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

## Abstract

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} (Σn^{r}P_{n}(p) ≈ |p_{c} — p|^{-γ} _{r} ^{P} 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.

### Keywords

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