Large Deviation Principles for Hypersingular Riesz Gases

We study N-particle systems in R^d whose interactions are governed by a hypersingular Riesz potential | x- y| ^-s, s> d, and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as N→ ∞ for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature β. We show that a large deviation principle holds with a rate function of the form ‘β-Energy + Entropy’, yielding that the microscopic behavior (on the scale N^-1/d) of such N-point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case s< d, where on the macroscopic scale N-point empirical measures have limiting density independent of β, the limiting density for s> d is strongly β-dependent.