Joint Moments of Characteristic Polynomials from the Orthogonal and Unitary Symplectic Groups

We establish asymptotic formulae for general joint moments of characteristic polynomials and their higher-order derivatives associated with matrices drawn randomly from the groups $\mathrm{USp}(2N)$ and $\mathrm{SO}(2N)$ in the limit as $N\to\infty$. This relates the leading-order asymptotic contribution in each case to averages over the Laguerre ensemble of random matrices. We uncover an exact connection between these joint moments and a solution of the $σ$-Painlevé V equation, valid for finite matrix size, as well as a connection between the leading-order asymptotic term and a solution of the $σ$-Painlevé III$'$ equation in the limit as $N \rightarrow \infty$. These connections enable us to derive exact formulae for joint moments for finite matrix size and for the joint moments of certain random variables arising from the Bessel point process in a recursive way. As an application, we provide a positive answer to a question proposed by Altuğ et al.