Cycles in Mallows random permutations

We study cycle counts in permutations of (Formula presented.) drawn at random according to the Mallows distribution. Under this distribution, each permutation (Formula presented.) is selected with probability proportional to (Formula presented.), where (Formula presented.) is a parameter and (Formula presented.) denotes the number of inversions of (Formula presented.). For (Formula presented.) fixed, we study the vector (Formula presented.) where (Formula presented.) denotes the number of cycles of length (Formula presented.) in (Formula presented.) and (Formula presented.) is sampled according to the Mallows distribution. When (Formula presented.) the Mallows distribution simply samples a permutation of (Formula presented.) 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 (Formula presented.). Here we show that if (Formula presented.) is fixed and (Formula presented.) then there are positive constants (Formula presented.) such that each (Formula presented.) has mean (Formula presented.) and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when (Formula presented.) 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 (Formula presented.) when (Formula presented.). Both (Formula presented.) and (Formula presented.) have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all (Formula presented.). 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 (Formula presented.) the expected number of 1-cycles tends to (Formula presented.) —which, curiously, differs from the value corresponding to (Formula presented.). In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for (Formula presented.) and (Formula presented.) odd versus (Formula presented.) even.