Exact threshold and limiting distribution for non-linear Hamilton cycles

TL;DR

This paper determines the precise threshold and distribution for the appearance of non-linear Hamilton cycles in Erdős–Rényi hypergraphs, revealing different behaviors depending on cycle parameters and confirming a conjecture.

Contribution

It provides the exact threshold and limiting distribution for non-linear Hamilton cycles in random hypergraphs, extending understanding of their probabilistic properties.

Findings

Distribution converges to Poisson for    when expectation diverges.

Normalized number of cycles converges to lognormal for  = 2.

Pinpoints the exact threshold for the emergence of non-linear Hamilton cycles.

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 Erd\H{o}s--R\'enyi 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.