Inequalities for γ and Related Critical Exponents in Short and Long Range Percolation

We relate the cluster size distribution P_n(p) at the percolation critical point, p = p|1c|0, to the critical exponent γ (ΣnP_n(p) ≈ |p_c — p|-γ as p ↑ p_c). If P_∞(p_c) > 0 (i.e., if P_∞(p) is discontinuous at p = p_c), then γ ≥ 2. If P_n(p_c) ≈ n-1–1/δ as n → ∞, then γ ≥ 2(1 – 1/δ). Related inequalities are yalid for γ_r (ΣnrP_n(p) ≈ |p_c — p|-γ_rP as p ↑ p_c) and γ_r' (defined analogously as p ↓ p_c) when r > 1/δ: γ_r, γ_r' ≥ 2(r — 1/δ). These results are yalid for Bernoulli site or bond percolation on d-dimensional lattices for any d with p the site or bond occupation probability. They are also valid for long range translation invariant Bernoulli bond percolation with p the occupation probability for bonds of some given length.