Lower bounds for bootstrap percolation on Galton-Watson trees

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 Bollobas, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.