Toppling and height probabilities in sandpiles

We study Abelian sandpiles numerically, using exact sampling. Our method uses a combination of Wilson's algorithm to generate uniformly distributed spanning trees, and Majumdar and Dhar's bijection with sandpiles. We study the probability of topplings of individual vertices in avalanches initiated at the centre of large cubic lattices in dimensions d = 2, 3 and 5. Based on these, we estimate the values of the toppling probability exponent in the infinite volume limit in dimensions d = 2, 3, and find good agreement with theoretical results on the mean-field value of the exponent in d ≥ 5. We also study the distribution of the number of waves in 2D avalanches. Our simulation method, combined with a variance reduction idea, lends itself well to studying some problems even in very high dimensions. We illustrate this with an estimation of the single site height probability distribution in d = 32, and compare this to the asymptotic behaviour as d → ∞.