An RDT based approach to large deviations of Wishart and Wigner matrices spectral edges

Abstract

We present a novel methodology for studying \emph{large deviations principles} (LDPs) of random matrices. By utilizing a partially lifted variant of \emph{random duality theory} (RDT), we develop a generic LDP framework that completely circumvents traditional random matrix theory (RMT) methods. To demonstrate the framework's simplicity and accuracy, we apply it to the Wishart and Wigner GOE classical statistical ensembles. In both cases, we obtain elegant LDP characterizations of the upper and lower spectral edges that fully match the results achieved through traditional \emph{Coulomb gas} methodologies in [85,95].