Local limits of descent-biased permutations and trees

In this talk, we will discuss two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation is a pair of consecutive elements where the first is greater than the second. Likewise, a descent in a rooted tree with labelled vertices is a pair of a parent vertex and a child such that the label of the parent is greater than the label of the child. For some nonnegative real number q, we consider the probability measures on permutations and on rooted labelled trees of a given size where each permutation or tree is chosen with a probability proportional to q^{number of descents}. In particular, we determine the asymptotic distribution of the first elements of permutations under this model. Different phases can be observed based on how q depends on the number of elements n in our permutations. Using the results on permutations, one can also characterize the local limit of descent-biased rooted labelled trees.