Electronic Communications in Probability

A note on large deviations for 2D Coulomb gas with weakly confining potential

Adrien Hardy

Full-text: Open access

Abstract

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.

Article information

Source
Electron. Commun. Probab. Volume 17 (2012), paper no. 19, 12 pp.

Dates
Accepted: 17 May 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263152

Digital Object Identifier
doi:10.1214/ECP.v17-1818

Mathematical Reviews number (MathSciNet)
MR2926763

Zentralblatt MATH identifier
1258.60027

Subjects
Primary: 60F10: Large deviations
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B21: Continuum models (systems of particles, etc.)

Keywords
Large deviations Coulomb gas Random matrices Compactification

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Hardy, Adrien. A note on large deviations for 2D Coulomb gas with weakly confining potential. Electron. Commun. Probab. 17 (2012), paper no. 19, 12 pp. doi:10.1214/ECP.v17-1818. https://projecteuclid.org/euclid.ecp/1465263152.


Export citation

References

  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5
  • R. Ash and W. Novinger, Complex variables, Dover publication, Second edition (2007).
  • Ben Arous, G.; Guionnet, A. Large deviations for Wigner's law and Voiculescu's non-commutative entropy. Probab. Theory Related Fields 108 (1997), no. 4, 517–542.
  • Cegrell, U.; Kolodziej, S.; Levenberg, N. Two problems on potential theory for unbounded sets. Math. Scand. 83 (1998), no. 2, 265–276.
  • Chau, Ling-Lie; Zaboronsky, Oleg. On the structure of correlation functions in the normal matrix model. Comm. Math. Phys. 196 (1998), no. 1, 203–247.
  • R. M. Dudley, Real analysis and probablity, Cambridge Studies in Advanced Mathematics Vol. 74, Cambridge, England (2002).
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Forrester, P. J. Log-gases and random matrices. London Mathematical Society Monographs Series, 34. Princeton University Press, Princeton, NJ, 2010. xiv+791 pp. ISBN: 978-0-691-12829-0
  • A. Hardy and A. B. J. Kuijlaars, Large deviations for a non-centered Wishart matrix, ARXIV1204.6261.
  • A. Hardy and A. B. J. Kuijlaars, Weakly admissible vector equilibrium problems, J. Approx. Theory 164 (2012), 854–868.
  • Hiai, Fumio; Petz, DÄ‚Å nes. The semicircle law, free random variables and entropy. Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8
  • Krishnapur, Manjunath. From random matrices to random analytic functions. Ann. Probab. 37 (2009), no. 1, 314–346.
  • Mehta, Madan Lal. Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991. xviii+562 pp. ISBN: 0-12-488051-7