A note on monotone subsequences and the RS image of invariant random permutations with macroscopic number of fixed points

TL;DR

This paper investigates the limiting shapes of invariant random permutations with many fixed points, extending classical results on Young diagram shapes and exploring Tracy-Widom universality for monotone subsequences.

Contribution

It extends the understanding of limiting shapes for permutations with many fixed points and enhances Tracy-Widom universality results for monotone subsequences.

Findings

Limiting shape is a scaled version of Vershik-Kerov-Logan-Shepp shape under certain conditions.

Identifies regimes similar to recent work by Chapuy, Louf, and Walsh.

Improves results on Tracy-Widom universality classes for $eta otin ext{(1,2,4)}$.

Abstract

The work of Vershik and Kerov [1977], Logan and Shepp [1977] established that the shape of the scaled random young diagram in Russian notation, as determined by the Plancherel measure, converges to a deterministic shape. In this article, we focus on the scenario where the number of fixed points is substantial. We provide evidence that, subject to specific requirements on the total number of cycles, the limiting shape is a scaled version of the Vershik-Kerov-Logan-Shepp limiting shape. Additionally, we identify certain limiting regimes that resemble those in Chapuy, Louf, and Walsh [2022]. Furthermore, we enhance the existing results on Tracy-Widom universality classes for $\beta \in \{ 1, 2, 4\}$ for monotone subsequences.