Cycles in Mallows random permutations

Abstract

We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $\pi \in S_n$ is selected with probability proportional to $q^{\text{inv}(\pi)}$, where $q>0$ is a parameter and $\text{inv}(\pi)$ denotes the number of inversions of $\pi$. For $\ell$ fixed, we study the vector $(C_1(\Pi_n),\dots,C_\ell(\Pi_n))$ where $C_i(\pi)$ denotes the number of cycles of length $i$ in $\pi$ and $\Pi_n$ is sampled according to the Mallows distribution. Here we show that if $0<q<1$ is fixed and $n\to\infty$ then there are positive constants $m_i$ such that each $C_i(\Pi_n)$ has mean $(1+o(1)) \cdot m_i\cdot n$ and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when $q>1$ there is striking difference between the behaviour of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behaviour depends on the parity of $n$ when $q>1$. Both $(C_1(\Pi_{2n}),C_3(\Pi_{2n}),\dots)$ and $(C_1(\Pi_{2n+1}),C_3(\Pi_{2n+1}),\dots)$ have discrete limiting distributions -- they do not need to be renormalized -- but the two limiting distributions are distinct for all $q>1$. We describe these limiting distributions in terms of Gnedin and Olshanski's bi-infinite extension of the Mallows model. We also investigate these limiting distributions, and study the behaviour of the constants involved in the Gaussian limit laws. We for example show that as $q\downarrow 1$ the expected number of 1-cycles tends to $1/2$ -- which, curiously, differs from the value corresponding to $q=1$. In addition we exhibit an interesting "oscillating" behaviour in the limiting probability measures for $q>1$ and $n$ odd versus $n$ even.