Abstract
We prove a large deviation principle for the sequence of push-forwards of empirical measures in the setting of Riesz potential interactions on compact subsets K in \(\mathbb {R}^d\) with continuous external fields. Our results are valid for base measures on K satisfying a strong Bernstein–Markov type property for Riesz potentials. Furthermore, we give sufficient conditions on K (which are satisfied if K is a smooth submanifold) so that a measure on K that satisfies a mass-density condition will also satisfy this strong Bernstein–Markov property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bloom, T., Levenberg, N., Piazzon, F., Wielonsky, F.: Bernstein–Markov: a survey. Dolomites Res. Note Approx. 8, 1–17 (2015)
Bloom, T., Levenberg, N., Wielonsky, F.: Vector energy and large deviation. J. d’Analyse Math. 125(1), 139–174 (2015)
Bloom, T., Levenberg, N., Wielonsky, F.: Logarithmic potential theory and large deviation. Comput. Methods Funct. Theory 15(4), 555–594 (2015)
Brauchart, J., Dragnev, P., Saff, E.: Riesz external field problems on the hypersphere and optimal point separation. Potential Anal. 41, 647–678 (2014)
Chafai, D., Gozlan, N., Zitt, P.-A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)
Dembo, A., Zeitouni, O.: Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin (2010)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley, Chichester (2003)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Landkof, N.S.: Foundations of Modern Potential Theory, Grundlehren der Mathematischen Wissenschaften, 180. Springer, Berlin (1972)
Leblé, T., Serfaty, T.: Large deviation principle for empirical fields of Log and Riesz gases. Invent. Math. (2017). https://doi.org/10.1007/s00222-017-0738-0
Saff, E., Totik, V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Doug Hardin.
Norman Levenberg was supported by Simons Foundation (grant no. 354549).
Rights and permissions
About this article
Cite this article
Bloom, T., Levenberg, N. & Wielonsky, F. A Large Deviation Principle for Weighted Riesz Interactions. Constr Approx 47, 119–140 (2018). https://doi.org/10.1007/s00365-017-9396-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-017-9396-0