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Introduction

Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$ $$V_{n,m}=\bigotimes_{k=1}^n \mathbb C^m$$ and the symmetric group $S_n$ acts on $V_{n,m}$ by permutation of factors. Now the space $V_{n,m}$ breaks into the direct sum of subspaces $V_{n,m}(\lambda)$ which are primary with respect to each of these actions and irreducible with respect to the joint action of $S_n\times \mathbf U(m)$ $$V_{n,m}=\bigoplus_{\lambda \in \mathbb{Y}(n,m)}V_{n,m}(\lambda)$$

Where $\lambda$ ranges over all Young diagrams of size $n$ with at most $m$ rows (So $\lambda \in \mathbb{Y}(n,m)$ means $\lambda=(\lambda_1,\dots ,\lambda_k)$ is a partition of $n$ with $k\le m$). The tensors from $V_{n,m}(\lambda)$ are said to have symmetry type $\lambda$. We define the relative dimensions $$d_{n,m}(\lambda)=\frac{\dim V_{n,m}(\lambda)}{\dim V_{n,m}}$$ which tell us how tensors are distributed into symmetry types.

Motivation and Question

I was reading Kerov's "Asymptotic representation theory of the symmetric group and it's aplications in analysis" and was trying to provide proofs for some of the results stated there. (He does give references, which I can't reach at the moment.)

The following two theorems are due to Kerov

Theorem 1 If for each $\lambda$ we associate $x=(x_1,\dots,x_m)$ with $x_k=\frac{\lambda ^{(n)}_k-n/m}{\sqrt{n}}$. The joint distribution of $x_k$'s as $n\to \infty$ with respect to the measure $d_{n,m}$ on $\mathbb{Y}(n,m)$ weakly converges to an absolutely continuous measure on the cone $C_m=\{x: x_1\geq x_2\geq\cdots \geq x_m \;;\;\sum x_k=0\}$ with density $$\phi _m(x)=c \prod _{i < j}(x_i-x_j)^2 e^{-m/2 \sum x_k^2}$$ where $$c=\frac{m^{(m-1)m/2}}{1!2!\cdots (m-1)!}\left(\frac{m}{2\pi}\right)^{(m-1)/2}$$

.

Theorem 2 Let $\lambda ^{(n)}\in \mathbb{Y}(n,m)$ be the Young diagram for which the tensors of type $\lambda ^{(n)}$ are most probable. Then $$\lim_{n\to \infty} \frac{\lambda ^{(n)}_k-n/m}{\sqrt{n/m}}=z_k$$ for each $k=1,2,\dots,m$ where $z_1,z_2,\dots z_m$ are the roots of the Hermite polynomial $H_m(z)$

I can prove Theorem 2 assuming Theorem 1, but I don't see a nice argument for proving the first theorem itself. Can anyone provide a sketch of the proof, or some hint how to approach Theorem 1?

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This is not a complete answer, but perhaps will help. The probability distribution on tuples that shows up in your Theorem 1 is well known: It is the joint probability density function for the eigenvalues of a random unitary matrix, in the standard ``Gaussian unitary ensemble". See equation (9) in the paper of Terry Tao and Van Vue

http://arxiv.org/abs/0906.0510v9

I have unfortunately not read that closely, but the title, "random matrices: universality of local eigenvalue statistics" suggests it might have something to say about why this distribution would appear in other places. I've looked more closely at the following paper of Okounkov.

http://arxiv.org/abs/math-ph/0309015

There he explains how a similar distribution does show up in a system of random partitions. See especially Section 1.4.2. There Okounkov uses the distribution coming from the ``Plancherel measure", which is slightly different then the distribution you describe: the probability of observing $\lambda$ is propositional to $\operatorname{dim} S(\lambda)^2$, where $S(\lambda)$ is the representation of the symmetric group corresponding to $\lambda$. You seem to have chosen the distribution where the probability of observing $\lambda$ is proportional to $\operatorname{dim} S(\lambda) \operatorname{dim} V(\lambda)$, where $V(\lambda)$ is an irreducible representation of $U(n)$. Also, he has $-1/2$ where you have $-m/2$ in the exponential part. But perhaps it is still related.

Anyway, if you are interested in this type of question about partitions, I highly recommend looking at these papers.

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Kuperberg in Random words, quantum statistics, central limits, random matrices attributes Theorem 1 to Johansson and he gives two additional interesting proofs of this result.

A proof of a more general result is presented in my joint work with Benoit Collins Representations of Lie groups and random matrices.

Asymptotics when both $n$ and $m$ tend to infinity was considered by Biane in Approximate factorization and concentration for characters of symmetric groups.

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