RANDOM PERMUTATIONS WITHOUT MACROSCOPIC CYCLES

• 成果类型: Article
• 署名作者: Betz, Volker; Schaefer, Helge; Zeindler, Dirk
• 署名单位: Technical University of Darmstadt; Lancaster University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1538
• 发表日期: 2020
• 页码: 1484-1505
• 关键词: NUMBER; THEOREM

摘要

We consider uniform random permutations of length n conditioned to have no cycle longer than n(beta) with 0 < beta < 1, in the limit of large n. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.