Support convergence for XY* in paired Gaussian matrices

Establish whether, for paired Gaussian matrices X and Y with corresponding entries jointly i.i.d. Gaussian, the support of the empirical spectral density μ_{XY*}^N converges to the support of the limiting spectral density μ_{XY*}.

Background

The paper studies the eigenvalue spectrum support for products of paired Gaussian random matrices, focusing on XY* (non-Hermitian Wishart-type product) and XY^\dagger (involving the Moore–Penrose pseudo-inverse). The authors establish exact support descriptions for XY* under various conditions and derive a circular-law-type support for XY^\dagger assuming a conjecture.

The conjecture concerns the absence of isolated outliers for XY*, equivalently the convergence of the support of the empirical spectral distribution to that of the limiting distribution. This is reminiscent of known results for the Ginibre ensemble, where outliers can be created by bounded-rank perturbations. The authors suggest that a proof might be obtained by showing continuity of the Brown measure with respect to the relevant topology of convergence, referencing Belinschi–Śniady–Speicher (2018).

The conjecture is also used as an assumption in proving the support for XY^\dagger. The authors explicitly state that the proof of Theorem 3 requires the yet unproven conjecture, underscoring its importance for the broader results in the paper.