A central limit theorem for cycles of Mallows permutations

TL;DR

This paper establishes a central limit theorem for cycle counts in Mallows permutations, revealing different asymptotic behaviors for even and odd cycles when the parameter q exceeds 1.

Contribution

It provides the first detailed analysis of cycle distributions in Mallows permutations for q>1, including Gaussian limits for even cycles and bounded limits for odd cycles.

Findings

Even cycles have mean and variance of order n and converge to Gaussian variables.

Odd cycles have bounded mean and variance, converging to limits depending on permutation parity.

The results extend understanding of Mallows permutations beyond the q<1 regime.

Abstract

Fix $q\neq 1$, and sample $w\in S_n$ from the Mallows measure. We study the distribution of $C_i(w)$, the number of $i$-cycles, as $n$ grows large. When $q<1$, they are jointly Gaussian, and this more or less follows from known ideas, but the regime $q>1$ behaves quite differently. In particular, we show that the even cycles $C_{2i}(w)$ have a mean and variance of order $n$, and jointly converge to Gaussian random variables, while the odd cycles $C_{2i+1}(w)$ have a bounded mean and variance, and converge to $C_{2i+1}(w^{even})$ or $C_{2i+1}(w^{odd})$ for some explicit random permutations $w^{even}$ and $w^{odd}$, depending on whether $n$ is even or odd. An extension to a larger class of functions is also given. The proof utilizes a two-sided stationary regenerative process associated to Mallows permutations constructed by Gnedin and Olshanski, extending the ideas of Basu and Bhatnagar.