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### Abstract

Let ${\left\{{U}_{t}^{N}\right\}}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}\left(N\right)$. For fixed $N\in \mathbb{N}$ and $t>0$, we give explicit almost-sure bounds on the ${L}_{1}$-Wasserstein distance between the empirical spectral measure of ${U}_{t}^{N}$ and the large-$N$ limiting measure. The bounds obtained are tight enough that we are able to use them to study the evolution of the eigenvalue process in time, bounding the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to obtain rates of convergence of the empirical spectral measures in classical random matrix ensembles, as well as recent estimates for the rates of convergence of moments of the ensemble-averaged spectral distribution.

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