FORMATION OF LARGE-SCALE RANDOM STRUCTURE BY COMPETITIVE EROSION

• 成果类型: Article
• 署名作者: Ganguly, Shirshendu; Levine, Lionel; Sarkar, Sourav
• 署名单位: University of California System; University of California Berkeley; Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1342
• 发表日期: 2019
• 页码: 3649-3704
• 关键词: 1st passage percolation; asymptotic-behavior; internal dla; fluctuations; densities

摘要

We study the following one-dimensional model of annihilating particles. Beginning with all sites of Z uncolored, a blue particle performs simple random walk from 0 until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from 0 until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after n blue and n red particles alternately perform such walks, the total number of colored sites is of order n(1/4). The resulting random color configuration, after rescaling by n(1/4) and taking n -> infinity, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).