Limit theorems for descents of Mallows permutations

The Mallows measure on the symmetric group gives a way to generate random permutations which are more likely to be sorted than not. There has been a lot of recent work to try and understand limiting properties of Mallows permutations. I'll discuss recent work on the joint distribution of descents, a statistic counting the number of "drops" in a permutation, and descents in its inverse, generalizing work of Chatterjee and Diaconis, and Vatutin. The proof is new even in the uniform case and uses Stein's method with a size-bias coupling as well as a regenerative representation of Mallows permutations.