Connectivity of a graph in the Erdős–Rényi model

In the lecture series on random graphs that I’m watching teacher has made the following statements:
if the probability of a branch between any two vertexes to be present in a graph is this function of number of vertexes: $p(n) = cfrac {log(n)}{n}$, then there are 3 cases depending on the value of $c$:

  1. if $c$ is > 1 then the probability that the graph is connected tends to 1 as $n rightarrow infty $
  2. if $c$ is < 1 then the probability that the graph is connected tends to 0 as $n rightarrow infty $
  3. if $c$ is = 1 then the probability that the graph is connected tends to $exp^{-1}$ as $n rightarrow infty $

The 1st and 2nd ones have been proven, but the 3rd one hasn’t due to its significantly higher difficulty in terms of means needed to carry out the proof. My question is where can I read about this proof and what are the prerequisites for understanding it?