Article Dans Une Revue Indiana University Mathematics Journal Année : 2024

Freeness of type B and conditional freeness for random matrices

Résumé

The asymptotic freeness of independent unitarily invariant N×N random matrices holds in expectation up to O(N−2). An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type B. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type B, conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles.

Dates et versions

hal-04871222 , version 1 (07-01-2025)

Identifiants

Citer

Guillaume Cébron, Antoine Dahlqvist, Franck Gabriel. Freeness of type B and conditional freeness for random matrices. Indiana University Mathematics Journal, 2024, 73 (3), pp.1207--1252. ⟨10.1512/iumj.2024.73.9916⟩. ⟨hal-04871222⟩
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