A novel approach to the giant component fluctuations

We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walks, introduced by Limic in 2019, that encodes the dynamics of the evolution of the sizes of the connected components of a large class of random graph processes. We limit our study to the variant of the Erdős–Rényi graph process ( G n ( s ) ) s ≥ 0 with n vertices where an edge connecting a pair of vertices appears at an exponential rate 1 waiting time, independently over pairs. We first use the properties of the simultaneous breadth-first walks to obtain an alternative and self-contained proof of the functional central limit theorem recently established by Enriquez, Faraud and Lemaire in the super-critical regime ( s = c n and c > 1 ). Next, to show the versatility of our approach, we prove a functional central limit theorem in the barely super-critical regime ( s = 1 + t ϵ n n where t > 0 and ( ϵ n ) n is a sequence of positive reals that converges to 0 such that ( n ϵ n 3 ) n tends to + ∞ ).