A geometric approach to conjugation-invariant random permutations

• 成果类型: Article
• 署名作者: Dubach, Victor
• 署名单位: Uppsala University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01449-7
• 发表日期: 2026-02
• 页码: 245-298
• 关键词: Random permutations; Conjugation-invariance; Longest increasing subsequences; Robinson-Schensted correspondence; records; patterns; LONGEST INCREASING SUBSEQUENCE; central-limit-theorem; plancherel measure; random partitions; large deviations; random-variables; statistics; number; sums

摘要

We propose a new approach to conjugation-invariant random permutations. Namely, we explain how to construct uniform permutations in given conjugacy classes from certain point processes in the plane. This enables the use of geometric tools to study various statistics of such permutations. For their longest decreasing subsequences, we prove universality of the 2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\sqrt{n}$$\end{document} asymptotic. For Robinson-Schensted shapes, we prove universality of the Vershik-Kerov-Logan-Shepp limit shape, thus solving a conjecture of Kammoun. For the number of records, we establish a phase transition phenomenon as the number of fixed points grows. For pattern counts, we obtain an asymptotic normality result, partially answering a conjecture of Hamaker and Rhoades.