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Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. (English) Zbl 0767.60098
Summary: We extend the theorem of R. M. Burton and 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 d or d × with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on d or on a slab d ×{0,,K} and prove both that percolation occurs and that the infinite component is unique for V= 2 ×{0,1} or larger.

MSC:
60K35Interacting random processes; statistical mechanics type models; percolation theory
82B43Percolation (equilibrium statistical mechanics)
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