Cycles in Mallows random permutations

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