) fraction of occupied sites (in the site problem) or bonds (in the bond problem). Two examples of technological processes (current generation in a fuel cell and self-propagating high-temperature synthesis in heterogeneous condensed systems) in which the surface of a percolation cluster plays a significant role are discussed.     

Note on the susceptibility exponent for two-dimensional percolation

Analyses of mean site content, mean bond content, mean perimeter and related quantities give a susceptibility diverging as (p _c –p)^–, with 2.41±0.025. This exponent disagrees with some earlier estimates but it is consistent with the (p _c –p)^–2.388... divergence predicted by the formulas of den Nijs, Nienhuis et al. and Pearson.     

Analysis of the percolation cluster structure

An implementation of algorithms for constructing and analyzing the cluster structure for a square quadruply connected lattice in the uncorrelated percolation problem is considered. Subsets of the complete superior hull and the skeleton of a percolation cluster are singled out using a modification of the Hoshen—Kopelman relabeling algorithm and the Bellman principle of optimality. The critical nature of the percolation process is demonstrated using the method for statistical tests, and the behavior of mass dimension is analyzed for various subsets of a percolation cluster.     

Search for logarithmic factors near the two-dimensional percolation threshold

Monte Carlo data of Eschbach et al. indicate z = 0.06 ± 0.06 for a possible logarithmic factor in the correlation length ξ ∝ (|p ? p_cz.sfnc; log^zz.sfnc;p ? p_cz.sfnc;)^?v.     

Note on the “susceptibility” exponent for two-dimensional percolation

Zeitschrift für Physik B Condensed Matter - New percolation series derived by Duarte and Ruskin are analysed with a recently developed method of series analysis which explicitly accounts for...     

The bond problem in a one-dimensional percolation theory for finite systems

The results of investigations of main characteristics of a one-dimensional percolation theory (percolation threshold, critical exponents of correlation radius and specific heat, and free energy) are presented for the bond and site problems. For the first time it is shown that for a finite-size system the stability condition is fulfilled while the scaling hypothesis is inacceptable for one-dimensional bond problem.     

Exact determination of the two-point cluster function for one-dimensional continuum percolation

The two-point cluster functionC ₂(r ₁,r ₂) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium,C ₂(r ₁,r ₂) is defined to be the probability of finding two points at positionsr ₁ andr ₂ in thesame cluster of one of the phases. An exact analytical expression is found for the two-point cluster functionC ₂(r ₁,r ₂) of a one-dimensional continuumpercolation model of Poisson-distributed rods (for an arbitrary number density) using renewal theory. We also give asymptotic formulas for the tail probabilities. Along the way we find exact results for other cluster statistics of this continuum percolation model, such as the cluster size distribution, mean number of clusters, and two-point blocking function.     

A finite fractal cluster theory of 1/f noise in percolation systems near metal-insulator transition

The problem of 1/f noise in thin metal films and metal-insulator composites in the scaling fractal regime near percolation threshold is considered. The correspondence between a percolation transition and a second order phase transition is extended from the point of view of electronic polarization and electrical fluctuations. The charge fluctuations on finite fractal clusters are argued to be analogous to spontaneous order parameter fluctuations in phase transitions, being correlated upto percolation correlation length. The charge relaxation times are shown to be related to the cluster sizes having distribution function of the formg()^–b , whereb is connected to Euclidean and fractal dimensionalities and critical exponents. This produces the 1/f noise spectrum. Below percolation threshold, the nodes-links-blobs picture is invoked such that the blobs represent metallic conductances of the finite clusters and the links are tunnelling conductances between them through narrowest barrier regions. Above threshold, the finite cluster network is visualized as connected to the infinite cluster through narrowest tunnelling regions. The correlated spontaneous charge fluctuation on finite fractal clusters is held responsible for conductance fluctuation on either side of the metal-insulator transition via tunnelling processes. Finally, the scaling behaviour of noise magnitude near percolation threshold is explained.