The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Abstract

The Mallows measure is a probability measure on S n where the probability of a permutation π is proportional to q l(π) with q > 0 being a parameter and l(π) the number of inversions in π. We show the convergence of the random empirical measure of the product of two independent permutations drawn from the Mallows measure, when q is a function of n and n(1 − q) has limit in R as n → ∞.Keywords Mallows measure · random permutation · convergence of measure Mathematics Subject Classification (2010) 60F05 · 60B15 · 05A05 1 − q i 1 − q .

Continuous-time Mallows processes

“…The study of its structural properties has gained a lot of interests in the last years and now contains literature related to various topics. Among the properties of these models that have attracted the most attention, it is worth mentioning the length of longest increasing subsequences [6,8,24,27] as well as the cycle and subsequence structure [10,18,22,28,29], substantially studied over the last decade. Other works studying exchangeability [20,19], random matchings [4], binary search trees [1], and colourings [23] have led to interesting insights on properties of these random permutations.…”

The length of the longest common subsequence of two independent mallows permutations

“…Results. Before stating the main theorem, we introduce the following lemma proved in Jin (2017), which shows the convergence of the empirical measure of a collection of random points defined by two independent Mallows permutations.…”