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.
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- Department of Mathematical Sciences - Senior Lecturer
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
- Probability Laboratory at Bath
Person: Research & Teaching