In a previous work, we showed that the 2D, extended-source internal DLA (IDLA) of Levine and Peres is δ^3/5-close to its scaling limit, if δ is the lattice size. In this paper, we investigate the scaling limits of the fluctuations themselves. Namely, we show that two naturally defined error functions, which measure the “lateness” of lattice points at one time and at all times, respectively, converge to geometry-dependent Gaussian random fields. We use these results to calculate point-correlation functions associated with the fluctuations of the flow. Along the way, we demonstrate similar δ^3/5 bounds on the fluctuations of the related divisible sandpile model of Levine and Peres, and we generalize the results of our previous work to a larger class of extended sources.

在之前的工作中，我们表明 Levine 和 Peres 的二维扩展源内部 DLA (IDLA) 为δ ^3/5 - 接近其缩放极限（如果δ是晶格尺寸）。在本文中，我们研究了波动本身的尺度限制。也就是说，我们证明了两个自然定义的误差函数，分别测量某一时刻和所有时刻格点的“迟到性”，收敛于几何相关的高斯随机场。我们使用这些结果来计算与流量波动相关的点相关函数。一路上，我们展示了类似的δ ^3/5Levine 和 Peres 的相关可分沙堆模型的波动范围，并且我们将之前工作的结果推广到更大类别的扩展源。