Random Graph is a good expander

Question

If a (n,d) random graph is a n-vertex graph defined as :

Choose d random permutations $\pi_1 \ldots \pi_d $, from [n] to [n]. Take edge (u,v) if $v = \pi_i(u)$ for some i. I am trying to prove that, for every n and $d \geq 2$, a (n,d) random graph is a (n,2d,1/10) edge expander with probability 1 - o(1).

For a given S such that $|S| \leq n/2$, I have found the expected value of the number of edges |E(S,S')| between S and V\S. But since this random variable is the addition of random variables which are not independent, I cannot apply the Chernoff bound to say that, the probability of |E(S,S')| being far from its expected value is low.

Is there any other way to prove this ?

Answer 1

Try proving directly that with high probability, no set of size at most $n/2$ has a small edge neighborhood. See for example Ellis's writeup for the complete details.

Another possibility is to try and use a more sophisticated large deviation bound to analyze $|E(S,S')|$. For example, there are versions of Chernoff's bound for negatively dependent random variables, and there is Azuma's bound, which is a martingale version of Chernoff's bound. The classical example is concentration of the chromatic number, appearing in Chapter 7 of Alon & Spencer as well as many other places, such as Bukh's write-up.