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 → ∞.

We give an asymptotic formula for the single site height distribution of Abelian sandpiles on Z d as d → ∞, in terms of Poisson(1) probabilities. We provide error estimates.

We consider a simple random walk on Z d started at the origin and stopped on its first exit time from (−L, L) d ∩ Z d . Write L in the form L = mN with m = m(N ) and N an integer going to infinity in such a way that L 2 ∼ AN d for some real constant A > 0. Our main result is that for d ≥ 3, the projection of the stopped trajectory to the N -torus locally converges, away from the origin, to an interlacement process at level Adσ 1 , where σ 1 is the exit time of a Brownian motion from the unit cube (−1, 1) d that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) andSznitman (2009).

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