Meckes, Elizabeth; Melcher, Tai
Convergence of the empirical spectral measure of unitary Brownian motion
Annales Henri Lebesgue, Volume 1 (2018), p. 247-265

Metadata

KeywordsUnitary Brownian motion, empirical spectral measure, heat kernel measure, concentration

Abstract

Let {U t N } t0 be a standard Brownian motion on 𝕌N. For fixed 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.


References

[AGZ10] Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer An Introduction to Random Matrices, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 118 (2010), xiv+492 pages | MR 2760897 | Zbl 1184.15023

[Bia97a] Biane, Philippe Free Brownian motion, free stochastic calculus and random matrices, Free probability theory (Waterloo, ON, 1995), American Mathematical Society (Fields Institute Communications) Volume 12 (1997), pp. 1-19 | MR 1426833 | Zbl 0873.60056

[Bia97b] Biane, Philippe 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

[CDK18] Collins, Benoît; Dahlqvist, Antoine; Kemp, Todd The spectral edge of unitary Brownian motion, Probab. Theory Relat. Fields, Volume 170 (2018) no. 1-2, pp. 49-93 | Article | MR 3748321 | Zbl 1384.15009

[DHK13] Driver, Bruce K.; Hall, Brian C.; Kemp, Todd The large-N limit of the Segal-Bargmann transform on 𝕌 N , J. Funct. Anal., Volume 265 (2013) no. 11, pp. 2585-2644 | Article | MR 3096985 | Zbl 1286.22010

[Gri99] Grigor’Yan, Alexander 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

[Gro87] Grove, Karsten Metric differential geometry, Differential geometry (Lyngby, 1985), Springer (Lecture Notes in Mathematics) Volume 1263 (1987), pp. 171-227 | Article | MR 905882 | Zbl 0622.53022

[Kat04] Katznelson, Yitzhak An introduction to harmonic analysis, Cambridge University Press, Cambridge Mathematical Library (2004), xviii+314 pages | Article | MR 2039503 | Zbl 1055.43001

[Kem15] Kemp, Todd Heat Kernel Empirical Laws on 𝕌 N and 𝔾𝕃 N , J. Theor. Probab., Volume 30 (2015) no. 2, pp. 1-55 | Zbl 06738999

[Led99] Ledoux, Michel Concentration of measure and logarithmic Sobolev inequalities, Séminaire de probabilités de Strasbourg XXXIII, Springer (Lecture Notes in Mathematics) Volume 1709 (1999), pp. 120-216 | Article | Numdam | MR 1767995 | Zbl 0957.60016

[Lév08] Lévy, Thierry Schur–Weyl duality and the heat kernel measure on the unitary group, Adv. Math., Volume 218 (2008) no. 2, pp. 537-575 | Article | MR 2407946 | Zbl 1147.60053

[LM10] Lévy, Thierry; Maïda, Mylène Central limit theorem for the heat kernel measure on the unitary group, J. Funct. Anal., Volume 259 (2010) no. 12, pp. 3163-3204 | Article | MR 2727643 | Zbl 1207.60018

[MM13a] Meckes, Elizabeth S.; Meckes, Mark W. Concentration and convergence rates for spectral measures of random matrices, Probab. Theory Relat. Fields, Volume 156 (2013) no. 1-2, pp. 145-164 | Article | MR 3055255 | Zbl 1291.60015

[MM13b] Meckes, Elizabeth S.; Meckes, Mark W. Spectral measures of powers of random matrices, Electron. Commun. Probab., Volume 18 (2013), 78, 13 pages (Art. ID 78, 13 pages) | Article | MR 3109633 | Zbl 1310.60003

[MM17] Meckes, Elizabeth S.; Meckes, Mark W. Rates of convergence for empirical spectral measures: a soft approach, Convexity and concentration, Springer (The IMA Volumes in Mathematics and its Applications) Volume 161 (2017), pp. 1-21 | MR 3837270 | Zbl 1376.15027

[Rai97] Rains, Eric M. Combinatorial properties of Brownian motion on the compact classical groups, J. Theor. Probab., Volume 10 (1997) no. 3, pp. 659-679 | Article | MR 1468398 | Zbl 1002.60504

[Riv81] Rivlin, Theodore J. An Introduction to the Approximation of Functions, Dover Publications, Dover Books on Advanced Mathematics (1981), viii+150 pages (Corrected reprint of the 1969 original) | MR 634509 | Zbl 0489.41001

[RV10] Rudelson, Mark; Vershynin, Roman Non-asymptotic theory of random matrices: extreme singular values, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency (2010), pp. 1576-1602 | MR 2827856 | Zbl 1227.60011

[SC94] Saloff-Coste, Laurent Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z., Volume 217 (1994) no. 4, pp. 641-677 | Article | MR 1306030 | Zbl 0815.60074

[Vil09] Villani, Cédric Optimal transport. Old and new, Springer, Grundlehren der Mathematischen Wissenschaften, Volume 338 (2009), xxii+973 pages | Article | MR 2459454 | Zbl 1156.53033

[Xu97] Xu, Feng A random matrix model from two-dimensional Yang-Mills theory, Commun. Math. Phys., Volume 190 (1997) no. 2, pp. 287-307 | Article | MR 1489573 | Zbl 0937.81043