Given a hypergraph , the -bootstrap process starts with an initial set of infected vertices of and, at each step, a healthy vertex v becomes infected if there exists a hyperedge of in which v is the unique healthy vertex. We say that the set of initially infected vertices percolates if every vertex of is eventually infected. We show that this process exhibits a sharp threshold when is a hypergraph obtained by randomly sampling hyperedges from an approximately d-regular r-uniform hypergraph satisfying some mild degree and codegree conditions; this confirms a conjecture of Morris. As a corollary, we obtain a sharp threshold for a variant of the graph bootstrap process for strictly 2-balanced graphs which generalises a result of Korándi, Peled and Sudakov. Our approach involves an application of the differential equations method.
Parts of this work were completed while the second author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648509) and by the Leverhulme Trust Early Career Fellowship ECF-2018-534.
This work was initiated while the authors were visiting Rob Morris at IMPA in 2016. We are grateful to Rob and IMPA for their hospitality and for providing a stimulating research environment. We would also like to thank Oliver Riordan for his careful reading of the proof as part of the first author’s DPhil thesis and for helpful comments regarding the exposition and presentation of the results. We would also like to thank an anonymous referee for reading the paper thoroughly and proposing several improvements and corrections.
"A sharp threshold for bootstrap percolation in a random hypergraph." Electron. J. Probab. 26 1 - 85, 2021. https://doi.org/10.1214/21-EJP650