A new relationship between Erdős-Rényi graphs, epidemic models and Brownian motion with parabolic drift

Abstract

In the Reed-Frost model, an example of an SIR epidemic model, one can examine a statistic that counts the number of concurrently infected individuals. This statistic can be reformulated as a statistic on the \ER random graph $G(n,p)$. Within the critical window of Aldous and Martin-Löf, i.e. when $p=p(n)=n^{-1}+\lambda n^{-4/3}$, the cumulative sum of this statistic converges weakly to the integral of a Brownian motion with parabolic drift. This same statistic exhibits a deterministic scaling limit when $p=(1+\lambda {\epsilon }_n)/n$ whenever ${\epsilon }_n\to 0$ and $n^{1/3}{\epsilon }_n\to \infty$.

arXiv comment25 pages, Corollary 3.4 is removed and the proof of Theorem 1.1 is changed