École Nationale Supérieure de
Techniques Avancées

Abstract

We revisit a well-known combinatorial analysis result found independently by Logan and Shepp, and Vershik and Kerov in 1977: the asymptotic convergence in probability of Young diagrams under the Plancherel measure to a certain function called limit shape. Neither of the authors nor any subsequent published article we have come across provides with the method-ology to compute said limit form. After an overview of the underlying linear representation theory behind the problem, we set ourselves to come up with the formula of the limit shape, introducing the notion of energy-entropy functionals for the upper border function of diagrams. We then relate them to a Riemann-Hilbert problem through a variational approach, which lead us to consider hypergeometric functions right before the internship ended in an inconclusive manner.