Geodesic
The asymptotics of traces of paths in the Young and Schur graphs
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 126-137 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $G$ be a graded graph with levels $V_0,V_1,\dots$. Fix $m$ and choose a vertex $v$ in $V_n$, where $n\ge m$. Consider the uniform measure on the paths from $V_0$ to the vertex $v$. Each such path has a unique vertex at the level $V_m$, and so a measure $\nu_v^m$ on $V_m$ is induced. It is natural to expect that such 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 proportion 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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     author = {F. V. Petrov},
     title = {The asymptotics of traces of paths in the {Young} and {Schur} graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {126--137},
     year = {2018},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a10/}
}
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F. V. Petrov. The asymptotics of traces of paths in the Young and Schur graphs. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 126-137. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a10/

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