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BIRS Workshop Lecture Videos

Canonical structures in traffic spaces: with a view toward random matrices Au, Benson


For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau)$ that extends the trace $\varphi$ [CDM16]. This construction comes equipped with some canonical independence structure: in a joint work in progress with Male, we show that $(\mathcal{G}(\mathcal{A}), \tau)$ can be realized as the free product of three natural subalgebras (in the sense of Voiculescu), and that there exists a canonical homomorphic conditional expectation $P$ onto a subalgebra intermediate to $\mathcal{A}$ and $\mathcal{G}(\mathcal{A})$. Combining this with the coherent convergence properties of $(\mathcal{G}(\mathcal{A}), \tau)$ proved in [CDM16], we show that free independence describes the asymptotic behaviour of a large class of dependent random matrices (in particular, we recover and explain a result of Bryc, Dembo, and Jiang on random Markov matrices [BDJ06]).

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