Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices

Abstract

Abstract In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions goes to infinity. We consider a general framework containing the cases where A and/or B are taken from an invariant ensemble or are fixed diagonal matrices. We show that the tilting method introduced in Guionnet and Maïda (2020 Electron. J. Probab. 25 1–24) can be extended to our general setting and is equivalent to the study of a spherical spin glass model specific to the operation—sum of symmetric matrices/product of symmetric matrices/sum of rectangular matrices—we are considering.