Local Limit Theorems for the Giant Component of Random Hypergraphs?

摘要

Let Hd(n, p) signify a random d-uniform hypergraph with n vertices in which each of the (nd) possible edges is present with probability p = p(n) independently, and let H_d(n,m) denoted 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 Hd(n,m) in the regime (d ? 1) p > 1 + ε, resp.d(d ? 1)m > 1 + ε, where ε > 0 is arbitrarily small but fixed as n→∞. The proofs are based on a purely probabilistic approach.