Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P_∞(p) is discontinuous at p_c, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nP_n(p) ∼ (P_c-P)^-γ as p ↑p_c) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓p_c) and δ [P_n(p_c) ∼ (n^-1-1/δ) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, P_x has no discontinuity at p_c.