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

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 εn2/3 ) is almost constant.

• Type: Journal
• Year: 2007
• Where: CPC
• Authors: Svante Janson, Joel Spencer
• Keywords: Asymptotic Behavior, CPC 2007, Point Process, fixed Real Number