A geometric approach to conjugation-invariant random permutations

Abstract

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 $2\sqrt{n}$ asymptotic. For Robinson--Schensted shapes, we prove universality of the Vershik--Kerov--Logan--Shepp limit curve, 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.

37 pages. V4: Section 5 has been revised to include more details

References in corpus (9)

• Random permutations with cycle weights
• Stein couplings for normal approximation
• Cycle structure of random permutations with cycle weights
• A Central Limit Theorem for Vincular Permutation Patterns
• Monotonous subsequences and the descent process of invariant random permutations
• Characters of local and regular permutation statistics
• A note on monotone subsequences and the RS image of invariant random permutations with macroscopic number of fixed points
• About universality of large deviation principles for conjugacy invariant permutations
• Asymptotic normality of pattern counts in conjugacy classes