The order of the giant component of random hypergraphs

Abstract

, where−1 is a constant. The proof relies on a new, purely probabilistic approach.

Asymptotic normality of the size of the giant component in a random hypergraph

“…Here we shall prove asymptotic normality throughout the supercritical regime, for all k 3 fixed. Note that our main result, while less precise than those of [8,2], has a much greater range of applicability. The proof is a (to us surprisingly) simple adaptation of the argument we gave for the case k = 2 in [4], itself based on exploration and martingale arguments using ideas of Nachmias and Peres [12], Martin-Löf [11], Karp [9] and Aldous [1].…”

The size of the giant high‐order component in random hypergraphs

“…The size of a j-component denotes the number of j-sets it contains. The case j = 1 is also known as vertex-connectedness, 1 and for j ≥ 2 we use the term high-order connectedness. 2 The case of vertex-connectedness is by far the most studied, not necessarily because it is a more natural definition, but because it is usually substantially easier to understand and analyze.…”

“…Recently, Cooley, Kang, and Person [11], and independently Lu and Peng [18], proved that for all 1 ≤ j ≤ k − 1, the phase transition for the largest j-component in H k (n, p) occurs at the critical probability threshold ofp g =p g (n, k, j) := 1 ( k j )−1 1 ( n k−j ) .…”

“…While the first group already determined the size of the largest j-component up to a multiplicative constant even when ε = o (1), the second group studied only the simpler regime when ε > 0 is a constant. 1 This notion is not to be confused with the (vertex-)connectivity of a (hyper-)graph H measuring the size of the smallest vertex-separator in H. 2 We remark that j-connectedness of a k-uniform hypergraph H is equivalent to vertex-connectedness of the auxiliary k j -uniform hypergraph H � whose vertices are the j-sets of H and with an edge consisting of k j such j-sets if they lie in a common edge of H.…”

Exploring hypergraphs with martingales

“…It is easy to check that λ * = λ(1 − ρ λ ), (2) and that for any A > 1 there exist C > c > 0 such that λ = 1 + ε ∈ (1, A] implies 1 − Cε λ * 1 − cε.…”