Average height for Abelian sandpiles and the looping constant on Sierpiński graphs

For the Abelian sandpile model on Sierpiński graphs, we investigate several statistics such as average height, height probabilities and looping constant. In particular, we calculate the expected average height of a recurrent sandpile on the finite iterations of the Sierpiński gasket and we also give an algorithmic approach for calculating the height probabilities of recurrent sandpiles under stationarity by using the connection between recurrent configurations of the Abelian sandpile Markov chain and uniform spanning trees. We also calculate the expected fraction of vertices of height i for i ∈ { 0 , 1 , 2 , 3 } of sandpiles under stationarity and relate the bulk average height to the looping constant on the Sierpiński gasket.

Asymptotic Height Distribution in High-Dimensional Sandpiles

We give an asymptotic formula for the single-site height distribution of Abelian sandpiles on $$\mathbb {Z}^d$$ Z d as $$d \rightarrow \infty $$ d → ∞ , in terms of $$\mathsf {Poisson}(1)$$ Poisson ( 1 ) probabilities. We provide error estimates.

Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach

In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054–3081, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalized Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $$(-\varDelta )^{-s/2} W$$ ( - Δ ) - s / 2 W for $$s>2$$ s > 2 and W a spatial white noise on the d-dimensional unit torus.

Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as $$t^{-1/2}$$t-1/2. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on $${\mathbb {Z}^d}$$Zd, $$d\ge 3$$d≥3, and other transient graphs.