On a d-ary tree, place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root.
Awake frogs perform simple random walk and wake any "sleepers" they encounter.
A longstanding open problem: Does every frog wake up?
It turns out this depends on d and the number of frogs.
The proof uses two different recursions and two different versions of stochastic domination.
Joint work with Christopher Hoffman and Tobias Johnson.