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Zbl 0767.60098
Gandolfi, A.; Keane, M.S.; Newman, C.M.
Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. (English)
[J] Probab. Theory Relat. Fields 92, No.4, 511-527 (1992). ISSN 0178-8051; ISSN 1432-2064/e

Summary: We extend the theorem of {\it R. M. Burton} and {\it M. Keane} [Commun. Math. Phys. 121, No. 3, 501-505 (1989; Zbl 0662.60113)] on uniqueness of the infinite component in dependent percolation to cover random graphs on $\bbfZ\sp d$ or $\bbfZ\sp d\times\bbfN$ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on $\bbfZ\sp d$ or on a slab $\bbfZ\sp d\times\{0,\dots,K\}$ and prove both that percolation occurs and that the infinite component is unique for $V=\bbfZ\sp 2\times\{0,1\}$ or larger.

MSC 2000:: *60K35 Interacting random processes
82B43 Percolation

Keywords: dependent percolation; random graphs; nearest-neighbor spin glasses; infinite component

Citations: Zbl 0662.60113

Cited in: Zbl 0827.60080

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