Schur function expansion in non-Hermitian ensembles and averages of characteristic polynomials

Abstract: We study $k$-point correlators of characteristic polynomials in non-Hermitian ensembles of random matrices, focusing on the real, complex and quaternion $N \times N$ Ginibre ensembles. Our approach is based on the technique of character expansions, which expresses the correlator as a sum over partitions involving Schur functions. We show how to re-sum the expansions in terms of representations which interchange the roles of $N$ and $k$. We also provide a probabilistic interpretation of the character expansion analogous to the Schur measure, linking the correlators to the distribution of the top row in certain Young diagrams. In more specific examples we evaluate these expressions explicitly in terms of $k \times k$ determinants or Pfaffians. We show that our approach extends to other ensembles, such as truncations of random unitary matrices.