Let $mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,sgeq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in $mathcal{G}(n,r,s)$. Finally we prove that for $ell = 2,ldots, s-1$ and for $r$ growing moderately as $ntoinfty$, the probability that $mathcal{G}(n,r,s)$ has a $ell$-overlapping Hamilton cycle tends to zero.
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