The frog model on the rooted $d$-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.
"The critical density for the frog model is the degree of the tree." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP29