### Abstract

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent γ associated with the expected cluster size x and the structure of the n-site connection probabilities τ=τ_{n}(x_{1},..., x_{n}). It is shown that quite generally γ≥ 1. The upper critical dimension, above which γ attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneous d-dimensional lattices with τ(x, y)=O(|x -y|^{-(d-2+η}), at p=p_{c}, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τ_{n} are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τ_{n}, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function τ_{2}(x, y).

Original language | English (US) |
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Pages (from-to) | 107-143 |

Number of pages | 37 |

Journal | Journal of Statistical Physics |

Volume | 36 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1 1984 |

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### Keywords

- Percolation
- cluster size distribution
- connectivity inequalities
- correlation functions
- critical exponents
- rigorous results
- upper critical dimension

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*36*(1-2), 107-143. https://doi.org/10.1007/BF01015729