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

Abstract

The Mallows measure is a probability measure on $S_{n}$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q>0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n\to\infty$.