TY - GEN A1 - Behrisch, Michael A1 - Coja-Oghlan, Amin A1 - Kang, Mihuyn T1 - Local Limit Theorems and the Number of Connected Hypergraphs N2 - Let H_d(n, p) signify a random d-uniform hypergraph with n vertices in which each of the 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 derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of H_d(n, p) and H_d(n,m). As an application, we obtain an asymptotic formula for the probability that H_d(n, p) is connected, and a corresponding formula for H_d(n,m). In addition, we infer a local limit theorem for the conditional distribution of the number of edges in H_d(n, p) given that H_d(n, p) is connected. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach. KW - random discrete structures KW - giant component KW - local limit theorems KW - connected hypergraphs Y1 - 2007 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/371 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-3711 ER -