Internal DLA and the Stefan problem

• 成果类型: Article
• 署名作者: Gravner, J; Quastel, J
• 署名单位: University of California System; University of California Davis; University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1528-1562
• 关键词: fluctuations

摘要

Generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as Poisson sources of particles which then perform symmetric random walks with an attractive zero-range interaction until they reach the first site which has been visited by fewer than a particles, at which point they stop. Sites on which particles are frozen constitute the occupied set. We prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one-phase Stefan problem. This is then used to study the asymptotic behavior of the occupied set. In two dimensions when the walks are independent with one source at the origin and alpha = 1, we obtain in particular that the occupied set is asymptotically a disc of radius K roott, where K is the solution of exp(-K-2/4) = piK(2), Settling a conjecture of Lawler, Bramson and Griffeath.