Scaling Limits of Fluctuations of Extended-Source Internal DLA

TL;DR

This paper studies the fine-scale fluctuations of the extended-source internal DLA model, showing they converge to Gaussian fields and providing bounds on their size, with implications for related sandpile models.

Contribution

It establishes the convergence of fluctuation error functions to Gaussian fields and extends bounds to the divisible sandpile model, advancing understanding of internal DLA fluctuations.

Findings

Fluctuations converge to geometry-dependent Gaussian fields.

Error functions are $ ext{delta}^{3/5}$-close to their limits.

Bounds on fluctuations apply to related sandpile models.

Abstract

In a previous work, we showed that the 2D, extended-source internal DLA (IDLA) of Levine and Peres is $\delta^{3/5}$-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 $\delta^{3/5}$ bounds on the fluctuations of the related divisible sandpile model of Levine and Peres.