Scaling limits in divisible sandpiles: a Fourier multiplier approach

TL;DR

This paper investigates the scaling limits of the odometer in divisible sandpiles on d-dimensional tori, using a Fourier multiplier approach to characterize the resulting Gaussian fields, including fractional Gaussian fields.

Contribution

It extends previous work by relaxing independence assumptions and introduces a Fourier multiplier method to generate and analyze generalized Gaussian fields in the scaling limit.

Findings

Recovered fractional Gaussian fields via Fourier multipliers

Extended the class of Gaussian fields in sandpile scaling limits

Provided a new analytical approach to study divisible sandpiles

Abstract

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 $(-\Delta)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.