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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

Critical Exponent Percolation Theory Percolation Model Cluster Size Distribution Bond Percolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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