Some critical exponent inequalities for percolation

Abstract

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP ∞(p) is discontinuous atp c , then the critical exponentγ (defined by the divergence of expected cluster size, ∑nP n (p) ∼ (P c −P)−γ asp ↑p c ) must satisfyγ ≥ 2. (2)γ orγ′ (defined analogously toγ, but asp ↓p c ) and δ [P n (p c ) ∼ (n −1−1/δ) asn → ∞ ] must satisfyγ,γ′ ≥ 2(1 − 1/δ). These inequalities forγ improve the previously known boundγ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P x has no discontinuity atp c .