Asymptotic characteristic polynomial for the spherical ensemble (ratio of Ginibre matrices)

Determine the high-dimensional asymptotic behavior of the characteristic polynomial det(AB^{-1} − z I_n) for the spherical ensemble M = AB^{-1}, where A and B are independent n×n complex Ginibre matrices. Specifically, establish convergence (under appropriate normalization) to a random analytic limit object and, as a consequence, derive a central limit theorem for the logarithmic potential obtained from the modulus of the characteristic polynomial.

Background

The paper studies the ratio M = AB^{-1} of two independent Girko matrices and proves universality for the high-dimensional fluctuation of the spectral radius, including a replacement principle and convergence to the infinite Ginibre process for local statistics.

In the Gaussian case (the spherical ensemble), where A and B are complex Ginibre matrices, the eigenvalues form a determinantal Coulomb gas with spherical symmetry. While spectral-radius fluctuations are established, the authors point out that understanding the asymptotic behavior of the characteristic polynomial of M, in the sense of convergence to a random analytic limit (as is known for related models), remains unresolved even in this integrable spherical case.

Resolving this would also yield, via the modulus and logarithm of the characteristic polynomial, a central limit theorem for the log-potential of the eigenvalue process.