Show that the phase transition for graph connectivity is at logn/n.
This is a problem that I would like to see a simple proof. The notes below are a significant first step. What needs to be done is to show that components of size two or greater disappear before isolated vertices.
Disappearance of isolated vertices (Lecture 3) threshold is log n n
.
Graph connectivity
1. Graph connected
2. Contains an isolated vertex
3. No isolated vertex but graph not connected
If the probability of (3) goes to zero before (2) then done. Threshold for graph connectivity is same as threshold for disappearance of isolated vertices.
Let x be the number of isolated vertices.
   n

1
 p
 n

1
If p were of the form c ln n n then lim n

 n lim
 n

1
 c ln n n
 n

1  n lim
 ne
 c ln n  lim n
 n
1
 c
If c>1 the expected value goes to zero and Prob[G has an isolated vertex]=0.
If c<1 the expected value goes to infinity and second moment argument say that
Prob[G has an isolated vertex]=1.
Disappearance of components of size k

2
E
 number of components of size k
  n
 

1
 p
 2
The less than or equal came about because we did not require that the k vertices form a connected component.

E
 n k k !

1
 c n ln
 kn  n k k e
 ck ln n  n k k !
n
 ck
For c>1 the expected value goes to zero and thus there are no components of size k.