Nonlocal energy functionals and determinantal point processes on non-smooth domains

Given a nonnegative integrable function $J$ on $\mathbb{R}^n$, we relate the asymptotic properties of the nonlocal energy functional \begin{equation*} \int_{\Omega} \int_{\Omega^c} J \bigg(\frac{x-y}{t}\bigg) \ dx dy \end{equation*} as $t \to 0^+$ with the boundary properties of a given domain $\Omega \subset \mathbb{R}^n$. Then, we use these asymptotic properties to study the fluctuations of many determinantal point processes, and show that their variances measure the Minkowski dimension of $\partial \Omega$.