Typical cluster size for two-dimensional percolation processes

Abstract

The typical cluster size for two-dimensional percolation models is discussed. It is shown that, forW ₀={xɛZ ²∶0→x}, [−lim _n→∞(1/n) logP _p (∣W ₀∣=n)]⁻¹≈∣p−p _c ∣^−Δ asp↑p _c , provided thatE _p (∣W ₀∣²)/E _p (∣W ₀∣)≈∣p−P _c ∣^−Δ asp↑p _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 asp→p _c .