Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theorems for longest monotone subsequences in random Mallows permutations

Riddhipratim Basu and Nayantara Bhatnagar

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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 $q^{\operatorname{inv}(\pi)}$ where $q$ is a positive parameter and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$.

In our main result we show that when $0<q<1$, then the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in (Probab. Theory Related Fields 161 (2015) 719–780). 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 limiting constant, answering a further open question in (Probab. Theory Related Fields 161 (2015) 719–780).

Résumé

Nous étudions les longueurs des sous-suites monotones de permutations aléatoires tirées sous la mesure de Mallows. La mesure de Mallows a été introduite par Mallows dans le contexte des problèmes de classement en statistique. Sous cette mesure la probabilité d’une permutation $\pi$ est proportionnelle à $q^{\operatorname{inv}(\pi)}$ où $q$ est un paramètre positif et $\operatorname{inv}(\pi)$ est le nombre d’inversions de $\pi$.

Notre résultat principal montre que lorsque $0<q<1$, la loi de la plus longue sous-suite croissante est Gaussienne, répondant ainsi à une question posée dans (Probab. Theory Related Fields 161 (2015) 719–780). Notons le contraste avec le cas $q=1$, où la loi limite de la plus longue sous-suite croissante proprement normalisée est la distribution du GUE Tracy–Widom. Nous obtenons aussi une loi des grands nombres pour la longueur de la plus longue sous-suite décroissante et identifions la limite, répondant ainsi à une autre question posée dans (Probab. Theory Related Fields 161 (2015) 719–780).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1934-1951.

Dates
Received: 23 January 2016
Revised: 29 June 2016
Accepted: 30 June 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773732

Digital Object Identifier
doi:10.1214/16-AIHP777

Mathematical Reviews number (MathSciNet)
MR3729641

Zentralblatt MATH identifier
1382.60018

Subjects
Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Keywords
Mallows permutations Longest increasing subsequence Central limit theorem

Citation

Basu, Riddhipratim; Bhatnagar, Nayantara. Limit theorems for longest monotone subsequences in random Mallows permutations. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1934--1951. doi:10.1214/16-AIHP777. https://projecteuclid.org/euclid.aihp/1511773732


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