On the Cover Time of the Emerging Giant

Abstract

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$, then with high probability (w.h.p.) $G_{n,p}$ has a unique giant largest component. We show that if in addition, $\varepsilon=\varepsilon(n)\to 0$, then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias, and Peres had shown this up to constant multiplicative factors.