Open Access
2016 Sandpiles and unicycles on random planar maps
Xin Sun, David B. Wilson
Electron. Commun. Probab. 21: 1-12 (2016). DOI: 10.1214/16-ECP4477


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.


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Xin Sun. David B. Wilson. "Sandpiles and unicycles on random planar maps." Electron. Commun. Probab. 21 1 - 12, 2016.


Received: 12 August 2015; Accepted: 25 July 2016; Published: 2016
First available in Project Euclid: 6 September 2016

zbMATH: 1348.82025
MathSciNet: MR3548769
Digital Object Identifier: 10.1214/16-ECP4477

Primary: 05C05 , 60C05 , 82B20

Keywords: abelian sandpile model , cycle-rooted spanning tree , hamburger-cheeseburger bijection , Random planar map

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