Entropy of Schur-Weyl measures at critical scale

Problem's Description

Consider the probability measure $\mathbb{P}_N^n(\lambda)=\frac{\dim E_\lambda}{N^n}$ on the set of Young diagrams with $n$ cells and at most $N$ rows, where $\dim E_\lambda$ is the dimension of the isotypic component corresponding to the Young diagram $\lambda$ in the tensor representation $(\mathbb{C}^N)^{\otimes n}$ of the group $GL(N,\mathbb{C})$.

It was proven in the referenced article that for any $c>0, c\neq 1$ there exists a positive number $H_c$ such that for any $\varepsilon>0$ we have $$\lim_{{n\rightarrow\infty}\\{N\rightarrow\infty}\\{\frac{\sqrt{n}}N\rightarrow c}}\mathbb{P}_N^n\left\{\lambda\in\mathbb{Y}_N^n:\left|-\frac 1{\sqrt n}\ln\frac{\dim E_\lambda}{N^n}-H_c\right|< \varepsilon\right\}=1,$$ where $\mathbb{Y}_N^n$ is the support of $\mathbb{P}_N^n$.

The constant $H_c$ can be interpreted as the entropy of the measure $\mathbb{P}_N^n$. The result should hold when $c=1$ as well, however this is still open.

The value $c=1$ is critical in the sense that at only that value the nature of fluctuations of a $\mathbb{P}_N^n$-distributed random Young diagram $\lambda$ near one of the edges is different.

1. Article Entropy of Schur-Weyl measures

   • Sevak Mkrtchyan