TITLE: Connectivity of the Erdős–Rényi random graph
QUESTION [54 upvotes]: It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is asymptotically almost surely connected.  The way I know how to prove this is (1) first counting the expected number of components of order $2, 3, \dots, \lfloor n/2 \rfloor$, and seeing that the expected number is tending to zero.  Then (2) showing the expected number of isolated vertices is also tending to zero.
This approach also allows more precise results, such as: if $p = (\log{n}+c) / n$ with $c \in \mathbb{R}$ constant, then Pr$[G(n,p)$ is connected] $\to e^{-e^{-c}}$ as $n \to \infty$, which follows once we know that in this regime the number of isolated vertices is approaching a Poisson distribution with mean $e^{-c}$.

I am wondering if it is possible to
  give an easier proof (of a coarser result) along the
  following lines. There are $n^{n-2}$
  spanning trees on the complete graph,
  and $G$ is connected if and only if
  one of these trees appears.  So the
  expected number of spanning trees
  is $n^{n-2}p^{n-1}$.  One might expect that if this
  function is growing quickly enough,
  then with
  high probability $G(n,p)$ is connected.

I think I remember reading somewhere that this approach doesn't quite work --- for example the variance is too large to apply Chebyshev’s inequality.  What I am wondering is if there is some way to fix this if we are willing to make $p$ a little bit bigger.  In particular, what about $p = C \log{n} / n$ for some large enough constant $C > 1$, or even $p = n^{-1 + \epsilon}$ for fixed but arbitrarily small $\epsilon >0$?

REPLY [7 votes]: this is not really an answer to your question but just a related matter. As you point out the expected number of trees in a random graph is 1 already when p=c/n and is very large when p=logn/n so the hope is that this can be used to show that with a large probability the random graph contains a tree. There is a collection of conjectured by Jeff Kahn and me trying to suggest a very general connection of this type. These conjectures are presented in the pape Thresholds and expectation thresholds by Kahn and me and are mentioned in this MO question. If true these conjectures will imply that the threshold for connectivity will be below logn/n (of course, we do not need it for this case...), and the proof will probably will at best be much much more complicated then existing proofs.
I should mention that the sharp threshold property which was proved by Erdos and Renyi for connectivity can be proved (with harder proofs) from more general principles: One is the Margulis-Talagrand theorem which applies to the threshold for random subgraphs of highly edge connected graphs and one is Friedgut's result which identify graph properties with coarse thresholds.