Convergence rates of the empirical spectral measure of unitary Brownian motion

Let $\{U^N_t\}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}(N)$. For fixed $N\in\mathbb{N}$ and $t>0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the ensemble-averaged spectral measure and to the large-$N$ limiting measure identified by Biane. 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.