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On the Differentiability of the Number of Clusters Per Vertex in the Percolation Model
Author(s) -
Grimmett G. R.
Publication year - 1981
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-23.2.372
Subject(s) - mathematics , combinatorics , square lattice , differentiable function , vertex (graph theory) , percolation (cognitive psychology) , cluster (spacecraft) , mathematical proof , lattice (music) , percolation threshold , statistical physics , physics , pure mathematics , graph , geometry , quantum mechanics , computer science , electrical resistivity and conductivity , neuroscience , ising model , biology , programming language , acoustics
The number λ( 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 [ p T , p H ]. Also λ′( 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.