Inequalities for γ and Related Critical Exponents in Short and Long Range Percolation
We relate the cluster size distribution Pn(p) at the percolation critical point, p = p|1c|0, to the critical exponent γ (ΣnPn(p) ≈ |pc — p|-γ as p ↑ pc). If P∞(pc) > 0 (i.e., if P∞(p) is discontinuous at p = pc), then γ ≥ 2. If Pn(pc) ≈ n-1–1/δ as n → ∞, then γ ≥ 2(1 – 1/δ). Related inequalities are yalid for γr (ΣnrPn(p) ≈ |pc — p|-γ r P as p ↑ pc) and γ r ' (defined analogously as p ↓ pc) when r > 1/δ: γr, 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.
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