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

Subhroshekhar Ghosh*, Alon Nishry

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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) β-Ginibre ensembles. We also discuss some possible directions for future investigations.

Original languageEnglish
Pages (from-to)101-136
Number of pages36
JournalConstructive Approximation
Issue number1
StatePublished - 1 Aug 2018


FundersFunder number
National Science FoundationDMS-1148711
Directorate for Mathematical and Physical Sciences1148711
Army Research OfficeW911NF-14-1-0094
National University of SingaporeR-146-000-250-133


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


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