Local Limit Theorems for the Giant Component of Random Hypergraphs

Exploring hypergraphs with martingales

“…As noted above, the corresponding result with ε = Θ(1) was proved recently by Behrisch, Coja-Oghlan and Kang [3]. Their formula for the quantity corresponding to ρ * r,λ coincides with ours, though the different notation obscures this.…”

“…The corresponding result for ε = Θ(1) was proved recently by Behrisch, Coja-Oghlan and Kang [3], as part of a stronger result, a local limit theorem. Their methods are completely different from ours, and seem very unlikely to adapt to the case ε → 0.…”

“…As in Remark 3 above, one can check that this constant depends only on r and the implicit constant in our assumption ε = O(1). For the largest component, much more precise results are known in the graph case, at least when [15] established a 'large deviation principle' tight up to a factor 1 + o(1) in the exponent in the 3 Suppose not. Then for each k = 1, 2, .…”

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

“…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.…”

“…In other words, while discovering this structure using , exactly one j ‐set would be seen twice.) This would be particularly interesting when aiming for a local limit theorem for the size of the largest j ‐component, because (for j = 1) in both and it proved crucial to closely investigate the interactions of small components with the giant component.…”