THE LONGEST EDGE OF THE RANDOM MINIMAL SPANNING TREE

• 成果类型: Article
• 署名作者: Penrose, Mathew D.
• 署名单位: Durham University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 1997
• 页码: 340-361
• 关键词: nearest-neighbor link; limit distribution; percolation; outliers; number

摘要

For n points placed uniformly at random on the unit square, suppose M-n (respectively, M-n(1)) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of n pi M-n(2) - log n converges weakly to the double exponential; we give a new proof of this. We show that P[M-n(1) = M-n] -> 1, so that the same weak convergence holds for M-n(1).