  In this paper, we are interested in the asymptotic size of rows and columns
of a random Young diagram under a natural deformation of the Plancherel measure
coming from Hecke algebras. The first lines of such diagrams are typically of
order $n$, so it does not fit in the context studied by P. Biane and P.
\'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov
and Olshanski, we are able to compute explicitly the first- and second-order
asymptotics of the length of the first rows. Our method works also for other
measures, for instance those coming from Schur-Weyl representations.
