The probability of unusually large components in the near-critical Erd\H{o}s-R\'enyi graph

The largest components of the critical Erd\H{o}s-R'enyi graph, $G(n,p)$ with $p=1\mathrm{/}n$, have size of order $n^{2\mathrm{/}3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size $a{n}^{2\mathrm{/}3}$ for large $a$. Our results, which extend work of Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$ around $1\mathrm{/}n$. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.