Large deviation principles for hypersingular Riesz gases

Abstract

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 $N\to \infty$ 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.