The window process of slightly subcritical frozen percolation

Abstract

We study the frozen percolation model on the complete graph $K_n$, where each edge appears at rate~$1/n$ and connected components are destroyed by lightning strikes hitting vertices at rate $λ_n$ (thus the rate at which a connected component is destroyed is linear in its size). We focus on the slightly subcritical sub-regime of the self-organized critical (SOC) regime, assuming $(\ln n)^{4/3}n^{-1/3} \ll λ_n \ll 1$. Our main object of study is the so-called window process, which allows us to study the fine details of the mechanism that drives the SOC behavior of this dynamic random graph. We prove two scaling limit theorems. First, we show that the fluctuations of the window process around its deterministic fluid limit converge to a stationary Ornstein--Uhlenbeck process, whose mean-reverting property captures the SOC nature of the model. Second, by zooming in on a finer space-time scale, we prove that the scaled increments of the window process converge to a tempered stable Lévy process.