This paper gives explicit formulas for derivatives of expressions that look like a determinant or Pfaffian divided by a Vandermonde determinant. Such ratios appear when researchers compute expectation values of products of characteristic polynomials of random matrices. These expectation values are used in several areas, from models of quantum systems and quantum chromodynamics to statistical studies related to the Riemann zeta function.

The authors start from known finite-size formulas in random matrix theory. For many ensembles, averages of products of characteristic polynomials are written as a determinant (for unitary symmetry) or a Pfaffian (for orthogonal and symplectic symmetry) of a kernel function, divided by a Vandermonde determinant. They prove several equivalent expressions for the derivatives of those ratios. First-order derivatives are given in terms of a Borel transform of the matrix or kernel. Higher-order and mixed derivatives are written as sums over partitions. Each term in those sums is a determinant built from derivatives of a transformed kernel, with combinatorial coefficients that the authors make explicit.

At a high level the technical problem is that the Vandermonde factor in the denominator prevents a straightforward differentiation. The paper shows how to remove that denominator by applying differential operators and by transforming the kernel (the Borel transform is one example). For mixed derivatives at different points they use tools from representation theory and combinatorics, such as Schur functions and Kostka numbers, to organise the many terms. A Pfaffian is explained and used where underlying matrices are antisymmetric; roughly, a Pfaffian plays the role that a determinant plays for symmetric square matrices.

Why this matters: the new formulas let one compute derivatives of characteristic polynomials at finite matrix size for a wide range of random matrix ensembles and for general weight functions that define the kernel. That is useful where derivatives of characteristic polynomials are of interest, for example in proposed connections to the statistics of zeros of the Riemann zeta function, in partition functions used in low-energy quantum chromodynamics, and in studies of non-Hermitian ensembles. The authors show how their general results apply to concrete examples, including the complex Ginibre ensemble and the Circular Unitary Ensemble (CUE).