### 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) |
---|---|

Pages (from-to) | 107-143 |

Number of pages | 37 |

Journal | Journal of Statistical Physics |

Volume | 36 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1984 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

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

**Tree graph inequalities and critical behavior in percolation models.** / Aizenman, Michael; Newman, Charles M.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 36, no. 1-2, pp. 107-143. https://doi.org/10.1007/BF01015729

}

TY - JOUR

T1 - Tree graph inequalities and critical behavior in percolation models

AU - Aizenman, Michael

AU - Newman, Charles M.

PY - 1984/7

Y1 - 1984/7

N2 - 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=τn(x1,..., xn). 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=pc, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τn are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τn, 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 τ2(x, y).

AB - 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=τn(x1,..., xn). 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=pc, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τn are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τn, 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 τ2(x, y).

KW - cluster size distribution

KW - connectivity inequalities

KW - correlation functions

KW - critical exponents

KW - Percolation

KW - rigorous results

KW - upper critical dimension

UR - http://www.scopus.com/inward/record.url?scp=0001639861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001639861&partnerID=8YFLogxK

U2 - 10.1007/BF01015729

DO - 10.1007/BF01015729

M3 - Article

VL - 36

SP - 107

EP - 143

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -