  Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define
a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u,
v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the
property of containing $\lfloor k/2 \rfloor$ Hamilton cycles, and one perfect
matching if $k$ is odd, all edge-disjoint. With an eigenvalue condition on $P$,
and conditions on its row sums, $G_{n, P}\in \mathcal{A}_k$ happens with high
probability if and only if $G_{n, P}$ has minimum degree $k$ whp. We also
provide a hitting time version. As a special case, the random graph process on
pseudorandom $(n, d, \mu)$-graphs with $\mu \le d(d/n)^\alpha$ for some
constant $\alpha > 0$ has property $\mathcal{A}_k$ as soon as it acquires
minimum degree $k$ with high probability.
