Average Winding Number for Determinantal Curves associated with 2-Matrix Models in the Class AIII

To classify one-dimensional disordered quantum systems with chiral symmetry, we analyse the winding number of the determinant of a parametrized non-Hermitian random matrix field over the unit circle modelling the off-diagonal block of a disordered chiral Hamiltonian. The associated partition function is computed explicitly for a broad class of additive two-matrix models extending beyond the Ginibre Unitary Ensemble. In the large-dimension limit, we derive an asymptotic expansion of the average winding number whose leading term exhibits universal features, up to the tail behaviour of the underlying random matrix ensemble, and identify a new correction term absent in the previously studied Ginibre case.