Let be a standard Brownian motion on . For fixed and , we give explicit almost-sure bounds on the -Wasserstein distance between the empirical spectral measure of and the large- 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.
[Bia97a] Free Brownian motion, free stochastic calculus and random matrices, Free probability theory (Waterloo, ON, 1995) (Fields Institute Communications), Volume 12, American Mathematical Society, 1997, pp. 1-19 | MR 1426833 | Zbl 0873.60056
[Bia97b] Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems, J. Funct. Anal., Volume 144 (1997) no. 1, pp. 232-286 | Article | MR 1430721 | Zbl 0889.47013
[Gri99] Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., Volume 36 (1999) no. 2, pp. 135-249 | Article | MR 1659871 | Zbl 0927.58019
[Led99] Concentration of measure and logarithmic Sobolev inequalities, Séminaire de probabilités de Strasbourg XXXIII (Lecture Notes in Mathematics), Volume 1709, Springer, 1999, pp. 120-216 | Article | MR 1767995 | Zbl 0957.60016
[MM17] Rates of convergence for empirical spectral measures: a soft approach, Convexity and concentration (The IMA Volumes in Mathematics and its Applications), Volume 161, Springer, 2017, pp. 1-21 | MR 3837270 | Zbl 1376.15027
[Riv81] An Introduction to the Approximation of Functions, Dover Books on Advanced Mathematics, Dover Publications, 1981, viii+150 pages (Corrected reprint of the 1969 original) | MR 634509 | Zbl 0489.41001