Constructing fractional Gaussian fields from long-range divisible sandpiles on the torus

In Cipriani et al. (2017), the authors proved that, with the appropriate rescaling, the odometer of the(nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we studyα-long-range divisible sandpiles, similar to those introduced in Frómeta and Jara (2018). We show that,forα∈(0,2), the limiting field is a fractional Gaussian field on the torus with parameterα/2. However,forα∈[2,∞), we recover the bi-Laplacian field. This provides an alternative construction of fractionalGaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on thegenerator of Lévy walks. The central tool for obtaining our results is a careful study of the spectrumof the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence ofthe eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtainprecise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine theorder of the expected maximum of the discrete fractional Gaussian field with parameterγ=min{α,2}andα∈R+\{2}on a finite grid