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.

DIVISIBLE SANDPILE ON SIERPINSKI GASKET GRAPHS

The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres [Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30(1) (2009) 1–27] as a tool to study internal diffusion limited aggregation. In this work, we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket [Formula: see text]. We show that the shape is a ball in the graph metric of [Formula: see text]. Moreover, we give an exact representation of the odometer function of the divisible sandpile.