Let G be a graded graph with levels V0, V1, . . .. Fix m and choose a vertex v in Vn where n ≥ m. Consider the uniform measure on the paths from V0 to v. Each such path has a unique vertex at the level Vm, so a measure \( {\nu}_v^m \) on Vm is induced. It is natural to expect that these measures have a limit as the vertex v goes to infinity in some “regular” way. We prove this (and compute the limit) for the Young and Schur graphs, for which regularity is understood as follows: the fraction of boxes contained in the first row and the first column goes to 0. For the Young graph, this was essentially proved by Vershik and Kerov in 1981; our proof is more straightforward and elementary.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 468, 2018, pp. 126–137.
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Petrov, F.V. Asymptotics of Traces of Paths in the Young and Schur Graphs. J Math Sci 240, 587–593 (2019). https://doi.org/10.1007/s10958-019-04377-9
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DOI: https://doi.org/10.1007/s10958-019-04377-9