# Measures on Young tableaux

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 ?

• Do you mean Young tableaux (those filled by numbers) or Young diagrams (only boxes, not filled)? – Fedor Petrov Aug 12 '15 at 15:56
• 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. – David Handelman Aug 12 '15 at 17:35