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Journal of the London Mathematical Society 1981 s2-23(2):372-384; doi:10.1112/jlms/s2-23.2.372
© 1981 by London Mathematical Society
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© Oxford University Press

On the Differentiability of the Number of Clusters Per Vertex in the Percolation Model

G. R. Grimmett

School of Mathematics, University of Bristol, University Walk Bristol, BS8 1TW

The number {lambda}(p) of clusters per vertex of the vertex percolation process on the two-dimensional square lattice is once differentiable for all p and is infinitely differentiate except possibly on the interval [pT, pH]. Also {lambda}'(p) may be expressed in terms of the mean number of black clusters containing vertices adjacent to the white origin. Easy proofs are given of two theorems concerning the boundary sizes of the black cluster containing the origin and the infinite black cluster, when it exists. A central limit theorem is established for the latter quantity. Similar results may be established for certain other two-dimensional lattices.


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