How long does it take for Internal DLA to forget its initial profile?

• 成果类型: Article
• 署名作者: Levine, Lionel; Silvestri, Vittoria
• 署名单位: Cornell University; University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0880-7
• 发表日期: 2019
• 页码: 1219-1271
• 关键词: fluctuations

摘要

Internal DLA is a discrete model of a moving interface. On the cylinder graph ZNxZ, a particle starts uniformly on ZNx{0} and performs simple random walk on the cylinder until reaching an unoccupied site in ZNxZ >= 0, which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order N2logN particles. For a lower bound, we show that at least order N2 particles are required to forget which of two independent typical subsets the process started from.