The Mean Field Bound for the Order Parameter of Bernoulli Percolation

We consider a general, translation invariant bond percolation model on ℤd with bonds characterized by couplings (J_x| X∈ℤd) and an inverse temperature parameter tf, with nontrivial critical value v_c. We prove several inequalities including: (Da differential inequality for the infinite cluster density, P_∞(v); and (2) an inequality relating the backbone density, Q_∞(v), to p_∞(v) and the expected size of finite clusters, χ’(v). If the above quantities exhibit critical scaling with exponents “defined” by P_∞(v) ~ |v — v_c|β, Q_∞(v) ~ |v — v_c|β_Q, and χ’(v) ~ |v — v_c|-γ’ as v↓v_c these inequalities imply the mean field bounds: β ≤ 1 and 2β ≤ β_Q≤ β + γ’. Furthermore, a magnetic backbone exponent, δ_Q, is defined analogously to the standard magnetic backbone exponent, δ. Again assuming critical scaling, our inequalities also imply the mean field bounds δ ≥ 2δ_Q and δ_Q ≥ 1.