Scaling limits in divisible sandpiles: a Fourier multiplier approach

In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised 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 $(-\mathrm{\Delta }{)}^{-(1+s)}W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.