Asymptotics of symmetry types of tensors

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.

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.