  We consider the abelian sandpile model and the uniform spanning unicycle on
random planar maps. We show that the sandpile density converges to 5/2 as the
maps get large. For the spanning unicycle, we show that the length and area of
the cycle converges to the hitting time and location of a simple random walk in
the first quadrant. The calculations use the "hamburger-cheeseburger"
construction of Fortuin--Kasteleyn random cluster configurations on random
planar maps.
