Large deviations for the largest eigenvalue of the sum of two random matrices. (English) Zbl 1440.15035

Guionnet, Alice; Maïda, Mylène

Electron. J. Probab. 25, Paper No. 14, 24 p. (2020).

Summary: In this paper, we consider the addition of two matrices in generic position, namely \(A+UBU^*\), 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 the case when \(A\) and \(B\) have no outliers.

MSC:

15B52	Random matrices (algebraic aspects)
60B20	Random matrices (probabilistic aspects)

Keywords:

random matrix; large deviations; extreme eigenvalues; free convolution

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