2024/11/20 by Byron Chin, Chin, Byron
Mathematics · #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2411.13452
openalex publication_date 2024/11/20 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28
For positive integers r > ℓ ≥ 1, an ℓ-cycle in an r-uniform hypergraph is a cycle where each edge consists of r vertices and each pair of consecutive edges intersect in ℓ vertices. For ℓ ≥ 2, we determine the limiting distribution of the number of Hamilton ℓ-cycles in an Erdős--Rényi random hypergraph. The behavior is distinguished in two cases: -When ℓ ≥ 3, the number of cycles concentrates when the expectation diverges and converges to a Poisson distribution when the expectation is constant. -When ℓ = 2, the normalized number of cycles converges to a lognormal distribution when the expectation diverges and converges to a lognormal mixture of Poisson distributions when the expectation is constant. As a result we pin down the exact threshold for the appearance of non-linear Hamilton cycles in random hypergraphs, confirming a conjecture of Narayanan and Schacht.