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

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. The matrices A and B are each assumed to be taken from an invariant (or bi-invariant) ensemble with a confining potential with a possible wall beyond which no eigenvalues/singular values are allowed. The introduction of this wall puts different models in a very generic framework. In particular, the case where the wall is exactly at the right edge of the limiting spectral density is equivalent, from the point of view of large deviations, to considering a fixed diagonal matrices, as studied previously in Ref.[1]. We show that that the tilting method introduced in Ref. [1] 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.

Spherical Integrals of Sublinear Rank

“…Because of the simple structure of the Hamiltonian in the spherical case, the computation of the free energy is closely tied to the behavior of the eigenvalues of a GOE matrix, which has been the studied extensively in random matrices. Random matrix techniques have been applied to study the fluctuations of the free energy and corresponding phase transtions in [6,7,5], the connection the large deviations of the top eigenvalue in [54], and the marginals of spherical spin glasses with correlated disorder matrices in [11].…”