Universality for random permutations and some other groups

TL;DR

This paper introduces Markovian methods to establish universality results for functions on the symmetric group, including CLTs for pattern occurrences and improvements on longest increasing subsequence results, with potential generalizations to other groups.

Contribution

It develops new Markovian approaches to prove universality for permutation statistics and extends existing results to broader classes of random permutations.

Findings

CLT for pattern occurrences in conjugation invariant permutations with few cycles

Improved bounds for the longest increasing subsequence

Potential generalizations to other algebraic structures

Abstract

We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the number of occurrences of a vincular patterns satisfies a CLT for conjugation invariant random permutations with few cycles and we improve the results already known for the longest increasing subsequence. The second approach is a suggestion of a generalization to other random permutations and other sets having a similar structure than the symmetric group.