The process of fluctuations of the giant component of an Erdős-Rényi graph

We consider the usual increasing process (ER(n,t∕n))0≤t≤n of Erdős-Rényi random graphs, with n vertices and mean degree t. It is well-known that when t>1, the graph ER(n,t∕n) admits a giant component whose number of vertices satisfies a central limit theorem for a fixed t when n goes to infinity. The purpose of this work is to provide a functional central limit theorem for the number of vertices of the giant component as t varies and to identify the covariance of the limiting Gaussian process.