Universal Characteristic Polynomials in Non-Hermitian Random Matrix Theory

by Tobias Winkler ( Universität Bielefeld)

Thursday Jul 23, 2026, 4:15 PM → 6:15 PM Europe/Berlin

U2-253 (UHG)

In this talk, we study 2N × 2N non-Hermitian random matrices with block-off-diagonal structure. Their N × N off-diagonal blocks are built from complex Gaussian random variables. Depending on the model, the independent entries are unconstrained, symmetric, or antisymmetric. A parameter interpolates between a genuinely non-Hermitian regime and a Hermitian limiting case. We compare how these three choices affect the spectral behavior while preserving the same overall block-off-diagonal form. Our main objects are mean values of products of characteristic polynomials of the 2N × 2N random matrix and their complex conjugates. To evaluate these mean values, we use Grassmann integral representations of determinants, followed by Gaussian integration and Hubbard–Stratonovich decoupling. This yields dual integral representations whose number of integration variables is independent of N. For one characteristic polynomial and one complex conjugate characteristic polynomial, the dual integrals can be evaluated explicitly for all N. For unconstrained and symmetric blocks, the resulting finite-N formulae can be rewritten as contour integral representations. In the large-N analysis, the distinction between the block constraints is reflected in the order of the relevant pole. For antisymmetric blocks, the finite-N formulae instead exhibit a parity dependence caused by zero modes in odd matrix dimension.