The Annals of Probability

Jeu de taquin dynamics on infinite Young tableaux and second class particles

Dan Romik and Piotr Śniady

Full-text: Open access


We study an infinite version of the “jeu de taquin” sliding game, which can be thought of as a natural measure-preserving transformation on the set of infinite Young tableaux equipped with the Plancherel probability measure. We use methods from representation theory to show that the Robinson–Schensted–Knuth (RSK) algorithm gives an isomorphism between this measure-preserving dynamical system and the one-sided shift dynamics on a sequence of independent and identically distributed random variables distributed uniformly on the unit interval. We also show that the jeu de taquin paths induced by the transformation are asymptotically straight lines emanating from the origin in a random direction whose distribution is computed explicitly, and show that this result can be interpreted as a statement on the limiting speed of a second-class particle in the Plancherel-TASEP particle system (a variant of the Totally Asymmetric Simple Exclusion Process associated with Plancherel growth), in analogy with earlier results for second class particles in the ordinary TASEP.

Article information

Ann. Probab., Volume 43, Number 2 (2015), 682-737.

First available in Project Euclid: 2 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 05E10: Combinatorial aspects of representation theory [See also 20C30] 37A05: Measure-preserving transformations

Jeu de taquin Young tableau Plancherel measure TASEP exclusion process second class particle dynamical system isomorphism of measure preserving systems representation theory of symmetric groups


Romik, Dan; Śniady, Piotr. Jeu de taquin dynamics on infinite Young tableaux and second class particles. Ann. Probab. 43 (2015), no. 2, 682--737. doi:10.1214/13-AOP873.

Export citation


  • Amir, G., Angel, O. and Valkó, B. (2011). The TASEP speed process. Ann. Probab. 39 1205–1242.
  • Angel, O., Holroyd, A. and Romik, D. (2009). The oriented swap process. Ann. Probab. 37 1970–1998.
  • Angel, O., Holroyd, A. E., Romik, D. and Virág, B. (2007). Random sorting networks. Adv. Math. 215 839–868.
  • Biane, P. (1995). Permutation model for semi-circular systems and quantum random walks. Pacific J. Math. 171 373–387.
  • Biane, P. (1998). Representations of symmetric groups and free probability. Adv. Math. 138 126–181.
  • Biane, P. (2001). Approximate factorization and concentration for characters of symmetric groups. Int. Math. Res. Not. IMRN 4 179–192.
  • Cator, E. and Dobrynin, S. (2006). Behavior of a second class particle in Hammersley’s process. Electron. J. Probab. 11 670–685 (electronic).
  • Cator, E. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks. Ann. Probab. 33 879–903.
  • Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273–1295.
  • Cator, E. and Pimentel, L. P. R. (2013). Busemann functions and the speed of a second class particle in the rarefaction fan. Ann. Probab. 41 2401–2425.
  • Ceccherini-Silberstein, T., Scarabotti, F. and Tolli, F. (2010). Representation Theory of the Symmetric Groups: The Okounkov–Vershik Approach, Character Formulas, and Partition Algebras. Cambridge Studies in Advanced Mathematics 121. Cambridge Univ. Press, Cambridge.
  • Coletti, C. F. and Pimentel, L. P. R. (2007). On the collision between two PNG droplets. J. Stat. Phys. 126 1145–1164.
  • Deuschel, J.-D. and Zeitouni, O. (1999). On increasing subsequences of i.i.d. samples. Combin. Probab. Comput. 8 247–263.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
  • Ferrari, P. A., Gonçalves, P. and Martin, J. B. (2009). Collision probabilities in the rarefaction fan of asymmetric exclusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 1048–1064.
  • Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. Henri Poincaré Probab. Stat. 31 143–154.
  • Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235–1254.
  • Fulton, W. (1997). Young Tableaux: With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts 35. Cambridge Univ. Press, Cambridge.
  • Greene, C., Nijenhuis, A. and Wilf, H. S. (1984). Another probabilistic method in the theory of Young tableaux. J. Combin. Theory Ser. A 37 127–135.
  • Jucys, A.-A. A. (1974). Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5 107–112.
  • Kerov, S. V. (1993). Transition probabilities of continual Young diagrams and the Markov moment problem. Funktsional. Anal. i Prilozhen. 27 32–49, 96.
  • Kerov, S. (1999). A differential model for the growth of Young diagrams. In Proceedings of the St. Petersburg Mathematical Society, Vol. IV. Amer. Math. Soc. Transl. Ser. 2 188 111–130. Amer. Math. Soc., Providence, RI.
  • Kerov, S. V. and Vershik, A. M. (1986). The characters of the infinite symmetric group and probability properties of the Robinson–Schensted–Knuth algorithm. SIAM J. Algebraic Discrete Methods 7 116–124.
  • Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York.
  • Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222.
  • Mountford, T. and Guiol, H. (2005). The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 1227–1259.
  • O’Connell, N. (2003). A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. Amer. Math. Soc. 355 3669–3697 (electronic).
  • O’Connell, N. and Yor, M. (2002). A representation for non-colliding random walks. Electron. Commun. Probab. 7 1–12 (electronic).
  • Romik, D. (2004). Explicit formulas for hook walks on continual Young diagrams. Adv. in Appl. Math. 32 625–654.
  • Romik, D. (2014). The Surprising Mathematics of Longest Increasing Subsequences. Cambridge Univ. Press, Cambridge. To appear. Available at
  • Romik, D. and Śniady, P. (2013). Limit shapes of bumping routes in the Robinson–Schensted correspondence. Preprint. Available at arXiv:1304.7589.
  • Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • Sagan, B. E. (2001). The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed. Graduate Texts in Mathematics 203. Springer, New York.
  • Schützenberger, M. P. (1963). Quelques remarques sur une construction de Schensted. Math. Scand. 12 117–128.
  • Schützenberger, M.-P. (1977). La correspondance de Robinson. In Combinatoire et Représentation du Groupe Symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976). Lecture Notes in Math. 579 59–113. Springer, Berlin.
  • Seppäläinen, T. (1998). Large deviations for increasing sequences on the plane. Probab. Theory Related Fields 112 221–244.
  • Silva, C. E. (2008). Invitation to Ergodic Theory. Student Mathematical Library 42. Amer. Math. Soc., Providence, RI.
  • Śniady, P. (2006a). Asymptotics of characters of symmetric groups, genus expansion and free probability. Discrete Math. 306 624–665.
  • Śniady, P. (2006b). Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Related Fields 136 263–297.
  • Śniady, P. (2014). Robinson–Schensted–Knuth algorithm, jeu de taquin and Kerov–Vershik measures of infinite tableaux. SIAM J. Discrete Math. To appear. Available at arXiv:1307.5645.
  • Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246–290.
  • Stanley, R. P. (1999). Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics 62. Cambridge Univ. Press, Cambridge.
  • Vershik, A. M. and Kerov, S. V. (1977). Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233 1024–1027.
  • Vershik, A. M. and Kerov, S. V. (1981). Asymptotic theory of the characters of a symmetric group. Funktsional. Anal. i Prilozhen. 15 15–27, 96.
  • Vershik, A. M. and Kerov, S. V. (1985). Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group. Funktsional. Anal. i Prilozhen. 19 25–36, 96.