Convergence of the empirical spectral measure of unitary Brownian motion

Meckes, Elizabeth; Melcher, Tai

Annales Henri Lebesgue, Tome 1 (2018), pp. 247-265.

Soit ${\left\{U_t^N\right\}}_{t\ge 0}$ un mouvement brownien standard sur $𝕌\left(N\right)$. Étant donnés $N\in \mathbb{N}$ et $t>0$, nous donnons des bornes presque sûres explicites sur la distance de Wasserstein $L^1$ entre la mesure spectrale empirique de $U_t^N$ et la mesure limite en $N$. Nos bornes sont assez précises pour permettre l’étude de l’évolution du processus des valeurs propres, en bornant la vitesse de convergence de chemins de mesures sur des intervalles de temps compacts. Les preuves reposent sur des outils développés par le premier auteur pour obtenir des vitesses de convergence sur la mesure spectrale empirique dans des ensembles de matrices aléatoires classiques, ainsi que des estimées récentes sur la vitesse de convergence des moments pour la distribution spectrale moyennée sur l’ensemble.

Let ${\left\{U_t^N\right\}}_{t\ge 0}$ be a standard Brownian motion on $𝕌\left(N\right)$. 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.

Classification : 60B20, 58J65
Mots-clés : Unitary Brownian motion, empirical spectral measure, heat kernel measure, concentration

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