Scaling limit of the odometer of correlated Gaussians on the torus

Abstract

In the divisible sandpile model, we consider a collection of i.i.d. Gaussian heights on a finite graph. It was shown by Levine et al. (2015) that the odometer function in this case equals a discrete, bi-Laplacian field. Subsequently, Cipriani et al. (2016) proved that the scaling limit of the odometer is a continuum bi-Laplacian field, this time on the unit torus. In this thesis, we will determine the scaling limit of a divisible sandpile with an initial configuration of correlated Gaussians, where the covariance is given by a stationary covariance function K(x-y). We show that after appropriate scaling, the odometer still converges to a bi-Laplacian field on the unit torus.