Probability Theory and Related Fields

, Volume 92, Issue 4, pp 511–527

Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

  • A. Gandolfi
  • M. S. Keane
  • C. M. Newman

DOI: 10.1007/BF01274266

Cite this article as:
Gandolfi, A., Keane, M.S. & Newman, C.M. Probab. Th. Rel. Fields (1992) 92: 511. doi:10.1007/BF01274266


We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤd or ℤd × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤd or on a slab ℤd × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • A. Gandolfi
    • 1
  • M. S. Keane
    • 2
  • C. M. Newman
    • 3
  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  2. 2.Delft University of TechnologyDelftThe Netherlands
  3. 3.Courant Institute of Mathematical SciencesNew YorkUSA

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