Geodesic
Second class particles and limit shapes of evacuation and sliding paths for random tableaux.
Documenta mathematica, Tome 27 (2022), pp. 2183-2273.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the \textit{second class particle} converges (as the size of the system tends to infinity) to a random arc of an ellipse. \par In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) \textit{sliding paths} and \textit{evacuation paths} in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random \textit{meridian} connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.
Classification : 60C05, 05E10, 60K35, 82C22
Keywords: second class particles, interacting particle systems, TASEP, random Young tableaux, limit shape, jeu de taquin, promotion, Schützenberger's evacuation, square Young tableaux 
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     title = {Second class particles and limit shapes of evacuation and sliding paths for random tableaux.},
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Maślanka, Łukasz; Śniady, Piotr. Second class particles and limit shapes of evacuation and sliding paths for random tableaux.. Documenta mathematica, Tome 27 (2022), pp. 2183-2273. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a11/