A Combinatorial Approach to Mixed Ratios of Characteristic Polynomials

We provide a combinatorial derivation of an asymptotic formula for averages of mixed ratios of characteristic polynomials over the unitary group, where mixed ratios are products of ratios and/or logarithmic derivatives. Our proof of this formula is a generalization of Bump and Gamburd’s elegant combinatorial proof of Conrey, Forrester and Snaith’s formula for averages of ratios of characteristic polynomials over the unitary group. One application of this formula is an asymptotic expression for sums over zeros of a random characteristic polynomial from the unitary group, which we call an explicit formula for eigenvalues in an analogy to what is called an explicit formula in the context of $L$-functions.