FROM LOGARITHMIC TO SUBDIFFUSIVE POLYNOMIAL FLUCTUATIONS FOR INTERNAL DLA AND RELATED GROWTH MODELS

• 成果类型: Article
• 署名作者: Asselah, Amine; Gaudilliere, Alexandre
• 署名单位: Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP762
• 发表日期: 2013
• 页码: 1115-1159
• 关键词: diffusion-limited aggregation

摘要

We consider a cluster growth model on Z(d), called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d >= 2. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachere in our Appendix) on the expected time spent by a random walk inside an annulus.