Limit Theorems for Longest Monotone Subsequences in Random Mallows Permutations

TL;DR

This paper investigates the asymptotic behavior of the longest monotone subsequences in permutations drawn from the Mallows measure, revealing Gaussian limits for increasing subsequences when q<1 and precise constants for decreasing subsequences.

Contribution

It establishes the Gaussian limiting distribution for the LIS in Mallows permutations with q<1, and determines the exact growth rate for LDS, answering open questions.

Findings

LIS converges to a Gaussian distribution for 0<q<1.

LDS length has a precise law of large numbers.

Contrasts with the Tracy-Widom distribution at q=1.

Abstract

We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $\pi$ is proportional to $q^{{\rm inv}(\pi)}$ where $q$ is a positive parameter and ${\rm inv}(\pi)$ is the number of inversions in $\pi$. In our main result we show that when $0<q<1$, the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the case when $q=1$ where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the precise constant in the order of the expectation, answering a further open question in [Bhatnagar and Peled, PTRF, 2015].