Monotonous subsequences and the descent process of invariant random permutations

TL;DR

This paper extends Tracy-Widom fluctuation results for the longest increasing subsequence from uniform permutations to Ewens and conjugation-invariant distributions, also analyzing the limiting descent process.

Contribution

It proves Tracy-Widom fluctuations for a broader class of random permutations and characterizes the limiting descent process as stationary, one-dependent, and determinantal.

Findings

Tracy-Widom fluctuations hold for Ewens and conjugation-invariant permutations.

Convergence of Young tableaux components to the Airy Ensemble.

Limiting descent process is stationary, one-dependent, and determinantal.

Abstract

It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting descent process of a large class of random permutations is stationary, one-dependent and determinantal.