The Annals of Probability

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

Dan Romik and Piotr Śniady

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Abstract

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

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

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1422885573

Digital Object Identifier
doi:10.1214/13-AOP873

Mathematical Reviews number (MathSciNet)
MR3306003

Zentralblatt MATH identifier
1360.60028

Subjects
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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aop/1422885573


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