Internal DLA on mated-CRT maps

TL;DR

This paper establishes a shape theorem for internal diffusion limited aggregation on mated-CRT maps, linking it to Liouville quantum gravity surfaces and harmonic balls, and extends results to the divisible sandpile.

Contribution

It provides the first shape theorem for internal DLA on mated-CRT maps, connecting discrete models to LQG geometry.

Findings

Proves shape theorem for internal DLA on mated-CRT maps

Identifies the limit shape as an LQG harmonic ball

Extends results to the divisible sandpile model

Abstract

We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is an LQG harmonic ball, which we constructed in a companion paper. We also prove an analogous result for the divisible sandpile.