Correlators of the Jacobi Unitary Ensemble on a General Interval and Hurwitz Numbers

Abstract

We study correlators of the Jacobi Unitary Ensemble $\mathbf J(p,q,α,β,N)$ 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 $\langle\prod_j \mathrm{tr}\,X^{λ_j}\rangle$ 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 $\mathbb C[\mathfrak S_n]$, we also obtain an explicit formula for the Schur--Selberg type integral $\int_{(p,q)^N}χ_λ(x)\,Δ^2(x)\prod_a (q-x_a)^β(x_a-p)^α\,\mathrm{d}^Nx$ 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 $c_A=c_B\to\infty$. 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.