# Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations

## TL;DR

This paper investigates the asymptotic behavior of the longest common subsequence (LCS) of two permutations drawn from Mallows measures, establishing Gaussian limiting laws and weak laws under various regimes.

## Contribution

It extends previous results on longest increasing subsequences to the LCS of Mallows permutations, providing new limit theorems and probabilistic laws.

## Key findings

- LCS length has a Gaussian limiting distribution for 0<q,q'<1.
- Weak law of large numbers for LCS when n(1-q) and n(1-q') tend to infinity.
- Results generalize prior work on increasing subsequences to LCS context.

## Abstract

The Mallows measure is measure on permutations which 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 $\pi$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of large numbers for the LCS. These results extend the results of \cite{Basu} and \cite{Naya} showing weak laws and a limiting law for the distribution of the longest increasing subsequence to showing corresponding results for the longest common subsequence.