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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 {\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.

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