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Claim. Corollary 2 of D. Aldous, Brownian excursions, critical random graphs and the multiplicative coalescent, Ann. Probab. 25 (1997), no. 2, 812--854. Fix a real $t$ and let $C_n^t(j)$ be the number of vertices of the $j$-th largest component of the random graph on $n$ vertices in which each edge is present independently with probability $n^{-1}+tn^{-4/3}$. Then the sequence $(n^{-2/3}C_n^t(j))_{j\ge1}$ converges in distribution, in $\ell^2$, to the ordered sequence of excursion lengths, above its running minimum, of the Brownian motion with parabolic drift $W(s)+ts-s^2/2$. At $t=0$ the edge probability is $1/n$, the parameter of Problem 745. The limit's second excursion length is finite and positive almost surely, so $n^{-2/3}L_2$ converges in law to an almost surely positive, finite limit and $L_2=\Theta_{\mathbb P}(n^{2/3})$, where $L_2$ is the number of vertices of the second largest component. This describes the size the problem asks for, so the claim's value is solved. T. Łuczak, B. Pittel and J. C. Wierman, The structure of a random graph at the point of the phase transition, Trans. Amer. Math. Soc. 341 (1994), 721--748, had described the components of order $n^{2/3}$ in the same window.

The statement follows the zbMATH review of the paper (Zbl 0877.60010, by J. Franchi), which gives the main result in this form together with the surplus of each component, and Theorem 1 of L. Addario-Berry, N. Broutin and C. Goldschmidt, The continuum limit of critical random graphs (arXiv:0903.4730), which restates it with the drift $W(t)+t\lambda-t^2/2$ and cites it as Corollary 2 of the paper. The paper is not held, and the page rests on these two statements.

Depends on. Nothing in this wiki.

Acceptance. Refereed: The Annals of Probability 25 (1997), no. 2. The issue is dated April 1997, and this page carries its first day. The Lean developments of Boris Alexeev and Jingxuan Ding reach the same order at the critical point, and Ding's development cites this corollary as its source for the critical window.