Mean-field avalanche size exponent for sandpiles on Galton-Watson trees. (English) Zbl 1434.60297
Summary: We show that in abelian sandpiles on infinite Galton-Watson trees, the probability that the total avalanche has more than \(t\) topplings decays as \(t^{-1/2}\). We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of B. Morris [ibid. 125, No. 2, 259–265 (2003; Zbl 1031.60035)], that was previously used by R. Lyons et al. [Electron. J. Probab. 13, 1702–1725 (2008; Zbl 1191.60016)] to study uniform spanning forests on \({\mathbb{Z}^d}\), \(d\ge 3\), and other transient graphs.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |
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