Asymptotic cyclic-conditional freeness of random matrices

2022/07/13 by Guillaume Cébron, Cébron, Guillaume, Nicolas Gilliers +1
Mathematics · #Random Matrices and Applications #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2207.06249

Abstract

Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on N× N random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix U_N. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where U_N has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector v_N. In the limit N → +∞, we show that N× N matrices randomly rotated by the matrix U_N are asymptotically conditionally free with respect to the normalized trace and the state vector v_N. To describe second order asymptotics, we define cyclic-conditional freeness, a new notion of independence unifying infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. The infinitesimal distribution in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for ordered freeness and for indented independence.

Related

• Freeness of type B and conditional freeness for random matrices
• Free probability and random matrices
• Traffic distributions and independence: permutation invariant random matrices and the three notions of independence
• Asymptotically liberating sequences of random unitary matrices
• A combinatorial result on asymptotic independence relations for random matrices with non-commutative entries