Correlators of the Jacobi Unitary Ensemble on a General Interval and Hurwitz Numbers
Abstract
We study correlators of the Jacobi Unitary Ensemble J(p,q,α,β,N) of Hermitian matrices with spectrum in an arbitrary real interval [p,q]. For (p,q)=(0,1), Gisonni, Grava and Ruzza expressed the large-N expansion of the correlators ⟨∏j tr Xλj⟩ through triple monotone Hurwitz numbers. We extend this genus expansion to arbitrary p,q with α/N and β/N fixed. Using the Frobenius characteristic map and the idempotent basis of the center of ℂ[Sn], we also obtain an explicit formula for the Schur--Selberg type integral ∫(p,q)Nχλ(x) Δ2(x)∏a (q−xa)β(xa−p)α dNx in terms of irreducible characters of the symmetric group and content products. As consequences, we describe the zero-mean ensemble, and we recover, term by term in the genus expansion, the polygon-gluing formula for the Gaussian Unitary Ensemble in the limit cA=cB→∞. Structurally, passing from [0,1] to [p,q] leaves the hypergeometric weight of the associated 2D Toda tau function unchanged and only changes the specialization of its second set of times.
Related
- Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
- Inhomogeneous Geometric Last Passage Percolation and the q-deformed Jacobi Unitary Ensemble
- Invariant pseudo-unitary matrix ensembles and local statistics and conductance fluctuations of the DMPK ensemble
- A Positive Proportion of the Reduced D'Arcais Polynomials is not Hurwitz
- Very high genus double Hurwitz numbers via mixing times
BibTeX
@misc{sushkov2026correlators,
title = {{Correlators of the Jacobi Unitary Ensemble on a General Interval and Hurwitz Numbers}},
author = {Matvey Sushkov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02795},
eprint = {2611.02795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02795},
}
RIS
TY - UNPB
TI - Correlators of the Jacobi Unitary Ensemble on a General Interval and Hurwitz Numbers
AU - Matvey Sushkov
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02795
UR - https://arxiv.org/abs/2611.02795
AB - We study correlators of the Jacobi Unitary Ensemble J(p,q,α,β,N) of Hermitian matrices with spectrum in an arbitrary real interval [p,q]. For (p,q)=(0,1), Gisonni, Grava and Ruzza expressed the large-N expansion of the correlators ⟨∏_j tr X^λ_j⟩ through triple monotone Hurwitz numbers. We extend this genus expansion to arbitrary p,q with α/N and β/N fixed. Using the Frobenius characteristic map and the idempotent basis of the center of ℂ[S_n], we also obtain an explicit formula for the Schur--Selberg type integral ∫_(p,q)^Nχ_λ(x) Δ^2(x)∏_a (q−x_a)^β(x_a−p)^α d^Nx in terms of irreducible characters of the symmetric group and content products. As consequences, we describe the zero-mean ensemble, and we recover, term by term in the genus expansion, the polygon-gluing formula for the Gaussian Unitary Ensemble in the limit c_A=c_B→∞. Structurally, passing from [0,1] to [p,q] leaves the hypergeometric weight of the associated 2D Toda tau function unchanged and only changes the specialization of its second set of times.
ER -