Tree graph inequalities and critical behavior in percolation models.

*(English)*Zbl 0586.60096Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent \(\gamma\) associated with the expected cluster size \(\chi\), and the structure of the n-side connection probabilities \(\tau =\tau_ n(X_ 1,...,X_ n)\). It is shown that quite generally \(\gamma\geq 1\). The upper critical dimension, above which \(\gamma\) 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 \(\tau (x,y)=O(| x- y|^{-(d-2+\eta)}),\) at \(p=p_ c\), our criterion shows that \(\gamma =1\) if \(\eta >(6-d)/3\).

For homogeneous d-dimensional lattices with \(\tau (x,y)=O(| x- y|^{-(d-2+\eta)}),\) at \(p=p_ c\), our criterion shows that \(\gamma =1\) if \(\eta >(6-d)/3\).

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

##### Keywords:

critical exponents; correlation functions; connectivity inequalities; cluster size distribution; critical behavior in independent percolation models
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\textit{M. Aizenman} and \textit{C. M. Newman}, J. Stat. Phys. 36, 107--144 (1984; Zbl 0586.60096)

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##### References:

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