The threshold probability for long cycles

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k + \omega_k(1)}{k}$, where $\omega_k(1)$ is any function tending to infinity with $k$, then $G_p$ asymptotically almost surely contains a cycle of length at least $k+1$. When we take $G$ to be the complete graph on $k+1$ vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.