Random Young diagrams and Jacobi Unitary Ensemble

2025/11/05 by Anton Nazarov, Nazarov, Anton, M. S. Sushkov +1
Mathematics · #15A52 #60C05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2511.03881

openalex publication_date 2025/11/05 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28

Abstract

We consider random Young diagrams with respect to the measure induced by the decomposition of the p-th exterior power of ℂⁿ⊗ ℂ^k into irreducible representations of GL_n× GL_k. We demonstrate that transition probabilities for these diagrams in the limit n,k,p→∞ with p∼ 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(ℂⁿ⊗ℂ^k) and discuss their relation to surface counting. We formulate several conjectures on the connection between the correlators in both random ensembles.

Citations

• Limit shapes and fluctuations for (GL_n, GL_k) skew Howe duality
• Gromov-Witten theory, Hurwitz theory, and completed cycles
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Approximate factorization and concentration for characters of symmetric groups
• Toda equations for Hurwitz numbers

Related

• Law of Large Numbers for Infinite Random Matrices over a Finite Field
• Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials
• A Quantized Analogue of the Markov-Krein Correspondence
• Kerov's central limit theorem for the Plancherel measure on Young diagrams
• Gaussian Unitary Ensemble in random lozenge tilings