Abstract
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.
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Behrisch, M., Coja-Oghlan, A., Kang, M. (2007). Local Limit Theorems for the Giant Component of Random Hypergraphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_25
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DOI: https://doi.org/10.1007/978-3-540-74208-1_25
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