In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I understand (both intuitively and from searching around) their results on connectivity still hold if one use the more standard $G(n,p)$ model. I have attempted to generalize their proof ideas to this setting, but was unsuccessful. I was hoping that someone could either sketch or refer me to a proof of their connectivity results in the $G(n,p)$ setting. Ideally this proof would be in the same spirit as the original proof, but I'd accept any paper that works.



A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
The most satisfying solution I've seen so far are [these](https://www.math.wisc.edu/~roch/teaching_files/833.f14/mdp-chap2-web.pdf) notes by Sabastian Roch (see section 2.2.3, and in particular claim 2.25). He first argues that the only components besides the giant one are isolated vertices, and he then shows that there is a positive probability of isolated vertices existing (with a refined estimation being performed as part of the exercises of the notes). 


