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

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)Nχ_λ(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.