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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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).


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Mallows permutations Longest increasing subsequence Central limit theorem


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.

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  • [1] D. J. Aldous and P. Diaconis. Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 (1995) 199–213.
  • [2] D. J. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. Available at
  • [3] F. J. Anscombe. Large-sample theory of sequential estimation. Math. Proc. Cambridge Philos. Soc. 48 (1952) 600–607.
  • [4] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (2005) 1643–1697.
  • [5] J. Baik, P. Deift and K. Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999) 1119–1178.
  • [6] J. Baik, E. M. Rains. The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 (2) (2001) 205–281.
  • [7] G. Ben Arous, I. Corwin. Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab. 39 (1) (2011) 104–138.
  • [8] I. Benjamini, N. Berger, C. Hoffman and E. Mossel. Mixing times of the biased card shuffling and the asymmetric exclusion process. Trans. Amer. Math. Soc. 357 (8) (2005) 3013–3029.
  • [9] N. Bhatnagar and R. Peled. Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields 161 (3–4) (2015) 719–780.
  • [10] A. Borodin, P. Diaconis and J. Fulman. On adding a list of numbers (and other one-dependent determinantal processes). Bull. Amer. Math. Soc. 47 (4) (2010) 639–670.
  • [11] P. Caputo. Energy gap estimates in ${X}{X}{Z}$ ferromagnets and stochastic particle systems. Markov Process. Related Fields 11 (2005) 189–210.
  • [12] S. Chaterjee and P. S. Dey. Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Related Fields 156 (2013) 613–663.
  • [13] D. Critchlow. Metric Methods for Analyzing Partially Ranked Data. Lecture Notes in Statistics 34. Springer, Berlin, 1985.
  • [14] J.-D. Deuschel and O. Zeitouni. Limiting curves for i.i.d. records. Ann. Probab. 23 (1995) 852–878.
  • [15] J.-D. Deuschel and O. Zeitouni. On increasing subsequences of i.i.d. samples. Combin. Probab. Comput. 8 (3) (1999) 247–263.
  • [16] P. Dey, M. Joseph and R. Peled. Longest increasing path within the critical strip. Preprint, 2016.
  • [17] P. Diaconis. Group Representations in Probability and Statistics. Lecture Notes – Monograph Series 11. Institute of Mathematical Statistics, Hayward, CA, 1988.
  • [18] P. Diaconis and A. Ram. Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 (1) (2000) 157–190.
  • [19] M. Figner and J. Verducci. Probability Models and Statistical Analyses for Ranking Data. Lecture Notes in Statistics 80. Springer, New York, 1993.
  • [20] A. Gladkich and R. Peled. On the cycle structure of random Mallows permutation. Preprint, 2016. Available at
  • [21] A. Gnedin and G. Olshanski. $q$-Exchangeability via quasi-invariance. Ann. Probab. 38 (2010) 2103–2135.
  • [22] A. Gnedin and G. Olshanski. The two-sided infinite extension of the Mallows model for random permutations. Adv. in Appl. Math. 48 (5) (2012) 615–639.
  • [23] A. Gut. Probability: A Graduate Course. Springer, Berlin, 2006.
  • [24] C. Houdré and Ü. Işlak. A central limit theorem for the length of the longest common subsequence in random words. Preprint, 2014. Available at arXiv:1408.1559.
  • [25] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001) 295–327.
  • [26] D. E. Knuth. Permutations, matrices and generalized Young tableaux. Pacific J. Math. 34 (3) (1970) 709–727.
  • [27] G. Lebanon and J. Lafferty. Cranking: Combining rankings using conditional probability models on permutations. In Proceedings of the Nineteenth International Conference on Machine Learning (ICML’02) 363–370. Morgan Kaufmann Publishers Inc., San Francisco, CA, 2002.
  • [28] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Adv. Math. 26 (1977) 206–222.
  • [29] C. L. Mallows. Non-null ranking models I. Biometrika 44 (1–2) (1957) 114–130.
  • [30] J. Marden. Analyzing and Modeling Ranking Data. Monographs on Statistics and Applied Probability 64. Chapman & Hall, London, 1993.
  • [31] C. Mueller and S. Starr. The length of the longest increasing subsequence of a random Mallows permutation. J. Theoret. Probab. 26 (2013) 514–540.
  • [32] S. Mukherjee. Estimation of parameters in non-uniform models on permutations. Preprint, 2013. Available at arXiv:1307.0978.
  • [33] G. de B. Robinson. On representations of the symmetric group. Amer. J. Math. 60 (1938) 745–760.
  • [34] C. Schensted. Longest increasing and decreasing subsequences. Canad. J. Math. 13 (1961) 179–191.
  • [35] R. Serfozo. Basics of Applied Stochastic Processes. Springer, Berlin, 2009.
  • [36] S. Starr. Thermodynamic limit for the mallows model on $S_{n}$. J. Math. Phys. 50 (9) (2009) 095208.
  • [37] S. Starr and M. Walters. Phase uniqueness for the Mallows measure on permutations. Preprint, 2015. Available at
  • [38] A. M. Vershik and S. V. Kerov. Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux. Dokl. Akad. Nauk SSSR (Sov. Math. Dokl.) 233 (6) (1977) 1024–1027.