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

F. V. Petrov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 126-137

Résumé

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.