Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family 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 Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.
"Lower bounds for bootstrap percolation on Galton-Watson trees." Electron. Commun. Probab. 19 1 - 7, 2014. https://doi.org/10.1214/ECP.v19-3315