On the fluctuations of Internal DLA on the Sierpinski gasket graph

TL;DR

This paper investigates the fluctuations of IDLA clusters on the Sierpinski gasket graph, providing bounds on how much the clusters deviate from their known asymptotic shape.

Contribution

It establishes bounds for the fluctuations of IDLA clusters on the Sierpinski gasket, a fractal graph, advancing understanding of their geometric behavior.

Findings

Bounds for cluster fluctuations are established.

The asymptotic shape is confirmed to be a ball in the graph metric.

Quantitative estimates of deviations from the shape are provided.

Abstract

Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph $G$, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.