Cycles in Mallows random permutations

Abstract

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