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Constructive Approximation

, Volume 48, Issue 1, pp 101–136 | Cite as

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

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Abstract

We consider particle systems (also known as point processes) on the line and in the plane and are particularly interested in “hole” events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) \(\beta \)-Ginibre ensembles. We also discuss some possible directions for future investigations.

Keywords

Point processes Particle systems Coulomb gases Random matrices Random polynomials Hole probabilities Large deviations Empirical measures 

Mathematics Subject Classification

Primary 60G55 Secondary 60F10 

Notes

Acknowledgements

We thank the authors of the paper [46] for allowing us to use the picture in Fig. 7. We thank Diego Ayala for allowing us to use the picture in Figure 8. We thank the anonymous referee for numerous helpful suggestions. The work of S.G. was supported in part by the ARO Grant W911NF-14-1-0094, the NSF Grant DMS-1148711 and the NUS Grant R-146-000-250-133.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National University of SingaporeSingaporeSingapore
  2. 2.Tel Aviv UniversityTel AvivIsrael

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