Local Limit Theorems for the Giant Component of Random Hypergraphs

Authors:
Michael Behrisch;Amin Coja-Oghlan;Mihyun Kang
Affiliations:
Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany;Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213, USA;Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany
Venue:
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Year:
2007

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Abstract

Let Hd(n,p) signify a random d-uniform hypergraph with nvertices in which each of the ${n}\choose{d}$ possible edges is present with probability p= p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with nvertices and medges. We establish a local limit theoremfor the number of vertices and edges in the largest component of Hd(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 Hd(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.