The window process of slightly subcritical frozen percolation
Abstract
We study the frozen percolation model on the complete graph Kn, where each edge appears at rate~1/n and connected components are destroyed by lightning strikes hitting vertices at rate λn (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)4/3n−1/3 ≪ λn ≪ 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.
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BibTeX
@misc{rath2026window,
title = {{The window process of slightly subcritical frozen percolation}},
author = {Balázs Ráth and Márton Szőke and Dominic Yeo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37320},
eprint = {2610.37320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37320},
}
RIS
TY - UNPB
TI - The window process of slightly subcritical frozen percolation
AU - Balázs Ráth
AU - Márton Szőke
AU - Dominic Yeo
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37320
UR - https://arxiv.org/abs/2610.37320
AB - 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)^4/3n^−1/3 ≪ λ_n ≪ 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.
ER -