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Sandpiles and unicycles on random planar maps. (English) Zbl 1348.82025

Summary: 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.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60C05 Combinatorial probability
05C05 Trees
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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