Limit theorems for longest monotone subsequences in random Mallows permutations

We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation pi is proportional to q(inv(pi)) where q is a positive parameter and inv(pi) is the number of inversions in pi. In our main result we show that when 0 < q < 1, then the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in (Probab. Theory Related Fields 161 (2015) 719-780). This is in contrast to the case when q = 1 where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the limiting constant, answering a further open question in (Probab. Theory Related Fields 161 (2015) 719-780).