Scaling Limits of Fluctuations of Extended-Source Internal DLA

Abstract: In a previous work, we showed that the 2D, extended-source internal DLA (IDLA) of Levine and Peres is $\delta<sup>{3/5}$-close to its scaling limit, if $\delta$ 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 $\delta<sup>{3/5}$ bounds on the fluctuations of the related divisible sandpile model of Levine and Peres.