Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

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...

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Bibliographic Details
Published in:Probability theory and related fields 2020-06, Vol.177 (1-2), p.369-396
Main Authors: Járai, Antal A., Ruszel, Wioletta M., Saada, Ellen
Format: Article
Language:English
Subjects:
Avalanches
Economics
Finance
Insurance
Management
Martingales
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Resistance
Statistics for Business
Theoretical
Trees
Trees (mathematics)
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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 Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z d , d ≥ 3 , and other transient graphs.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-019-00951-z