The asymptotics of traces of paths in the Young and Schur graphs

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.