Random Young diagrams and Jacobi Unitary Ensemble

TL;DR

The paper establishes a precise link between the Jacobi Unitary Ensemble and the limit shape of random Young diagrams under the skew Howe measure, via the Markov–Krein correspondence. It derives the JUE limiting density in a large-$N$ regime, connects it to the limit shape of the corresponding random diagrams, and computes Young–Jucys–Murphy characters to relate correlators to Riemann surface counting and Hurwitz numbers. The authors formulate conjectures that fluctuations, tau-function structures, and interlacing constructions align between the two ensembles, and provide numerical evidence and analytic groundwork toward a unified asymptotic picture of random partitions and random matrices with geometric content. This work thus links random partitions, integrable systems, and enumerative geometry, suggesting broad implications for asymptotic representation theory and matrix 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.

Figures (1)

• Figure 1: Histogram of JUE eigenvalues for $N=1000$ and $c_{\alpha}=0.95$, $c_{\beta}=1.235$ and limiting law (red). Note the discrete load at zero due to $c_\alpha < 1$.