Schur Function Expansion in Non-Hermitian Ensembles and Averages of Characteristic Polynomials

We study k-point correlators of characteristic polynomials in non-Hermitian ensembles of random matrices, focusing on the Ginibre and truncated unitary random matrices. 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 sum the expansions in terms of representations which interchange the role of k with the matrix size N. 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 in terms of kxk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \times k$$\end{document} determinants or Pfaffians.