Internal Aggregation Models on Comb Lattices

TL;DR

This paper investigates three cluster growth models on the comb lattice, providing shape descriptions for the divisible sandpile and bounds for IDLA and rotor-router aggregation, enhancing understanding of these processes on complex structures.

Contribution

The paper introduces shape results for the divisible sandpile on the comb lattice and derives inner bounds for IDLA and rotor-router aggregation based on these shapes.

Findings

Shape of the divisible sandpile cluster on $C_2$ described.

Inner bounds for IDLA and rotor-router models established.

Enhanced understanding of cluster growth on comb lattices.

Abstract

The two-dimensional comb lattice $C_2$ is a natural spanning tree of the Euclidean lattice $\mathbb{Z}^2$. We study three related cluster growth models on $C_2$: internal diffusion limited aggregation (IDLA), in which random walkers move on the vertices of $C_2$ until reaching an unoccupied site where they stop; rotor-router aggregation in which particles perform deterministic walks, and stop when reaching a site previously unoccupied; and the divisible sandpile model where at each vertex there is a pile of sand, for which, at each step, the mass exceeding 1 is distributed equally among the neighbours. We describe the shape of the divisible sandpile cluster on $C_2$, which is then used to give inner bounds for IDLA and rotor-router aggregation.