Random Young diagrams and Jacobi Unitary Ensemble

TL;DR

This paper studies the asymptotic behavior of random Young diagrams derived from tensor decompositions and reveals their connection to the eigenvalue distribution of the Jacobi Unitary Ensemble in random matrix theory.

Contribution

It introduces a new link between random Young diagrams from tensor decompositions and the Jacobi Unitary Ensemble, including character computations and conjectures on correlator relations.

Findings

Transition probabilities converge to Jacobi Unitary Ensemble laws

Characters of Young--Jucys--Murphy elements computed

Conjectures on correlator connections between ensembles

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 $GL_{n}\times GL_{k}$. 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.