Local Limit Theorems for the Giant Component of Random Hypergraphs†. (May 2014) - BEHRISCH, MICHAEL; COJA-OGHLAN, AMIN; KANG, MIHYUN
<x content-type="archive" xml:space="preserve">Abstract</x> Let Hd(n, p) signify a random d-uniform hypergraph with n vertices in which each of the <alternatives><inline-graphic xlink:href="ark:/27927/pgg5h7wzgsd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\binom{n}{d}$]]> </alternatives> possible edges is present with probability p=p(n) independently, and let Hd(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 Hd(n, p) and Hd(n, m) in the regime <alternatives><inline-graphic xlink:href="ark:/27927/pgg5h7wzgw2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$(d-1)\binom{n-1}{d-1}p>1+\varepsilon$]]> </alternatives>, resp. d(d−1)m/n>1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach.