  In this paper we give a simple new proof of a result of Pittel and Wormald
concerning the asymptotic value and (suitably rescaled) limiting distribution
of the number of vertices in the giant component of $G(n,p)$ above the scaling
window of the phase transition. Nachmias and Peres used martingale arguments to
study Karp's exploration process, obtaining a simple proof of a weak form of
this result. We use slightly different martingale arguments to obtain a much
sharper result with little extra work.
