November 2023 The Wasserstein distance to the circular law
Jonas Jalowy
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(4): 2285-2307 (November 2023). DOI: 10.1214/22-AIHP1317

Abstract

We investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the circular law. For Ginibre matrices, we obtain an optimal rate of convergence n1/2 in 1-Wasserstein distance. This shows that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster (due to the repulsive behaviour) compared to that of i.i.d. points, which is known to include a logarithmic factor. For non-Gaussian entry distributions with finite moments, we also show that the rate of convergence nearly attains this optimal rate.

Nous étudions la distance de Wasserstein entre la distribution spectrale empirique des matrices aléatoires non hermitiennes et la loi circulaire. Pour les matrices de Ginibre, nous obtenons un taux de convergence optimal n1/2 en distance 1-Wasserstein. Cela montre que d’espérance du coût de transport des valeurs propres complexes vers la mesure uniforme sur le disque unitaire décroît plus rapidement (en raison du comportement répulsif) par rapport à celui de points i.i.d. qui inclut un facteur logarithmique. Pour le cas des entrées avec loi non gaussienne à moments finis, nous montrons également que le taux de convergence atteint presque ce taux optimal.

Funding Statement

Supported by the German Research Foundation (DFG) through the SPP 2265 Random Geometric Systems.

Acknowledgments

I would like to thank Anna Gusakova, Martin Huesmann and Matthias Erbar for helpful discussions and valuable suggestions. Furthermore, I thank the referees for reading the manuscript so thoroughly and for all their great feedback.

Citation

Download Citation

Jonas Jalowy. "The Wasserstein distance to the circular law." Ann. Inst. H. Poincaré Probab. Statist. 59 (4) 2285 - 2307, November 2023. https://doi.org/10.1214/22-AIHP1317

Information

Received: 25 January 2022; Revised: 15 September 2022; Accepted: 19 September 2022; Published: November 2023
First available in Project Euclid: 3 November 2023

Digital Object Identifier: 10.1214/22-AIHP1317

Subjects:
Primary: 60B20
Secondary: 41A25 , 49Q22 , 60G55

Keywords: circular law , Ginibre matrices , Optimal transport , rate of convergence , Wasserstein distance

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré

Vol.59 • No. 4 • November 2023
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