Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits

Abstract

In \cite{Cipriani2016}, the authors proved that with the appropriate rescaling, the odometer of the (nearest neighbours) Divisible Sandpile in the unit torus converges to the bi-Laplacian field. Here, we study $\alpha$-long-range divisible sandpiles similar to those introduced in \cite{Frometa2018}. We obtain that for $\alpha \in (0,2)$, the limiting field is a fractional Gaussian field on the torus. However, for $\alpha \in [2,\infty)$, we recover the bi-Laplacian field. The central tool for our results is a careful study of the spectrum of the fractional Laplacian in the discrete torus. More specifically, we need the rate of divergence of such eigenvalues as we let the side length of the discrete torus goes to infinity. As a side result, we construct the fractional Laplacian built from a long-range random walk. Furthermore, we determine the order of the expected value of the odometer on the finite grid. \end{abstract}Comment: 35 pages, 4 figure