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 study $\alpha$-long-range divisible sandpiles, similar to those introduced in Frómeta and Jara (2018). We show that, for $\alpha \in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $\alpha ∕2$. However, for $\alpha \in [2,\infty )$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Lévy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter $\gamma =min\{\alpha ,2\}$ and $\alpha \in {\mathbb{R}}_{+}\setminus \left\{2\right\}$ on a finite grid.

在 Cipriani 等人。(2017)，作者证明，通过适当的重新缩放，单位环面上（最近的邻居）可分沙堆的里程表会聚到双拉普拉斯场。在这里，我们学习$\alpha$-长程可分沙堆，类似于 Frómeta 和 Jara（2018）中引入的沙堆。我们证明，对于$\alpha \in (0,2)$，限制场是环面上的分数高斯场，参数为 $\alpha ∕2$. 然而，对于$\alpha \in [2,\infty )$，我们恢复双拉普拉斯场。这提供了分数高斯场的替代构造，例如使用基于 Lévy 游走生成器的扩散的高斯自由场或膜模型。获得我们结果的核心工具是仔细研究离散环面上的分数拉普拉斯算子的频谱。更具体地说，我们需要特征值的发散率，因为我们让离散环的边长趋于无穷大。作为附带的结果，我们获得了离散分数拉普拉斯算子的特征值的精确渐近线。此外，我们用参数确定离散分数高斯场的预期最大值的顺序$\gamma =分钟\{\alpha ,2\}$ 和 $\alpha \in {\mathbb{电阻}}_{+}\setminus \left\{2\right\}$ 在有限网格上。