ON THE CYCLE STRUCTURE OF MALLOWS PERMUTATIONS

• 成果类型: Article
• 署名作者: Gladkich, Alexey; Peled, Ron
• 署名单位: Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1202
• 发表日期: 2018
• 页码: 1114-1169
• 关键词: transition; MODEL

摘要

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation pi is an element of S-n is proportional to q(inv(pi)) where q > 0 and inv(pi) is the number of inversions in pi. We focus on the case that q < 1 and show that the expected length of the cycle containing a given point is of order min{(1 - q)(-2), n}. This marks the existence of two asymptotic regimes: with high probability, when n tends to infinity with (1 - q)(-2) << n then all cycles have size o(n) whereas when n tends to infinity with (1 - q)(-2) >> n then macroscopic cycles, of size proportional to n, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson-Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices. Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation.