Sun, Xin; Wilson, David B. Sandpiles and unicycles on random planar maps. (English) Zbl 1348.82025 Electron. Commun. Probab. 21, Paper No. 57, 12 p. (2016). 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. Cited in 2 Documents 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 Keywords:hamburger-cheeseburger bijection; random planar map; abelian sandpile model; cycle-rooted spanning tree PDFBibTeX XMLCite \textit{X. Sun} and \textit{D. B. Wilson}, Electron. Commun. Probab. 21, Paper No. 57, 12 p. (2016; Zbl 1348.82025) Full Text: DOI arXiv Euclid