Large Deviation Principles for Hypersingular Riesz Gases
We study N-particle systems in $$\mathbb {R}^d$$
 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 $$N\rightarrow \infty $$
 for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature $$\beta $$
 . We show that a large deviation principle holds with a rate function of the form ‘
 $$\beta $$
 -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 $$\beta $$
 , the limiting density for $$s>d$$
 is strongly $$\beta $$
 -dependent.