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Concentration for Coulomb gases on compact manifolds. (English) Zbl 1412.60011

Summary: We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric \(n\)-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of D. Chafaï et al. [J. Funct. Anal. 275, No. 6, 1447–1483 (2018; Zbl 1407.82045)] on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.

MSC:

60B05 Probability measures on topological spaces
82D05 Statistical mechanics of gases
26D10 Inequalities involving derivatives and differential and integral operators
35K05 Heat equation

Citations:

Zbl 1407.82045
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References:

[1] Thierry Aubin: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xviii+395 pp. · Zbl 0896.53003
[2] Daniel Azagra, Juan Ferrera Cuesta, Fernando López-Mesas and Yenny Rangel: Smooth approximation of Lipschitz functions on Riemannian manifolds. Journal of Mathematical Analysis and Applications326, (2007), no. 2, 1370-1378. · Zbl 1116.49011
[3] Carlos Beltrán, Nuria Corral and Juan G. Criado del Rey: Discrete and continuous Green energy on compact manifolds. Journal of Approximation Theory237, (2019), 160-185. · Zbl 1402.31005
[4] Djalil Chafaï, Adrien Hardy, and Mylène Maïda: Concentration for Coulomb gases and Coulomb transport inequalities. Journal of Functional Analysis275, (2018), no. 6, 1447-1483. · Zbl 1407.82045
[5] Isaac Chavel: Eigenvalues in Riemannian geometry. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. · Zbl 0551.53001
[6] David García-Zelada: A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds. To appear in Annales de l’Institut Henri Poincaré,arXiv:1703.02680 · Zbl 1466.60059
[7] Alexander Grigor’yan: Heat kernels on weighted manifolds and applications. The ubiquitous heat kernel, 93-191, Contemporary Mathematics 398, American Mathematical Society, Providence, RI, 2006.
[8] Elton P. Hsu: Stochastic analysis on manifolds. Graduate Studies in Mathematics 38. American Mathematical Society, Providence, RI, 2002. xiv+281 pp. · Zbl 0994.58019
[9] Manjunath Ramanatha Krishnapur: Zeros of random analytic functions. Ph. D. Thesis, U. C. Berkley, 2006. 76 pp. · Zbl 1120.82007
[10] Serge Lang: Introduction to Arakelov theory. Springer-Verlag, New York, 1988. x+187 pp. · Zbl 0667.14001
[11] Michel Ledoux: On optimal matching of Gaussian samples. Zapiski Nauchnykh Seminarov POMI457, (2017), 226-264.
[12] Elliott Lieb and Michael Loss: Analysis. Second edition. Graduate Studies in Mathematics 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp. · Zbl 0966.26002
[13] Mylène Maïda and Édouard Maurel-Segala: Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices. Probability Theory and Related Fields159, (2014), no. 1-2, 329-356. · Zbl 1305.46057
[14] Nicolas Rougerie and Sylvia Serfaty: Higher-dimensional Coulomb gases and renormalized energy functionals. Communications on Pure and Applied Mathematics69, (2016), no. 3, 519-605. · Zbl 1338.82043
[15] Sylvia Serfaty: Systems of Points with Coulomb Interactions. Proceedings of the International Congress of Mathematicians 20181, 935-978, Rio de Janeiro, 2018. arXiv:1712.04095 · Zbl 1417.81166
[16] Cédric Villani: Topics in optimal transportation. Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI, 2003. xvi+370 pp. · Zbl 1106.90001
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