Small cycle structure for words in conjugation invariant random permutations

TL;DR

This paper investigates the cycle structure of words in conjugation invariant random permutations, establishing universal laws for small cycles under certain conditions, extending previous results from permutation products to general words.

Contribution

It extends prior work by analyzing the cycle structure of arbitrary words in conjugation invariant permutations, providing universal limiting laws for small cycles.

Findings

Universal limiting law for small cycles in words with at least two different letters

Extension of previous results from permutation products to general words

Generalization from uniform to conjugation invariant permutations

Abstract

We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word w still contains at least two different letters, then we get a universal limiting joint law for small cycles for the word in these permutations. These results can be seen as an extension of our previous work [Kammoun and Ma\"ida, 2020] from the product of permutations to any non-trivial word in the permutations and also as an extension of the results of [Nica, 1994] from uniform permutations to general conjugation invariant random permutations.