  Recently, we adapted random walk arguments based on work of Nachmias and
Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic
normality of the size of the giant component in the random graph $G(n,p)$ above
the phase transition. Here we show that the same method applies to the
analogous model of random $k$-uniform hypergraphs, establishing asymptotic
normality throughout the (sparse) supercritical regime. Previously, asymptotic
normality was known only towards the two ends of this regime.
