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 $\mathrm{GL}_{n}\times\mathrm{GL}_{k}$. 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.
Key words and phrases:random Young diagrams, Jacobi unitary ensemble, skew Howe duality, limit shape, transition probability, Markov–Krein correspondence, Young–Jucys–Murphy elements, Riemann surfaces.
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