Scaling theory of percolation clusters

Abstract

For beginners : This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied with probability p or empty with probability 1− p. An s-cluster is a group of s occupied sites connected by nearest-neighbor distances; the number of empty nearest neighbors of cluster sites is the perimeter t. For p above p c also one infinite cluster percolates through the lattice. How do the properties of s-clusters depend on s, and how do they feel the influence of the phase transition at p = p c? The answers to these questions are given by various methods (in particular computer simulations) and are interpreted by the so-called scaling theory of phase transitions. The results presented here suggest a qualitative difference of cluster structures above and below p c: Above p c some cluster properties suggest the existence of a cluster surface varying as s 2 3 in three dimensions, but below p c these “surface” contributions are proportional to s. We suggest therefore that very large clusters above p c (but not at and below p c) behave like large clusters of Swiss cheese: Inspite of many internal holes the overall cluster shape is roughly spherical, similar to raindrops. For experts : Scaling theory suggests for large clusters near the percolation threshold p c that the average cluster numbers n s vary as s −τƒ(z) , with z ≡ ( p − p c) s σ . Analogously the average cluster perimeter is t s = s · (1 − p)/ p + s σ · ψ 1( z), the average cluster radius R s varies as s σv · R 1( z), and the density profile D s ( r), which depends also on the distance r from the cluster center, varies as s −1 δ · D ̃ 1(rs −σv, z) . These assumptions relate the seven critical exponents α,β,γ,δ, v,σ,τ in d dimensions through the well-known five scaling laws 2 − α = γ + 2β = βδ + β = dv = β + 1 σ = (τ − 1)/σ , leaving only two exponents as independent variables to be fitted by “experiment” and not predicted by scaling theory. For the lattice “animals”, i.e. the number g st of geometrically different cluster configurations, a modified scaling assumption is derived: g sts st 1/(s + t) s + 1 ∝ s −τ− 1 2 · ƒ(z) , with z ∝ ( a c − t/ s) s σ and a c = (1 − p c)/ p c. All these expressions are variants of the general scaling idea for second-order phase transitions that a function g( x,y) of two critical variables takes the homogeneous form x cG ( x/y b ) near the critical point, with two free exponents b and c and a scaling function G of a single variable. These assumptions, which may be regarded as generalizations of the Fisher droplet model, are tested “experimentally” by Monte Carlo simulation, series expansion, renormalization group technique, and exact inequalities. In particular, detailed Monte Carlo evidence of Hoshen et al. and Leath and Reich is presented for the scaling of cluster numbers in two and three dimensions. If the cluster size s goes to infinity at fixed concentration p, not necessarily close to p c, three additional exponents ξ, θ, ϱ are defined by: cluster numbers ∝ s − θ exp(−const · s ξ ) and cluster radii ∝ s ϱ. These exponents are different on both sides of the phase transition; for example ξ( p < p c) = 1 and ξ( p > pc) = 1 − 1/ d was found from inequalities, series and Monte Carlo data. The behavior of θ and of ϱ( p < p c) remains to be explained by scaling theory. This article does not cover experimental applications, correlation functions and “classical” (mean field, Bethe lattice, effective medium) theories. For the reader to whom this abstract is too short and the whole article is too long we recommend sections 1 and 3.