Diameter of a random graph


I’m considering the standard Erdös/Renyi $ G(n,p)$ model where we have $ n$ nodes and each possible edge is sampled independently with probability $ p = \frac{1}{n^\epsilon}$ .

It is relatively straightforward to show that, starting from any node $ u$ , the expected number of hops to reach every other node is $ 1/\epsilon$ . However, this does not say much about the probability of having a diameter of $ 1/\epsilon$ . (I could apply Markov’s inequality, but that gives a rather weak bound.)

Thus I’m looking for a reference to a concentration result that states that the diameter of such a random graph is $ O(1/\epsilon)$ with probability at least $ 1 – o(1)$ .