Unusually large components in near-critical Erdős-Rényi graphs via ballot theorems

We consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and A is large, \begin{equation*}\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}},\end{equation*} where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows A and $\lambda$ to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.