Large deviations for the largest eigenvalue of the sum of two random matrices

Alice Guionnet (1, 2) , Mylène Maïda (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 CNRS - Centre National de la Recherche Scientifique
3 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.

Domaines: Probabilités [math.PR]

Alice Guionnet, Mylène Maïda. Large deviations for the largest eigenvalue of the sum of two random matrices. Electronic Journal of Probability, 2020, 25, pp.14. ⟨hal-01887673v2⟩