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 exit location and exit time 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.
"Sandpiles and unicycles on random planar maps." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP4477