I have seen that on the set of Young tableau the Plancherel measure was quite natural to define. I was wondering if other measures were also studied. In particular, a simple exemple which comes to my mind is that given a positive integer $n$, we can consider a sequence of positive reals $x_{n}=(x_{n}(i))_{i\geq 1}$, one could associate a weight for a Young tableau $\lambda$ with $n$ boxes by only considering the weight $\prod_{i} x_{n}(\lambda_i)$. Thus one could consider the probability measure such that $\mathbb{P}^{x}_n(\lambda)$ is proportional to the weight defined. Did some people already study this kind of probability measure, and the asymptotic of these probability measures when $x_{n}$ is chosen properly and when $n$ goes to infinity ?

$\begingroup$ Do you mean Young tableaux (those filled by numbers) or Young diagrams (only boxes, not filled)? $\endgroup$ – Fedor Petrov Aug 12 '15 at 15:56

1$\begingroup$ Work of Vershik and Kerov in the 1980s and 1990s is probably what you want. These papers were originally published in Russian, so are not as widely known as they should be. $\endgroup$ – David Handelman Aug 12 '15 at 17:35
You might take a look at the answers here: Random RSK and Plancherel Measure
One class of measures generalizing some of those described therein are the Schur measures introduced by Okounkov in Infinite wedge and random partitions. Borodin and Gorin have a nice survey that discusses them, called Lectures on integrable probability.
Fedor Petrov, I was thinking more about Young diagrams, sorry.
I will try to have a look to these papers, do you know if they were translated after?

$\begingroup$ Works of Vershik and Kerov are certainly translated. $\endgroup$ – Fedor Petrov Aug 12 '15 at 20:36