Inequalities for critical exponents in d-dimensional sandpiles

Antal Jarai, Jack Hanson, Sandeep Bhupatiraju

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

Consider the Abelian sandpile measure on Z d   , d≥2  , obtained as the L→∞  limit of the stationary distribution of the sandpile on [−L,L] d ∩Z d   . When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In d=2,  we show that for any 1≤k<∞  , the last k  waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the tail of the radius distribution.
Original languageEnglish
Article number85
Number of pages51
JournalElectronic Journal of Probability
Volume22
DOIs
Publication statusPublished - 14 Oct 2017

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