Internal Diffusion Limited Aggregation on Discrete Groups Having Exponential Growth

Abstract

The internal diffusion limited aggregation (DLA) has been introduced by Diaconis and Fulton [Rend. Sem. Mat. Univ. Pol. Torino 49, 95–119 (1991)]. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We prove a shape theorem for the internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.