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BIRS Workshop Lecture Videos

Asymptotic freeness of Biane-Perelemov-Popov matrices Novak, Jonathan


Biane-Perelemov-Popov matrices are a family of quantum random matrices which quantize uniformly random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors.  BPP matrices depend on a semiclassical parameter which controls the degree of noncommutativity of their entries.  Improving on results of Biane and Collins-Sniady, and resolving a conjecture of Bufetov-Gorin, we demonstrate that classically independent BPP matrices become asymptotically free provided the semiclassical parameter decays as the dimension increases.  This result has consequences in asymptotic representation theory, implying in particular that $GL_N(\mathbb{C})$ tensor products are described by free probability in any and all semiclassical limits.  This is joint work with Collins and Sniady.

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