A Point Process Describing the Component Sizes in the Critical Window of the Random Graph Evolution

Source abstract

We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G ( n , p ) in the critical window, that is, for p = n −1 + λ n −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 ε n 2/3 ) is almost constant.