Exceptional times of the critical dynamical Erdős-Rényi graph (Q1617126)

An Erdős-Rényi graph \(\mathrm{ER}(n,p)\) is a random graph on \(n\) vertices \(1,\ldots, n\) in which each pair of vertices is connected by an edge with probability \(p\), independently of all other pairs of vertices. Starting with a critical Erdős-Rényi graph \(\mathrm{ER}(n,1/n)\) the authors consider a process of graphs \((G_{t}:t\in [0,1])\) which evolves forward in time by resampling each edge independently at rate 1. They show that the size of the largest connected component appearing during the time interval \([0,1]\) is of order \(n^{2/3}\log^{1/3}n\) with high probability. In contrast, for each fixed \(t \geq 0\), \(G_{t}\) is a realisation of \(\mathrm{ER}(n,1/n)\) and its largest connected component is of order \(n^{2/3}\).