Jucys-Murphy element and walks on modified Young graph
Biane found out that irreducible decomposition of some representations of the symmetric group admits concentration at specific isotypic components in an appropriate large $n$ scaling limit. This deepened the result on the limit shape of Young diagrams due to Vershik-Kerov and Logan-Shepp in a wider framework. In particular, it is remarkable that asymptotic behavior of the Littlewood-Richardson coefficients in this regime was characterized in terms of an operation in free probability of Voiculescu. These phenomena are well understood through highest order analysis in the Kerov-Olshanski algebra of polynomial functions on Young diagrams with respect to the weight degree. Taking this point of view of highest order analysis into account, we show an asymptotic formula for moments of the Jucys-Murphy element by considering an appropriate graph structure on the Young diagrams which parametrize the conjugacy classes.