On Weak Hamiltonicity of a Random Hypergraph

A {\it weak (Berge) cycle} is an alternating sequence of vertices and (hyper)edges $C=(v_0,e_1,v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_{\mathrm{\ell }-1},e_{\mathrm{\ell }},v_{\mathrm{\ell }}=v_0)$ such that the vertices $v_0,\mathrm{.}\mathrm{.}\mathrm{.},v_{\mathrm{\ell }-1}$ are distinct with $v_k, v_{k+1} \in e_{k}$foreach$k$,buttheedges$e_1, …, e_\ell$ are not necessarily distinct. We prove that the main barrier to the random $d$-uniform hypergraph $H_d(n,p),$ where each of the potential edges of cardinality $d$ is present with probability $p$, developing a weak Hamilton cycle is the presence of isolated vertices. In particular, for $d\ge 3$ fixed and $p=(d-1)! \frac{\ln n + c}{n^{d-1}}$,theprobabilitythat$H_d(n, p)$ has a weak Hamilton cycle tends to $e^{-e^{-c}}$, which is also the limiting probability that $H_d(n,p)$ has no isolated vertices. As a consequence, the probability that the random hypergraph $H_d(n,m=\frac{n(\ln n+c)}{d}),$ where $m$ potential edges are chosen uniformly at random to be present, is weak Hamiltonian also tends to $e^{-e^{-c}}$.