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 language  English 

Article number  85 
Number of pages  51 
Journal  Electronic Journal of Probability 
Volume  22 
DOIs  
Publication status  Published  14 Oct 2017 
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Antal Jarai
 Department of Mathematical Sciences  Senior Lecturer
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching