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

We consider the near-critical Erdos-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+λ/n4/3 and A is large, {equation presented} where Cmax is the largest connected component of the graph. Our result allows A and λ 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 Erdos-Rényi graphs, together with analytic estimates.