Cycles in Mallows random permutations

Abstract

We study cycle counts in permutations of 1,…,n$$ 1,\dots, n $$ drawn at random according to the Mallows distribution. Under this distribution, each permutation π∈Sn$$ \pi \in {S}_n $$ is selected with probability proportional to qinv(π)$$ {q}^{\mathrm{inv}\left(\pi \right)} $$ , where q>0$$ q>0 $$ is a parameter and inv(π)$$ \mathrm{inv}\left(\pi \right) $$ denotes the number of inversions of π$$ \pi $$ . For ℓ$$ \ell $$ fixed, we study the vector (C1(Πn),…,Cℓ(Πn))$$ \left({C}_1\left({\Pi}_n\right),\dots, {C}_{\ell}\left({\Pi}_n\right)\right) $$ where Ci(π)$$ {C}_i\left(\pi \right) $$ denotes the number of cycles of length i$$ i $$ in π$$ \pi $$ and Πn$$ {\Pi}_n $$ is sampled according to the Mallows distribution. When q=1$$ q=1 $$ the Mallows distribution simply samples a permutation of 1,…,n$$ 1,\dots, n $$ uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means 1,12,13,…,1ℓ$$ 1,\frac{1}{2},\frac{1}{3},\dots, \frac{1}{\ell } $$ . Here we show that if 01$$ q>1 $$ there is a striking difference between the behavior 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 behavior depends on the parity of n$$ n $$ when q>1$$ q>1 $$ . Both (C1(Π2n),C3(Π2n),…)$$ \left({C}_1\left({\Pi}_{2n}\right),{C}_3\left({\Pi}_{2n}\right),\dots \right) $$ and (C1(Π2n+1),C3(Π2n+1),…)$$ \left({C}_1\left({\Pi}_{2n+1}\right),{C}_3\left({\Pi}_{2n+1}\right),\dots \right) $$ have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all q>1$$ q>1 $$ . We describe these limiting distributions in terms of Gnedin and Olshanski's bi‐infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as q↓1$$ q\downarrow 1 $$ the expected number of 1‐cycles tends to 1/2$$ 1/2 $$ —which, curiously, differs from the value corresponding to q=1$$ q=1 $$ . In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for q>1$$ q>1 $$ and n$$ n $$ odd versus n$$ n $$ even.