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Convergence of the empirical spectral measure of unitary Brownian motion

Abstract

Let {UtN}t0\{U^N_t\}_{t\ge 0} be a standard Brownian motion on U(N)\mathbb{U}(N). For fixed NNN\in\mathbb{N} and t>0t>0, we give explicit bounds on the L1L_1-Wasserstein distance of the empirical spectral measure of UtNU^N_t to both the ensemble-averaged spectral measure and to the large-NN limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.Comment: 17 pages; rate of convergence for fixed tt sharpened and proof simplified; new result on convergence of paths of empirical measures on compact time interval

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