Electronic Journal of Probability

Fixed points and cycle structure of random permutations

Sumit Mukherjee

Full-text: Open access

Abstract

Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall’s Tau, $\mu $ random permutations introduced in [11], as well as a class of exponential families introduced in [15].

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 40, 18 pp.

Dates
Received: 12 October 2015
Accepted: 24 May 2016
First available in Project Euclid: 15 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465991838

Digital Object Identifier
doi:10.1214/16-EJP4622

Mathematical Reviews number (MathSciNet)
MR3515570

Zentralblatt MATH identifier
1343.05011

Subjects
Primary: 05A05: Permutations, words, matrices 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Keywords
combinatorial probability Mallows model permutation limit fixed points cycle structure

Rights
Creative Commons Attribution 4.0 International License.

Citation

Mukherjee, Sumit. Fixed points and cycle structure of random permutations. Electron. J. Probab. 21 (2016), paper no. 40, 18 pp. doi:10.1214/16-EJP4622. https://projecteuclid.org/euclid.ejp/1465991838


Export citation

References

  • [1] Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approximation and the Chen-Stein method. Statist. Sci. 5, 4, 403–434. With comments and a rejoinder by the authors.
  • [2] Basu, R. and Bhatnagar, N. (2016). Limit Theorems for Longest Monotone Subsequences in Random Mallows Permutations. Available at http://arxiv.org/pdf/1601.02003.
  • [3] Bhattachara, B. and Mukherjee, S. (2015). Degree sequence of random permutation graphs. Ann. Appl. Probab., to appear.
  • [4] Bhattachara, B. and Mukherjee, S. (2015). Inference in Ising models. Available at http://arxiv.org/abs/1507.07055.
  • [5] Bhatnagar, N. and Peled, R. (2015). Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields 161, 3–4, 719–780.
  • [6] Borodin, A., Diaconis, P., and Fulman, J. (2010). On adding a list of numbers (and other one-dependent determinantal processes). Bull. Amer. Math. Soc. (N.S.) 47, 4, 639–670.
  • [7] Chatterjee, S., Diaconis, P., and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2, 64–106.
  • [8] Diaconis, P. (1988). Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA.
  • [9] Diaconis, P. and Ram, A. (2000) Analysis of Systematic Scan Metropolis Algorithms Using Iwahori-Hecke Algebra Techniques. Michigan Math. J. 48, 1, 157–190.
  • [10] Gladkich, A. and Peled, R. (2016) On the cycle structure of Mallows permutations. Available at http://arxiv.org/pdf/1601.06991.
  • [11] Hoppen, C., Kohayakawa, Y., Moreira, C. G., Ráth, B., and Menezes Sampaio, R. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103, 1, 93–113.
  • [12] Kenyon, R., Král, D., Radin, C., and Winkler, P.(2015). A variational principle for permutations. Available at http://arxiv.org/pdf/1506.02340.
  • [13] Mallows, C. L. (1957). Non-null ranking models. I. Biometrika 44, 114–130.
  • [14] Mueller, C. and Starr, S. (2013). The length of the longest increasing subsequence of a random Mallows permutation. J. Theoret. Probab. 26, 2, 514–540.
  • [15] Mukherjee, S. (2016). Estimation in exponential families on permutations. Ann. Statist. 44, 2, 853–875.
  • [16] Nelsen, R. B. (2006). An introduction to copulas, Second ed. Springer Series in Statistics. Springer, New York.
  • [17] Starr, S. (2009). Thermodynamic limit for the Mallows model on $S_n$. J. Math. Phys. 50, 9, 095208, 15.
  • [18] Walters, M. and Starr, S. (2015). A note on mixed matrix moments for the complex Ginibre ensemble. J. Math. Phys. 56, 1, 013301, 20.
  • [19] Trashorras, J. (2008). Large deviations for symmetrised empirical measures. J. Theoret. Probab. 21, 2, 397–412.