In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016. https://doi.org/10.1007/s00023-015-0433-x) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

中文翻译：

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可分沙堆中里程表的标度限制

在最近的一项工作中，莱文等人。(Ann Henri Poincaré 17:1677–1711, 2016. https://doi.org/10.1007/s00023-015-0433-x) 证明有限图上可分沙堆模型的里程表函数可以表示为位移离散双拉普拉斯高斯场。对于离散环面，他们提出了里程表的标度限制可能与连续双拉普拉斯场有关的可能性。在这项工作中，我们表明，在任何维度上，重新缩放的里程表都会收敛到单位环面上的连续双拉普拉斯场。