What's the probability for a random graph with degrees greater than 1 to be Hamiltonian?
The probability that a random graph with $n$ nodes and $cn\log n$ edges contains a Hamiltonian circuit tends to $1$ as $n\rightarrow\infty$ (and for sufficiently large $c$) (Pósa 1976).  Since an ER random graph has $\Omega(n^2)$ edges, it is almost certainly Hamiltonian as $n\rightarrow \infty$, even without the constraint on the minimal degree.