An RDT based confirmation of Lehner's formula for Kronecker-Gaussian matrices

Abstract

Remarkable breakthroughs [25,43] established the so-called strong asymptotic freeness between classical Gaussian random ensembles and their semicircular free counterparts. Along the same lines, the Lehner formula [31], associated with the free counterpart, precisely determines the spectral edges of Kronecker-Gaussian matrices. We here revisit and study this formula without the utilization of random matrix theory and spectral methods. In particular, relying on concepts utilized within \emph{Random Duality Theory} (RDT) [45,46,51], we reconfirm Lehner's formula and effectively reprove the key asymptotic freeness results obtained via spectral methods in [25,43].