Derivative relations for determinants, Pfaffians and characteristic polynomials in random matrix theory
Explicit expressions are proven for derivatives of the ratio of a determinant or Pfaffian determinant and a Vandermonde determinant. Such ratios appear for example in general group integrals of Harish-Chandra--Itzykson--Zuber type and in expectation values of products of characteristic polynomials in random matrix theory. In the latter case we start from known results for general non-Hermitian and Hermitian ensembles for expectation values without derivatives, at finite matrix size. They are given in terms of the determinant or Pfaffian of the corresponding kernel, for unitary or orthogonal and symplectic ensembles, respectively. Several equivalent expressions are proven for general ratios of determinants, starting from first order derivatives containing the Borel transform of the corresponding matrix or kernel. Higher order derivatives are expressed as sums over partitions containing determinants of derivatives of these, with coefficients given in terms of combinatorial expressions. Our most general result is valid for mixed higher order derivatives of ratios of determinants in several variables. This generalises previous findings, e.g. for mixed moments in specific ensembles of random matrices, relevant in applications to the Riemann $\zeta$-function. Applications of our results to several examples are presented, including the complex Ginibre ensemble and the circular unitary ensemble.