Exact threshold and limiting distribution for non-linear Hamilton cycles
Byron Chin
TL;DR
The paper resolves the sharp threshold and distributional behavior for non-linear Hamilton $\ell$-cycles in random $r$-uniform hypergraphs. It combines a refined second-moment analysis with small subgraph conditioning to isolate the impact of dense clusters of short subgraphs, yielding a lognormal limit for $\ell=2$ and Poisson-type limits in the diverging-mean and constant-mean regimes. The results confirm Narayanan and Schacht's conjecture, pin down the exact threshold when $\mathbb{E}[Z]$ tends to infinity, and provide precise distributional limits across regimes, advancing the understanding of spanning substructures in random hypergraphs. The work employs planted and double-planted models, Gaussian limits for subgraph counts, and explicit constants $A_k$ arising from path overlaps, delivering sharp threshold results and distributional characterizations with potential applications to related combinatorial structures.
Abstract
For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the limiting distribution of the number of Hamilton $\ell$-cycles in an Erdős--Rényi random hypergraph. The behavior is distinguished in two cases: -When $\ell \geq 3$, the number of cycles concentrates when the expectation diverges and converges to a Poisson distribution when the expectation is constant. -When $\ell = 2$, 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.