Lower bounds for bootstrap percolation on Galton-Watson trees

Karen Gunderson; Michał Przykucki

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arXiv · 1402.4462

Lower bounds for bootstrap percolation on Galton-Watson trees

Karen Gunderson·Michał Przykucki

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Abstract

Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.

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Karen Gunderson, Michał Przykucki. 2014-02-18. Lower bounds for bootstrap percolation on Galton-Watson trees. https://arxiv.org/abs/1402.4462

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