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

  • C. M. Newman
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 8)

Abstract

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| 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| 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.

Keywords

Percolat 

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Copyright information

© Springer-Verlag New York, Inc. 1987

Authors and Affiliations

  • C. M. Newman
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

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