Abstract: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 walk, introduced by Limic in 2019, that encodes the dynamic 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 $(G_n(s))_{s\geq 0}$ 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 walk 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 ($s=\frac{c}{n}$ and $c>1$). Next, to show the versatility of our approach, we prove a functional central limit theorem in the barely super-critical regime ($s=\frac{1+t\epsilon_n}{n}$ where $t>0$ and $(\epsilon_n)_n$ is a sequence of positive reals that converges to 0 such that $(n\epsilon_n^3)_n$ tends to $+\infty$).
From: Josué Corujo Rodríguez [view email]
[v1]
Mon, 9 Dec 2024 21:03:42 UTC (380 KB)
[v2]
Wed, 16 Jul 2025 14:25:00 UTC (190 KB)
[v3]
Tue, 14 Jul 2026 10:52:42 UTC (279 KB)
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