Robinson-Schensted-Knuth algorithm, jeu de taquin and Kerov-Vershik measures on infinite tableaux

TL;DR

This paper explores the connections between the RSK algorithm, jeu de taquin, and Vershik-Kerov measures on infinite Young tableaux, revealing isomorphisms between certain dynamical systems and extending known results on random walks.

Contribution

It demonstrates that the recording tableau in RSK creates an isomorphism between i.i.d. random sequences and infinite Young tableaux under jeu de taquin, linking combinatorics and dynamical systems.

Findings

RSK recording tableau induces an isomorphism of dynamical systems

Establishes a connection between Bernoulli shifts and infinite tableaux

Recovers results on non-colliding random walks and Pitman transform

Abstract

We investigate Robinson-Schensted-Knuth algorithm (RSK) and Sch\"utzenberger's jeu de taquin in the infinite setup. We show that the recording tableau in RSK defines an isomorphism of the following two dynamical systems: (i) a sequence of i.i.d. random letters equipped with Bernoulli shift, and (ii) a random infinite Young tableau (with the distribution given by Vershik-Kerov measure, corresponding to some Thoma character of the infinite symmetric group) equipped with jeu de taquin transformation. As a special case we recover the results on non-colliding random walks and multidimensional Pitman transform.