Electronic Communications in Probability

Sandpiles and unicycles on random planar maps

Xin Sun and David B. Wilson

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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.

Article information

Electron. Commun. Probab. Volume 21 (2016), paper no. 57, 12 pp.

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

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Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60C05: Combinatorial probability 05C05: Trees

hamburger-cheeseburger bijection random planar map abelian sandpile model cycle-rooted spanning tree

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Sun, Xin; Wilson, David B. Sandpiles and unicycles on random planar maps. Electron. Commun. Probab. 21 (2016), paper no. 57, 12 pp. doi:10.1214/16-ECP4477. https://projecteuclid.org/euclid.ecp/1473186614.

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