Nous établissons des vitesses de convergence pour la loi du cercle de l’ensemble de Ginibre complexe. Plus précisément, nous donnons des bornes supérieurs pour les distances de Wasserstein d’ordre entre la mesure spectrale empirique de l’ensemble de Ginibre complexe normalisée et la mesure uniform du disque, en espérance et presque sûrement. Si , les bornes sont de la taille , à des facteurs logarithmiques près.
We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the -Wasserstein distances between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For , the bounds are of the order , up to logarithmic factors.
@article{AFST_2015_6_24_1_93_0,
author = {Elizabeth S. Meckes and Mark W. Meckes},
title = {A rate of convergence for the circular law for the complex {Ginibre} ensemble},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {93--117},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 24},
number = {1},
year = {2015},
doi = {10.5802/afst.1443},
mrnumber = {3325952},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1443/}
}
TY - JOUR AU - Elizabeth S. Meckes AU - Mark W. Meckes TI - A rate of convergence for the circular law for the complex Ginibre ensemble JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2015 SP - 93 EP - 117 VL - 24 IS - 1 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1443/ DO - 10.5802/afst.1443 LA - en ID - AFST_2015_6_24_1_93_0 ER -
%0 Journal Article %A Elizabeth S. Meckes %A Mark W. Meckes %T A rate of convergence for the circular law for the complex Ginibre ensemble %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2015 %P 93-117 %V 24 %N 1 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1443/ %R 10.5802/afst.1443 %G en %F AFST_2015_6_24_1_93_0
Elizabeth S. Meckes; Mark W. Meckes. A rate of convergence for the circular law for the complex Ginibre ensemble. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 93-117. doi : 10.5802/afst.1443. https://afst.centre-mersenne.org/articles/10.5802/afst.1443/
[1] Anderson (G. W.), Guionnet (A.), and Zeitouni (O.).— An Introduction to Random Matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. | MR | Zbl
[2] Bai (Z. D.).— Circular law. Ann. Probab., 25(1), p. 494-529 (1997). | MR | Zbl
[3] Bordenave (C.) and Chafaï (D.).— Around the circular law. Probab. Surv., 9, p. 1-89 (2012). | MR | Zbl
[4] Dallaporta (S.).— Eigenvalue variance bounds for Wigner and covariance random matrices. Random Matrices: Theory Appl., 1:1250007, 2012. | MR | Zbl
[5] Feller (W.).— An Introduction to Probability Theory and its Applications. Vol. I. Third edition. John Wiley & Sons, Inc., New York-London-Sydney, 1968. | MR | Zbl
[6] Forrester (P. J.).— Log-gases and random matrices, volume 34 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2010. | MR | Zbl
[7] Ginibre (J.).— Statistical ensembles of complex, quaternion, and real matrices. J. Mathematical Phys., 6, p. 440-449 (1965). | MR | Zbl
[8] Gustavsson (J.).— Gaussian uctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Statist., 41(2), p. 151-178 (2005). | Numdam | MR | Zbl
[9] Hough (J. B.), Krishnapur (M.), Peres (Y.), and Virág (B.).— Determinantal processes and independence. Probab. Surv., 3, p. 206-229 (2006). | MR | Zbl
[10] Khoruzhenko (B. A.) and Sommers (H.).— Non-Hermitian ensembles. In The Oxford Handbook of Random Matrix Theory, pages 376-397. Oxford Univ. Press, Oxford, 2011. | MR | Zbl
[11] Meckes (E. S.) and Meckes (M. W.).— Spectral measures of powers of random matrices. Electron. Commun. Probab., 18, no. 78, 13, 2013. | MR
[12] Mehta (M. L.).— Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York-London, 1967. | MR | Zbl
[13] Mehta (M. L.).— Random Matrices, volume 142 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, third edition, 2004. | MR | Zbl
[14] Rougerie (N.) and Serfaty (S.).— Higher dimensional Coulomb gases and renormalized energy functionals. Preprint, available at http://arxiv.org/abs/1307.2805, 2013.
[15] Sandier (E.) and Serfaty (S.).— 2D Coulomb gases and the renormalized energy. Preprint, available at http://arxiv.org/abs/1201.3503, 2012.
[16] Talagrand (M.).— The Generic Chaining: Upper and Lower Bounds of Stochastic Processes. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. | MR | Zbl
[17] Tao (T.) and Vu (V.).— Random matrices: the circular law. Commun. Contemp. Math., 10(2), p. 261-307 (2008). | MR | Zbl
[18] Tao (T.) and Vu (V.).— Random matrices: universality of ESDs and the circular law. Ann. Probab., 38(5), p. 2023-2065 (2010). With an appendix by M. Krishnapur. | MR | Zbl
[19] Villani (C.).— Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. | MR | Zbl
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