Isolated vertices in the Erdős-Rényi model

Let $N$ be the number of isolated vertices in $G(n,p)$. Indicator variables giveand, because a specified pair is simultaneously isolated precisely when the $2n-3$ incident edges are absent,

For $p=c\log n/n$, the number $N$ of isolated vertices satisfiesThe upper side follows from the first-moment bound. On the lower side, $\mathbb{E}N\to \infty$ and $\mathrm{var}(N)/(\mathbb{E}N{)}^2\to 0$, so the second-moment method applies.