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In this paper, we consider a deformation of Plancherel measure linked to Jackpolynomials. Our main result is the description of the first and second-orderasymptotics of the bulk of a random Young diagram under this distribution,which extends celebrated results of Vershik-Kerov and Logan-Shepp (for thefirst order asymptotics) and Kerov (for the second order asymptotics). Thisgives more evidence of the connection with Gaussian $\beta$-ensemble, alreadysuggested by some work of Matsumoto.  Our main tool is a polynomiality result for the structure constant of somequantities that we call Jack characters, recently introduced by Lassalle. Webelieve that this result is also interested in itself and we give several otherapplications of it.

Gaussian fluctuations of Young diagrams and structure constants of Jack characters