A point process describing the component sizes in the critical window of the random graph evolution

Abstract. We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n −1 + λn −4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ε (a scaled version of the number of vertices in components of size greater than εn 2/3) is almost constant. 1