Random Young diagrams and Jacobi Unitary Ensemble

Abstract

We consider random Young diagrams with respect to the measure induced by the decomposition of the $p$-th exterior power of ${\mathbb{C}}^n\otimes {\mathbb{C}}^k$ into irreducible representations of $G{L}_n\times G{L}_k$. We demonstrate that transition probabilities for these diagrams in the limit $n,k,p\to \infty$ with $p\sim nk$ converge to the large $N$ limiting law for the eigenvalues of random matrices in Jacobi Unitary Ensemble. We compute the characters of Young--Jucys--Murphy elements in ${\bigwedge }^p({\mathbb{C}}^n\otimes {\mathbb{C}}^k)$ and discuss their relation to surface counting. We formulate several conjectures on the connection between the correlators in both random ensembles.

16 pages, 3 figures, submitted to Zapiski Nauchnykh Seminarov POMI