Let H d (n,p) signify a random d-uniform hypergraph with n vertices in which each of the \({n}\choose{d}\) possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We establish a local limit theorem for the number of vertices and edges in the largest component of H d (n,p) in the regime, thereby determining the joint distribution of these parameters precisely. As an application, we derive an asymptotic formula for the probability that H d (n,m) is connected, thus obtaining a formula for the asymptotic number of connected hypergraphs with a given number of vertices and edges. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.


Joint Distribution Connected Graph Random Graph Large Component Random Structure 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Michael Behrisch
    • 1
  • Amin Coja-Oghlan
    • 2
  • Mihyun Kang
    • 1
  1. 1.Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 BerlinGermany
  2. 2.Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213USA

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