Inequalities for critical exponents in d-dimensional sandpiles

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.