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

• Published in: Probability theory and related fields Vol. 177; no. 1-2; pp. 369 - 396
• Main Authors: Járai, Antal A., Ruszel, Wioletta M., Saada, Ellen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2020; Springer Nature B.V; Springer Verlag
• 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); Uniform spanning tree; 82C20; Conductance martingale; 60K35; Abelian sandpile; Wired spanning forest
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-019-00951-z

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.