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We present a survey of points of view on the problem of the asymptotic shapeof a path between two large Young diagrams, and introduce a modification of theTASEP process related to it. This representation allows to write explicitly thefunctional, counting the asymptotics of the number of Young tableau close to agiven one, as well as to see the sine-process on the boundary shape of a largerandom Young diagram.

Limit shapes of large skew Young tableaux and a modification of the TASEP process