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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. [ABSTRACT FROM AUTHOR]

For general dimension d, we prove the equidistribution of energy at the micro-scale in $$\mathbb {R}^{d}$$ for the optimal point configurations appearing in Coulomb gases at zero temperature. More precisely, we show that, after blow-up at the scale corresponding to the interparticle distance, the value of the energy in any large enough set is completely determined by the macroscopic density of points. This uses the 'jellium energy' which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for 2-dimensional log-gases. For Riesz gases with interaction potentials $$g(x)=|x|^{-s}, s\in ]\min \{0,d-2\},d[$$ , we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case, we use the Caffarelli-Silvestre description of the nonlocal fractional Laplacians in $$\mathbb {R}^{d}$$ to render the problem local. [ABSTRACT FROM AUTHOR]

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method. [ABSTRACT FROM AUTHOR]