Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups

Source abstract

We study joint moments of characteristic polynomials and their second derivatives for Haar-distributed matrices from the unitary symplectic and orthogonal groups. We establish their large-$N$ asymptotics throughout the full admissible range of the moment parameters and identify the limiting coefficients probabilistically in terms of the first inverse power sum of the Bessel point process. This extends the corresponding results of Assiotis, Gunes, Keating, and Wei [Proc. Lond. Math. Soc., 132 (2026), e70136] to the full admissible parameter range. Throughout this full range, we also express these limits in terms of a distinguished solution of the $σ$-Painlevé III' equation. Our approach reduces the problem to singular inverse-power statistics of the Jacobi unitary ensemble and combines their finite-$N$ $σ$-Painlevé V representation with a refined Riemann-Hilbert analysis yielding precise small-$x$ asymptotics and uniform derivative estimates for the finite-$N$ correction term. A key ingredient in the probabilistic identification is a kernel-based proof that the rescaled first inverse power sum of the Jacobi ensemble converges in distribution to its Bessel point process counterpart.