Trajectories of jeu de taquin and second class particles
Final Report Abstract
We study various aspects of the scaling limit of Robinson–Schensted–Knuth (RSK) correspondence applied to random input. In particular, if RSK is applied to a sequence of independent random variables with the uniform distribution on the unit interval then: • the (scaled down) bumping routes occuring in each insertion step converge in probability to a family of explicit, deterministic curves; • the (scaled down) dynamics of the insertion tableau converges in probability to a deterministic, steady–state flow on the plane. If RSK is applied to a sequence of independent random letters from a carefully selected alphabet consisting of row and column letters, with probability distribution fulfulling some additional assumptions, then RSK becomes an isomorphism between the simple product probability space and the probability space of random infinite Young tableau with the distribution related to some Thoma character of the infinite symmetric group. Furthermore, it is an isomorphism of dynamical systems: the dynamical system of the Bernoulli shift on one side and the dynamical system of jeu de taquin transformation on the set of infinite Young tableaux on the other.
Publications
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Limit shapes of bumping routes in the Robinson-Schensted correspondence
Dan Romik, Piotr Śniady
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Partial transpose of random quantum states: exact formulas and meanders. J. Math. Phys. 54, 042202 (2013)
Fukuda, Motohisa & Śniady, Piotr
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Dimensions of components of tensor products of representations of linear groups with applications to Beurling-Fourier algebras. Studia Math. 220 (2014), no. 3, 221–241
Collins, Benoît; Lee, Hun Hee & Śniady, Piotr
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Robinson–Schensted–Knuth algorithm, jeu de taquin, and Kerov–Vershik measures on infinite tableaux. SIAM J. Discrete Math. 28 (2014), no. 2, 598–630
Śniady, Piotr
