HamiltonianCycles in Random Graphs

Question

Lets say we consider the Erdős-Renyi undirected random graph $G(n,p)$ with $V(G) = \{1,2,\cdots,n\}$ and $\displaystyle{P((u,v)\in E(G)) = p} \quad \forall u,v \in V $.

Is there anything we can say about the probability of the $G$ containing a $HAM$ Cycle?

This seems like a helpful quantity to figure out for certain computations

Generating random graphs and seeing what fraction contains a $HAM$ cycle can of-course be done.

What is the rate of increase of $P(G\text{ contains }HAMCYC)$ as $p$ increases?